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Theorem poimirlem16 36123
Description: Lemma for poimir 36140 establishing the vertices of the simplex of poimirlem17 36124. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (πœ‘ β†’ 𝑁 ∈ β„•)
poimirlem22.s 𝑆 = {𝑑 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ⦋if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‘)) ∘f + ((((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))}
poimirlem22.1 (πœ‘ β†’ 𝐹:(0...(𝑁 βˆ’ 1))⟢((0...𝐾) ↑m (1...𝑁)))
poimirlem22.2 (πœ‘ β†’ 𝑇 ∈ 𝑆)
poimirlem18.3 ((πœ‘ ∧ 𝑛 ∈ (1...𝑁)) β†’ βˆƒπ‘ ∈ ran 𝐹(π‘β€˜π‘›) β‰  𝐾)
poimirlem18.4 (πœ‘ β†’ (2nd β€˜π‘‡) = 0)
Assertion
Ref Expression
poimirlem16 (πœ‘ β†’ 𝐹 = (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0))) ∘f + (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})))))
Distinct variable groups:   𝑓,𝑗,𝑛,𝑝,𝑑,𝑦   πœ‘,𝑗,𝑛,𝑦   𝑗,𝐹,𝑛,𝑦   𝑗,𝑁,𝑛,𝑦   𝑇,𝑗,𝑛,𝑦   πœ‘,𝑝,𝑑   𝑓,𝐾,𝑗,𝑛,𝑝,𝑑   𝑓,𝑁,𝑝,𝑑   𝑇,𝑓,𝑝   𝑓,𝐹,𝑝,𝑑   𝑑,𝑇   𝑆,𝑗,𝑛,𝑝,𝑑,𝑦
Allowed substitution hints:   πœ‘(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem16
StepHypRef Expression
1 poimirlem22.2 . . 3 (πœ‘ β†’ 𝑇 ∈ 𝑆)
2 fveq2 6847 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (2nd β€˜π‘‘) = (2nd β€˜π‘‡))
32breq2d 5122 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (𝑦 < (2nd β€˜π‘‘) ↔ 𝑦 < (2nd β€˜π‘‡)))
43ifbid 4514 . . . . . . . 8 (𝑑 = 𝑇 β†’ if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)))
5 2fveq3 6852 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (1st β€˜(1st β€˜π‘‘)) = (1st β€˜(1st β€˜π‘‡)))
6 2fveq3 6852 . . . . . . . . . . . 12 (𝑑 = 𝑇 β†’ (2nd β€˜(1st β€˜π‘‘)) = (2nd β€˜(1st β€˜π‘‡)))
76imaeq1d 6017 . . . . . . . . . . 11 (𝑑 = 𝑇 β†’ ((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)))
87xpeq1d 5667 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) Γ— {1}) = (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}))
96imaeq1d 6017 . . . . . . . . . . 11 (𝑑 = 𝑇 β†’ ((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)))
109xpeq1d 5667 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}) = (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))
118, 10uneq12d 4129 . . . . . . . . 9 (𝑑 = 𝑇 β†’ ((((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})) = ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))
125, 11oveq12d 7380 . . . . . . . 8 (𝑑 = 𝑇 β†’ ((1st β€˜(1st β€˜π‘‘)) ∘f + ((((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) = ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))
134, 12csbeq12dv 3869 . . . . . . 7 (𝑑 = 𝑇 β†’ ⦋if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‘)) ∘f + ((((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) = ⦋if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))
1413mpteq2dv 5212 . . . . . 6 (𝑑 = 𝑇 β†’ (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ⦋if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‘)) ∘f + ((((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))) = (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ⦋if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))))
1514eqeq2d 2748 . . . . 5 (𝑑 = 𝑇 β†’ (𝐹 = (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ⦋if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‘)) ∘f + ((((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ⦋if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))))
16 poimirlem22.s . . . . 5 𝑆 = {𝑑 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ⦋if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‘)) ∘f + ((((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))}
1715, 16elrab2 3653 . . . 4 (𝑇 ∈ 𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ⦋if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))))
1817simprbi 498 . . 3 (𝑇 ∈ 𝑆 β†’ 𝐹 = (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ⦋if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))))
191, 18syl 17 . 2 (πœ‘ β†’ 𝐹 = (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ⦋if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))))
20 elrabi 3644 . . . . . . . . . . . 12 (𝑇 ∈ {𝑑 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ⦋if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‘)) ∘f + ((((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))} β†’ 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)))
2120, 16eleq2s 2856 . . . . . . . . . . 11 (𝑇 ∈ 𝑆 β†’ 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)))
221, 21syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)))
23 xp1st 7958 . . . . . . . . . 10 (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) β†’ (1st β€˜π‘‡) ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}))
2422, 23syl 17 . . . . . . . . 9 (πœ‘ β†’ (1st β€˜π‘‡) ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}))
25 xp1st 7958 . . . . . . . . 9 ((1st β€˜π‘‡) ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) β†’ (1st β€˜(1st β€˜π‘‡)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
2624, 25syl 17 . . . . . . . 8 (πœ‘ β†’ (1st β€˜(1st β€˜π‘‡)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
27 elmapfn 8810 . . . . . . . 8 ((1st β€˜(1st β€˜π‘‡)) ∈ ((0..^𝐾) ↑m (1...𝑁)) β†’ (1st β€˜(1st β€˜π‘‡)) Fn (1...𝑁))
2826, 27syl 17 . . . . . . 7 (πœ‘ β†’ (1st β€˜(1st β€˜π‘‡)) Fn (1...𝑁))
2928adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (1st β€˜(1st β€˜π‘‡)) Fn (1...𝑁))
30 1ex 11158 . . . . . . . . . 10 1 ∈ V
31 fnconstg 6735 . . . . . . . . . 10 (1 ∈ V β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))))
3230, 31ax-mp 5 . . . . . . . . 9 (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1)))
33 c0ex 11156 . . . . . . . . . 10 0 ∈ V
34 fnconstg 6735 . . . . . . . . . 10 (0 ∈ V β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)))
3533, 34ax-mp 5 . . . . . . . . 9 (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))
3632, 35pm3.2i 472 . . . . . . . 8 ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) ∧ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)))
37 xp2nd 7959 . . . . . . . . . . . . 13 ((1st β€˜π‘‡) ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) β†’ (2nd β€˜(1st β€˜π‘‡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})
3824, 37syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ (2nd β€˜(1st β€˜π‘‡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})
39 fvex 6860 . . . . . . . . . . . . 13 (2nd β€˜(1st β€˜π‘‡)) ∈ V
40 f1oeq1 6777 . . . . . . . . . . . . 13 (𝑓 = (2nd β€˜(1st β€˜π‘‡)) β†’ (𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁) ↔ (2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1-ontoβ†’(1...𝑁)))
4139, 40elab 3635 . . . . . . . . . . . 12 ((2nd β€˜(1st β€˜π‘‡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)} ↔ (2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
4238, 41sylib 217 . . . . . . . . . . 11 (πœ‘ β†’ (2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
43 dff1o3 6795 . . . . . . . . . . . 12 ((2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1-ontoβ†’(1...𝑁) ↔ ((2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–ontoβ†’(1...𝑁) ∧ Fun β—‘(2nd β€˜(1st β€˜π‘‡))))
4443simprbi 498 . . . . . . . . . . 11 ((2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1-ontoβ†’(1...𝑁) β†’ Fun β—‘(2nd β€˜(1st β€˜π‘‡)))
4542, 44syl 17 . . . . . . . . . 10 (πœ‘ β†’ Fun β—‘(2nd β€˜(1st β€˜π‘‡)))
46 imain 6591 . . . . . . . . . 10 (Fun β—‘(2nd β€˜(1st β€˜π‘‡)) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) ∩ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))))
4745, 46syl 17 . . . . . . . . 9 (πœ‘ β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) ∩ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))))
48 elfznn0 13541 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ 𝑦 ∈ β„•0)
49 nn0p1nn 12459 . . . . . . . . . . . . . . 15 (𝑦 ∈ β„•0 β†’ (𝑦 + 1) ∈ β„•)
5048, 49syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ (𝑦 + 1) ∈ β„•)
5150nnred 12175 . . . . . . . . . . . . 13 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ (𝑦 + 1) ∈ ℝ)
5251ltp1d 12092 . . . . . . . . . . . 12 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ (𝑦 + 1) < ((𝑦 + 1) + 1))
53 fzdisj 13475 . . . . . . . . . . . 12 ((𝑦 + 1) < ((𝑦 + 1) + 1) β†’ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = βˆ…)
5452, 53syl 17 . . . . . . . . . . 11 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = βˆ…)
5554imaeq2d 6018 . . . . . . . . . 10 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2nd β€˜(1st β€˜π‘‡)) β€œ βˆ…))
56 ima0 6034 . . . . . . . . . 10 ((2nd β€˜(1st β€˜π‘‡)) β€œ βˆ…) = βˆ…
5755, 56eqtrdi 2793 . . . . . . . . 9 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = βˆ…)
5847, 57sylan9req 2798 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) ∩ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) = βˆ…)
59 fnun 6619 . . . . . . . 8 ((((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) ∧ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) ∧ (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) ∩ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) = βˆ…) β†’ ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))))
6036, 58, 59sylancr 588 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))))
61 imaundi 6107 . . . . . . . . 9 ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1...(𝑦 + 1)) βˆͺ (((𝑦 + 1) + 1)...𝑁))) = (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)))
62 nnuz 12813 . . . . . . . . . . . . . 14 β„• = (β„€β‰₯β€˜1)
6350, 62eleqtrdi 2848 . . . . . . . . . . . . 13 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ (𝑦 + 1) ∈ (β„€β‰₯β€˜1))
64 peano2uz 12833 . . . . . . . . . . . . 13 ((𝑦 + 1) ∈ (β„€β‰₯β€˜1) β†’ ((𝑦 + 1) + 1) ∈ (β„€β‰₯β€˜1))
6563, 64syl 17 . . . . . . . . . . . 12 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ ((𝑦 + 1) + 1) ∈ (β„€β‰₯β€˜1))
66 poimir.0 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝑁 ∈ β„•)
6766nncnd 12176 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝑁 ∈ β„‚)
68 npcan1 11587 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„‚ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
6967, 68syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
7069adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
71 elfzuz3 13445 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ (𝑁 βˆ’ 1) ∈ (β„€β‰₯β€˜π‘¦))
72 eluzp1p1 12798 . . . . . . . . . . . . . . 15 ((𝑁 βˆ’ 1) ∈ (β„€β‰₯β€˜π‘¦) β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜(𝑦 + 1)))
7371, 72syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜(𝑦 + 1)))
7473adantl 483 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜(𝑦 + 1)))
7570, 74eqeltrrd 2839 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ 𝑁 ∈ (β„€β‰₯β€˜(𝑦 + 1)))
76 fzsplit2 13473 . . . . . . . . . . . 12 ((((𝑦 + 1) + 1) ∈ (β„€β‰₯β€˜1) ∧ 𝑁 ∈ (β„€β‰₯β€˜(𝑦 + 1))) β†’ (1...𝑁) = ((1...(𝑦 + 1)) βˆͺ (((𝑦 + 1) + 1)...𝑁)))
7765, 75, 76syl2an2 685 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (1...𝑁) = ((1...(𝑦 + 1)) βˆͺ (((𝑦 + 1) + 1)...𝑁)))
7877imaeq2d 6018 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑁)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1...(𝑦 + 1)) βˆͺ (((𝑦 + 1) + 1)...𝑁))))
79 f1ofo 6796 . . . . . . . . . . . 12 ((2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1-ontoβ†’(1...𝑁) β†’ (2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–ontoβ†’(1...𝑁))
80 foima 6766 . . . . . . . . . . . 12 ((2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–ontoβ†’(1...𝑁) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑁)) = (1...𝑁))
8142, 79, 803syl 18 . . . . . . . . . . 11 (πœ‘ β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑁)) = (1...𝑁))
8281adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑁)) = (1...𝑁))
8378, 82eqtr3d 2779 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1...(𝑦 + 1)) βˆͺ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
8461, 83eqtr3id 2791 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
8584fneq2d 6601 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) ↔ ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn (1...𝑁)))
8660, 85mpbid 231 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn (1...𝑁))
87 ovexd 7397 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (1...𝑁) ∈ V)
88 inidm 4183 . . . . . 6 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
89 eqidd 2738 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) β†’ ((1st β€˜(1st β€˜π‘‡))β€˜π‘›) = ((1st β€˜(1st β€˜π‘‡))β€˜π‘›))
90 eqidd 2738 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›))
9129, 86, 87, 87, 88, 89, 90offval 7631 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))) = (𝑛 ∈ (1...𝑁) ↦ (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›))))
92 oveq1 7369 . . . . . . . . . 10 (1 = if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0) β†’ (1 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)) = (if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0) + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)))
9392eqeq2d 2748 . . . . . . . . 9 (1 = if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = (1 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)) ↔ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = (if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0) + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›))))
94 oveq1 7369 . . . . . . . . . 10 (0 = if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0) β†’ (0 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)) = (if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0) + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)))
9594eqeq2d 2748 . . . . . . . . 9 (0 = if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = (0 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)) ↔ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = (if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0) + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›))))
96 1p0e1 12284 . . . . . . . . . . . . . 14 (1 + 0) = 1
9796eqcomi 2746 . . . . . . . . . . . . 13 1 = (1 + 0)
98 f1ofn 6790 . . . . . . . . . . . . . . . . . 18 ((2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1-ontoβ†’(1...𝑁) β†’ (2nd β€˜(1st β€˜π‘‡)) Fn (1...𝑁))
9942, 98syl 17 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (2nd β€˜(1st β€˜π‘‡)) Fn (1...𝑁))
10099adantr 482 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (2nd β€˜(1st β€˜π‘‡)) Fn (1...𝑁))
101 fzss2 13488 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (β„€β‰₯β€˜(𝑦 + 1)) β†’ (1...(𝑦 + 1)) βŠ† (1...𝑁))
10275, 101syl 17 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (1...(𝑦 + 1)) βŠ† (1...𝑁))
103 eluzfz1 13455 . . . . . . . . . . . . . . . . . 18 ((𝑦 + 1) ∈ (β„€β‰₯β€˜1) β†’ 1 ∈ (1...(𝑦 + 1)))
10463, 103syl 17 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ 1 ∈ (1...(𝑦 + 1)))
105104adantl 483 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ 1 ∈ (1...(𝑦 + 1)))
106 fnfvima 7188 . . . . . . . . . . . . . . . 16 (((2nd β€˜(1st β€˜π‘‡)) Fn (1...𝑁) ∧ (1...(𝑦 + 1)) βŠ† (1...𝑁) ∧ 1 ∈ (1...(𝑦 + 1))) β†’ ((2nd β€˜(1st β€˜π‘‡))β€˜1) ∈ ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))))
107100, 102, 105, 106syl3anc 1372 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡))β€˜1) ∈ ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))))
108 fvun1 6937 . . . . . . . . . . . . . . . 16 (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) ∧ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) ∧ ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) ∩ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) = βˆ… ∧ ((2nd β€˜(1st β€˜π‘‡))β€˜1) ∈ ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1})β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)))
10932, 35, 108mp3an12 1452 . . . . . . . . . . . . . . 15 (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) ∩ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) = βˆ… ∧ ((2nd β€˜(1st β€˜π‘‡))β€˜1) ∈ ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1)))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1})β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)))
11058, 107, 109syl2anc 585 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1})β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)))
11130fvconst2 7158 . . . . . . . . . . . . . . 15 (((2nd β€˜(1st β€˜π‘‡))β€˜1) ∈ ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) β†’ ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1})β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = 1)
112107, 111syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1})β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = 1)
113110, 112eqtrd 2777 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = 1)
114 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((πœ‘ ∧ 𝑛 ∈ (1...(𝑁 βˆ’ 1))) β†’ 𝑛 ∈ (1...(𝑁 βˆ’ 1)))
11566nnzd 12533 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (πœ‘ β†’ 𝑁 ∈ β„€)
116 peano2zm 12553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ β„€ β†’ (𝑁 βˆ’ 1) ∈ β„€)
117115, 116syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (πœ‘ β†’ (𝑁 βˆ’ 1) ∈ β„€)
118 1z 12540 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ β„€
119117, 118jctil 521 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (πœ‘ β†’ (1 ∈ β„€ ∧ (𝑁 βˆ’ 1) ∈ β„€))
120 elfzelz 13448 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (1...(𝑁 βˆ’ 1)) β†’ 𝑛 ∈ β„€)
121120, 118jctir 522 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 ∈ (1...(𝑁 βˆ’ 1)) β†’ (𝑛 ∈ β„€ ∧ 1 ∈ β„€))
122 fzaddel 13482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1 ∈ β„€ ∧ (𝑁 βˆ’ 1) ∈ β„€) ∧ (𝑛 ∈ β„€ ∧ 1 ∈ β„€)) β†’ (𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 βˆ’ 1) + 1))))
123119, 121, 122syl2an 597 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((πœ‘ ∧ 𝑛 ∈ (1...(𝑁 βˆ’ 1))) β†’ (𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 βˆ’ 1) + 1))))
124114, 123mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((πœ‘ ∧ 𝑛 ∈ (1...(𝑁 βˆ’ 1))) β†’ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 βˆ’ 1) + 1)))
12569oveq2d 7378 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (πœ‘ β†’ ((1 + 1)...((𝑁 βˆ’ 1) + 1)) = ((1 + 1)...𝑁))
126125adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((πœ‘ ∧ 𝑛 ∈ (1...(𝑁 βˆ’ 1))) β†’ ((1 + 1)...((𝑁 βˆ’ 1) + 1)) = ((1 + 1)...𝑁))
127124, 126eleqtrd 2840 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ 𝑛 ∈ (1...(𝑁 βˆ’ 1))) β†’ (𝑛 + 1) ∈ ((1 + 1)...𝑁))
128127ralrimiva 3144 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ βˆ€π‘› ∈ (1...(𝑁 βˆ’ 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁))
129 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((πœ‘ ∧ 𝑦 ∈ ((1 + 1)...𝑁)) β†’ 𝑦 ∈ ((1 + 1)...𝑁))
130 peano2z 12551 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (1 ∈ β„€ β†’ (1 + 1) ∈ β„€)
131118, 130ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 + 1) ∈ β„€
132115, 131jctil 521 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (πœ‘ β†’ ((1 + 1) ∈ β„€ ∧ 𝑁 ∈ β„€))
133 elfzelz 13448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ((1 + 1)...𝑁) β†’ 𝑦 ∈ β„€)
134133, 118jctir 522 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ((1 + 1)...𝑁) β†’ (𝑦 ∈ β„€ ∧ 1 ∈ β„€))
135 fzsubel 13484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((1 + 1) ∈ β„€ ∧ 𝑁 ∈ β„€) ∧ (𝑦 ∈ β„€ ∧ 1 ∈ β„€)) β†’ (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 βˆ’ 1) ∈ (((1 + 1) βˆ’ 1)...(𝑁 βˆ’ 1))))
136132, 134, 135syl2an 597 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((πœ‘ ∧ 𝑦 ∈ ((1 + 1)...𝑁)) β†’ (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 βˆ’ 1) ∈ (((1 + 1) βˆ’ 1)...(𝑁 βˆ’ 1))))
137129, 136mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((πœ‘ ∧ 𝑦 ∈ ((1 + 1)...𝑁)) β†’ (𝑦 βˆ’ 1) ∈ (((1 + 1) βˆ’ 1)...(𝑁 βˆ’ 1)))
138 ax-1cn 11116 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ β„‚
139138, 138pncan3oi 11424 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((1 + 1) βˆ’ 1) = 1
140139oveq1i 7372 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((1 + 1) βˆ’ 1)...(𝑁 βˆ’ 1)) = (1...(𝑁 βˆ’ 1))
141137, 140eleqtrdi 2848 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((πœ‘ ∧ 𝑦 ∈ ((1 + 1)...𝑁)) β†’ (𝑦 βˆ’ 1) ∈ (1...(𝑁 βˆ’ 1)))
142133zcnd 12615 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ((1 + 1)...𝑁) β†’ 𝑦 ∈ β„‚)
143 elfznn 13477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (1...(𝑁 βˆ’ 1)) β†’ 𝑛 ∈ β„•)
144143nncnd 12176 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (1...(𝑁 βˆ’ 1)) β†’ 𝑛 ∈ β„‚)
145 subadd2 11412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑦 ∈ β„‚ ∧ 1 ∈ β„‚ ∧ 𝑛 ∈ β„‚) β†’ ((𝑦 βˆ’ 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
146138, 145mp3an2 1450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 ∈ β„‚ ∧ 𝑛 ∈ β„‚) β†’ ((𝑦 βˆ’ 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
147146bicomd 222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 ∈ β„‚ ∧ 𝑛 ∈ β„‚) β†’ ((𝑛 + 1) = 𝑦 ↔ (𝑦 βˆ’ 1) = 𝑛))
148 eqcom 2744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑛 + 1) = 𝑦 ↔ 𝑦 = (𝑛 + 1))
149 eqcom 2744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 βˆ’ 1) = 𝑛 ↔ 𝑛 = (𝑦 βˆ’ 1))
150147, 148, 1493bitr3g 313 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑦 ∈ β„‚ ∧ 𝑛 ∈ β„‚) β†’ (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 βˆ’ 1)))
151142, 144, 150syl2an 597 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ∈ ((1 + 1)...𝑁) ∧ 𝑛 ∈ (1...(𝑁 βˆ’ 1))) β†’ (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 βˆ’ 1)))
152151ralrimiva 3144 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ((1 + 1)...𝑁) β†’ βˆ€π‘› ∈ (1...(𝑁 βˆ’ 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 βˆ’ 1)))
153152adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((πœ‘ ∧ 𝑦 ∈ ((1 + 1)...𝑁)) β†’ βˆ€π‘› ∈ (1...(𝑁 βˆ’ 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 βˆ’ 1)))
154 reu6i 3691 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦 βˆ’ 1) ∈ (1...(𝑁 βˆ’ 1)) ∧ βˆ€π‘› ∈ (1...(𝑁 βˆ’ 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 βˆ’ 1))) β†’ βˆƒ!𝑛 ∈ (1...(𝑁 βˆ’ 1))𝑦 = (𝑛 + 1))
155141, 153, 154syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ 𝑦 ∈ ((1 + 1)...𝑁)) β†’ βˆƒ!𝑛 ∈ (1...(𝑁 βˆ’ 1))𝑦 = (𝑛 + 1))
156155ralrimiva 3144 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ βˆ€π‘¦ ∈ ((1 + 1)...𝑁)βˆƒ!𝑛 ∈ (1...(𝑁 βˆ’ 1))𝑦 = (𝑛 + 1))
157 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1))
158157f1ompt 7064 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)):(1...(𝑁 βˆ’ 1))–1-1-ontoβ†’((1 + 1)...𝑁) ↔ (βˆ€π‘› ∈ (1...(𝑁 βˆ’ 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁) ∧ βˆ€π‘¦ ∈ ((1 + 1)...𝑁)βˆƒ!𝑛 ∈ (1...(𝑁 βˆ’ 1))𝑦 = (𝑛 + 1)))
159128, 156, 158sylanbrc 584 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ (𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)):(1...(𝑁 βˆ’ 1))–1-1-ontoβ†’((1 + 1)...𝑁))
160 f1osng 6830 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ β„• ∧ 1 ∈ V) β†’ {βŸ¨π‘, 1⟩}:{𝑁}–1-1-ontoβ†’{1})
16166, 30, 160sylancl 587 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ {βŸ¨π‘, 1⟩}:{𝑁}–1-1-ontoβ†’{1})
16266nnred 12175 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (πœ‘ β†’ 𝑁 ∈ ℝ)
163162ltm1d 12094 . . . . . . . . . . . . . . . . . . . . . . . . 25 (πœ‘ β†’ (𝑁 βˆ’ 1) < 𝑁)
164117zred 12614 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (πœ‘ β†’ (𝑁 βˆ’ 1) ∈ ℝ)
165164, 162ltnled 11309 . . . . . . . . . . . . . . . . . . . . . . . . 25 (πœ‘ β†’ ((𝑁 βˆ’ 1) < 𝑁 ↔ Β¬ 𝑁 ≀ (𝑁 βˆ’ 1)))
166163, 165mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . 24 (πœ‘ β†’ Β¬ 𝑁 ≀ (𝑁 βˆ’ 1))
167 elfzle2 13452 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ (1...(𝑁 βˆ’ 1)) β†’ 𝑁 ≀ (𝑁 βˆ’ 1))
168166, 167nsyl 140 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ Β¬ 𝑁 ∈ (1...(𝑁 βˆ’ 1)))
169 disjsn 4677 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...(𝑁 βˆ’ 1)) ∩ {𝑁}) = βˆ… ↔ Β¬ 𝑁 ∈ (1...(𝑁 βˆ’ 1)))
170168, 169sylibr 233 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ ((1...(𝑁 βˆ’ 1)) ∩ {𝑁}) = βˆ…)
171 1re 11162 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1 ∈ ℝ
172171ltp1i 12066 . . . . . . . . . . . . . . . . . . . . . . . . . 26 1 < (1 + 1)
173171, 171readdcli 11177 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (1 + 1) ∈ ℝ
174171, 173ltnlei 11283 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (1 < (1 + 1) ↔ Β¬ (1 + 1) ≀ 1)
175172, 174mpbi 229 . . . . . . . . . . . . . . . . . . . . . . . . 25 Β¬ (1 + 1) ≀ 1
176 elfzle1 13451 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ ((1 + 1)...𝑁) β†’ (1 + 1) ≀ 1)
177175, 176mto 196 . . . . . . . . . . . . . . . . . . . . . . . 24 Β¬ 1 ∈ ((1 + 1)...𝑁)
178 disjsn 4677 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((1 + 1)...𝑁) ∩ {1}) = βˆ… ↔ Β¬ 1 ∈ ((1 + 1)...𝑁))
179177, 178mpbir 230 . . . . . . . . . . . . . . . . . . . . . . 23 (((1 + 1)...𝑁) ∩ {1}) = βˆ…
180 f1oun 6808 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)):(1...(𝑁 βˆ’ 1))–1-1-ontoβ†’((1 + 1)...𝑁) ∧ {βŸ¨π‘, 1⟩}:{𝑁}–1-1-ontoβ†’{1}) ∧ (((1...(𝑁 βˆ’ 1)) ∩ {𝑁}) = βˆ… ∧ (((1 + 1)...𝑁) ∩ {1}) = βˆ…)) β†’ ((𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)) βˆͺ {βŸ¨π‘, 1⟩}):((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁})–1-1-ontoβ†’(((1 + 1)...𝑁) βˆͺ {1}))
181179, 180mpanr2 703 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)):(1...(𝑁 βˆ’ 1))–1-1-ontoβ†’((1 + 1)...𝑁) ∧ {βŸ¨π‘, 1⟩}:{𝑁}–1-1-ontoβ†’{1}) ∧ ((1...(𝑁 βˆ’ 1)) ∩ {𝑁}) = βˆ…) β†’ ((𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)) βˆͺ {βŸ¨π‘, 1⟩}):((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁})–1-1-ontoβ†’(((1 + 1)...𝑁) βˆͺ {1}))
182159, 161, 170, 181syl21anc 837 . . . . . . . . . . . . . . . . . . . . 21 (πœ‘ β†’ ((𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)) βˆͺ {βŸ¨π‘, 1⟩}):((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁})–1-1-ontoβ†’(((1 + 1)...𝑁) βˆͺ {1}))
18330a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (πœ‘ β†’ 1 ∈ V)
18466, 62eleqtrdi 2848 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜1))
18569, 184eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (πœ‘ β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜1))
186 uzid 12785 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 βˆ’ 1) ∈ β„€ β†’ (𝑁 βˆ’ 1) ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
187 peano2uz 12833 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 βˆ’ 1) ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)) β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
188117, 186, 1873syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (πœ‘ β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
18969, 188eqeltrrd 2839 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
190 fzsplit2 13473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜1) ∧ 𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1))) β†’ (1...𝑁) = ((1...(𝑁 βˆ’ 1)) βˆͺ (((𝑁 βˆ’ 1) + 1)...𝑁)))
191185, 189, 190syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . . 25 (πœ‘ β†’ (1...𝑁) = ((1...(𝑁 βˆ’ 1)) βˆͺ (((𝑁 βˆ’ 1) + 1)...𝑁)))
19269oveq1d 7377 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (πœ‘ β†’ (((𝑁 βˆ’ 1) + 1)...𝑁) = (𝑁...𝑁))
193 fzsn 13490 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„€ β†’ (𝑁...𝑁) = {𝑁})
194115, 193syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (πœ‘ β†’ (𝑁...𝑁) = {𝑁})
195192, 194eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (πœ‘ β†’ (((𝑁 βˆ’ 1) + 1)...𝑁) = {𝑁})
196195uneq2d 4128 . . . . . . . . . . . . . . . . . . . . . . . . 25 (πœ‘ β†’ ((1...(𝑁 βˆ’ 1)) βˆͺ (((𝑁 βˆ’ 1) + 1)...𝑁)) = ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}))
197191, 196eqtr2d 2778 . . . . . . . . . . . . . . . . . . . . . . . 24 (πœ‘ β†’ ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}) = (1...𝑁))
198 iftrue 4497 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑁 β†’ if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
199198adantl 483 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ 𝑛 = 𝑁) β†’ if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
20066, 183, 197, 199fmptapd 7122 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ ((𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) βˆͺ {βŸ¨π‘, 1⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
201 eleq1 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = 𝑁 β†’ (𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↔ 𝑁 ∈ (1...(𝑁 βˆ’ 1))))
202201notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 𝑁 β†’ (Β¬ 𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↔ Β¬ 𝑁 ∈ (1...(𝑁 βˆ’ 1))))
203168, 202syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (πœ‘ β†’ (𝑛 = 𝑁 β†’ Β¬ 𝑛 ∈ (1...(𝑁 βˆ’ 1))))
204203necon2ad 2959 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (πœ‘ β†’ (𝑛 ∈ (1...(𝑁 βˆ’ 1)) β†’ 𝑛 β‰  𝑁))
205204imp 408 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((πœ‘ ∧ 𝑛 ∈ (1...(𝑁 βˆ’ 1))) β†’ 𝑛 β‰  𝑁)
206 ifnefalse 4503 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 β‰  𝑁 β†’ if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
207205, 206syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((πœ‘ ∧ 𝑛 ∈ (1...(𝑁 βˆ’ 1))) β†’ if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
208207mpteq2dva 5210 . . . . . . . . . . . . . . . . . . . . . . . 24 (πœ‘ β†’ (𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)))
209208uneq1d 4127 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ ((𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) βˆͺ {βŸ¨π‘, 1⟩}) = ((𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)) βˆͺ {βŸ¨π‘, 1⟩}))
210200, 209eqtr3d 2779 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = ((𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)) βˆͺ {βŸ¨π‘, 1⟩}))
211191, 196eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ (1...𝑁) = ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}))
212 uzid 12785 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ β„€ β†’ 1 ∈ (β„€β‰₯β€˜1))
213 peano2uz 12833 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ (β„€β‰₯β€˜1) β†’ (1 + 1) ∈ (β„€β‰₯β€˜1))
214118, 212, 213mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 + 1) ∈ (β„€β‰₯β€˜1)
215 fzsplit2 13473 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1 + 1) ∈ (β„€β‰₯β€˜1) ∧ 𝑁 ∈ (β„€β‰₯β€˜1)) β†’ (1...𝑁) = ((1...1) βˆͺ ((1 + 1)...𝑁)))
216214, 184, 215sylancr 588 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ (1...𝑁) = ((1...1) βˆͺ ((1 + 1)...𝑁)))
217 fzsn 13490 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (1 ∈ β„€ β†’ (1...1) = {1})
218118, 217ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1...1) = {1}
219218uneq1i 4124 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1...1) βˆͺ ((1 + 1)...𝑁)) = ({1} βˆͺ ((1 + 1)...𝑁))
220219equncomi 4120 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...1) βˆͺ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) βˆͺ {1})
221216, 220eqtrdi 2793 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ (1...𝑁) = (((1 + 1)...𝑁) βˆͺ {1}))
222210, 211, 221f1oeq123d 6783 . . . . . . . . . . . . . . . . . . . . 21 (πœ‘ β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-ontoβ†’(1...𝑁) ↔ ((𝑛 ∈ (1...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)) βˆͺ {βŸ¨π‘, 1⟩}):((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁})–1-1-ontoβ†’(((1 + 1)...𝑁) βˆͺ {1})))
223182, 222mpbird 257 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
224 f1oco 6812 . . . . . . . . . . . . . . . . . . . 20 (((2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1-ontoβ†’(1...𝑁) ∧ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-ontoβ†’(1...𝑁)) β†’ ((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
22542, 223, 224syl2anc 585 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ ((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
226 dff1o3 6795 . . . . . . . . . . . . . . . . . . . 20 (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-ontoβ†’(1...𝑁) ↔ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–ontoβ†’(1...𝑁) ∧ Fun β—‘((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))))
227226simprbi 498 . . . . . . . . . . . . . . . . . . 19 (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-ontoβ†’(1...𝑁) β†’ Fun β—‘((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))))
228225, 227syl 17 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ Fun β—‘((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))))
229 imain 6591 . . . . . . . . . . . . . . . . . 18 (Fun β—‘((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) ∩ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁))))
230228, 229syl 17 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) ∩ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁))))
23148nn0red 12481 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ 𝑦 ∈ ℝ)
232231ltp1d 12092 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ 𝑦 < (𝑦 + 1))
233 fzdisj 13475 . . . . . . . . . . . . . . . . . . . 20 (𝑦 < (𝑦 + 1) β†’ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = βˆ…)
234232, 233syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = βˆ…)
235234imaeq2d 6018 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ βˆ…))
236 ima0 6034 . . . . . . . . . . . . . . . . . 18 (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ βˆ…) = βˆ…
237235, 236eqtrdi 2793 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = βˆ…)
238230, 237sylan9req 2798 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) ∩ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁))) = βˆ…)
239 imassrn 6029 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...𝑁)) βŠ† ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
240 f1of 6789 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-ontoβ†’(1...𝑁) β†’ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)⟢(1...𝑁))
241 frn 6680 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)⟢(1...𝑁) β†’ ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) βŠ† (1...𝑁))
242223, 240, 2413syl 18 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) βŠ† (1...𝑁))
243239, 242sstrid 3960 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...𝑁)) βŠ† (1...𝑁))
244243adantr 482 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...𝑁)) βŠ† (1...𝑁))
245 elfz1end 13478 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ β„• ↔ 𝑁 ∈ (1...𝑁))
24666, 245sylib 217 . . . . . . . . . . . . . . . . . . . . 21 (πœ‘ β†’ 𝑁 ∈ (1...𝑁))
247 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
248198, 247, 30fvmpt 6953 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (1...𝑁) β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))β€˜π‘) = 1)
249246, 248syl 17 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))β€˜π‘) = 1)
250249adantr 482 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))β€˜π‘) = 1)
251 f1ofn 6790 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-ontoβ†’(1...𝑁) β†’ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁))
252223, 251syl 17 . . . . . . . . . . . . . . . . . . . . 21 (πœ‘ β†’ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁))
253252adantr 482 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁))
254 fzss1 13487 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 + 1) ∈ (β„€β‰₯β€˜1) β†’ ((𝑦 + 1)...𝑁) βŠ† (1...𝑁))
25563, 254syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ ((𝑦 + 1)...𝑁) βŠ† (1...𝑁))
256255adantl 483 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑦 + 1)...𝑁) βŠ† (1...𝑁))
257 eluzfz2 13456 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (β„€β‰₯β€˜(𝑦 + 1)) β†’ 𝑁 ∈ ((𝑦 + 1)...𝑁))
25875, 257syl 17 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ 𝑁 ∈ ((𝑦 + 1)...𝑁))
259 fnfvima 7188 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁) ∧ ((𝑦 + 1)...𝑁) βŠ† (1...𝑁) ∧ 𝑁 ∈ ((𝑦 + 1)...𝑁)) β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))β€˜π‘) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...𝑁)))
260253, 256, 258, 259syl3anc 1372 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))β€˜π‘) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...𝑁)))
261250, 260eqeltrrd 2839 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ 1 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...𝑁)))
262 fnfvima 7188 . . . . . . . . . . . . . . . . . 18 (((2nd β€˜(1st β€˜π‘‡)) Fn (1...𝑁) ∧ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...𝑁)) βŠ† (1...𝑁) ∧ 1 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...𝑁))) β†’ ((2nd β€˜(1st β€˜π‘‡))β€˜1) ∈ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...𝑁))))
263100, 244, 261, 262syl3anc 1372 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡))β€˜1) ∈ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...𝑁))))
264 imaco 6208 . . . . . . . . . . . . . . . . 17 (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...𝑁)))
265263, 264eleqtrrdi 2849 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡))β€˜1) ∈ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)))
266 fnconstg 6735 . . . . . . . . . . . . . . . . . 18 (1 ∈ V β†’ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) Fn (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)))
26730, 266ax-mp 5 . . . . . . . . . . . . . . . . 17 ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) Fn (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦))
268 fnconstg 6735 . . . . . . . . . . . . . . . . . 18 (0 ∈ V β†’ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}) Fn (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)))
26933, 268ax-mp 5 . . . . . . . . . . . . . . . . 17 ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}) Fn (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁))
270 fvun2 6938 . . . . . . . . . . . . . . . . 17 ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) Fn (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) ∧ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}) Fn (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) ∧ (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) ∩ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁))) = βˆ… ∧ ((2nd β€˜(1st β€˜π‘‡))β€˜1) ∈ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)))) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)))
271267, 269, 270mp3an12 1452 . . . . . . . . . . . . . . . 16 ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) ∩ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁))) = βˆ… ∧ ((2nd β€˜(1st β€˜π‘‡))β€˜1) ∈ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁))) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)))
272238, 265, 271syl2anc 585 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)))
27333fvconst2 7158 . . . . . . . . . . . . . . . 16 (((2nd β€˜(1st β€˜π‘‡))β€˜1) ∈ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) β†’ (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = 0)
274265, 273syl 17 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = 0)
275272, 274eqtrd 2777 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = 0)
276275oveq2d 7378 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (1 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1))) = (1 + 0))
27797, 113, 2763eqtr4a 2803 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = (1 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1))))
278 fveq2 6847 . . . . . . . . . . . . 13 (𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)))
279 fveq2 6847 . . . . . . . . . . . . . 14 (𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›) = ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)))
280279oveq2d 7378 . . . . . . . . . . . . 13 (𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1) β†’ (1 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)) = (1 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1))))
281278, 280eqeq12d 2753 . . . . . . . . . . . 12 (𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = (1 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)) ↔ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)) = (1 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜((2nd β€˜(1st β€˜π‘‡))β€˜1)))))
282277, 281syl5ibrcom 247 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = (1 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›))))
283282imp 408 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1)) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = (1 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)))
284283adantlr 714 . . . . . . . . 9 ((((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1)) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = (1 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)))
285 eldifsn 4752 . . . . . . . . . . . . . 14 (𝑛 ∈ ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 β‰  ((2nd β€˜(1st β€˜π‘‡))β€˜1)))
286 df-ne 2945 . . . . . . . . . . . . . . 15 (𝑛 β‰  ((2nd β€˜(1st β€˜π‘‡))β€˜1) ↔ Β¬ 𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1))
287286anbi2i 624 . . . . . . . . . . . . . 14 ((𝑛 ∈ (1...𝑁) ∧ 𝑛 β‰  ((2nd β€˜(1st β€˜π‘‡))β€˜1)) ↔ (𝑛 ∈ (1...𝑁) ∧ Β¬ 𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1)))
288285, 287bitri 275 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ Β¬ 𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1)))
289 fnconstg 6735 . . . . . . . . . . . . . . . . . 18 (1 ∈ V β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))))
29030, 289ax-mp 5 . . . . . . . . . . . . . . . . 17 (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1)))
291290, 35pm3.2i 472 . . . . . . . . . . . . . . . 16 ((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) ∧ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)))
292 imain 6591 . . . . . . . . . . . . . . . . . 18 (Fun β—‘(2nd β€˜(1st β€˜π‘‡)) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))))
29345, 292syl 17 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))))
294 fzdisj 13475 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 + 1) < ((𝑦 + 1) + 1) β†’ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = βˆ…)
29552, 294syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = βˆ…)
296295imaeq2d 6018 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2nd β€˜(1st β€˜π‘‡)) β€œ βˆ…))
297296, 56eqtrdi 2793 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = βˆ…)
298293, 297sylan9req 2798 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) = βˆ…)
299 fnun 6619 . . . . . . . . . . . . . . . 16 ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) ∧ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}) Fn ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) ∧ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) = βˆ…) β†’ ((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))))
300291, 298, 299sylancr 588 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))))
301 imaundi 6107 . . . . . . . . . . . . . . . . 17 ((2nd β€˜(1st β€˜π‘‡)) β€œ (((1 + 1)...(𝑦 + 1)) βˆͺ (((𝑦 + 1) + 1)...𝑁))) = (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)))
302 fzpred 13496 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ (β„€β‰₯β€˜1) β†’ (1...𝑁) = ({1} βˆͺ ((1 + 1)...𝑁)))
303184, 302syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (πœ‘ β†’ (1...𝑁) = ({1} βˆͺ ((1 + 1)...𝑁)))
304 uncom 4118 . . . . . . . . . . . . . . . . . . . . . . . 24 ({1} βˆͺ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) βˆͺ {1})
305303, 304eqtrdi 2793 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ (1...𝑁) = (((1 + 1)...𝑁) βˆͺ {1}))
306305difeq1d 4086 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ ((1...𝑁) βˆ– {1}) = ((((1 + 1)...𝑁) βˆͺ {1}) βˆ– {1}))
307 difun2 4445 . . . . . . . . . . . . . . . . . . . . . . 23 ((((1 + 1)...𝑁) βˆͺ {1}) βˆ– {1}) = (((1 + 1)...𝑁) βˆ– {1})
308 disj3 4418 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((1 + 1)...𝑁) ∩ {1}) = βˆ… ↔ ((1 + 1)...𝑁) = (((1 + 1)...𝑁) βˆ– {1}))
309179, 308mpbi 229 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 + 1)...𝑁) = (((1 + 1)...𝑁) βˆ– {1})
310307, 309eqtr4i 2768 . . . . . . . . . . . . . . . . . . . . . 22 ((((1 + 1)...𝑁) βˆͺ {1}) βˆ– {1}) = ((1 + 1)...𝑁)
311306, 310eqtrdi 2793 . . . . . . . . . . . . . . . . . . . . 21 (πœ‘ β†’ ((1...𝑁) βˆ– {1}) = ((1 + 1)...𝑁))
312311adantr 482 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((1...𝑁) βˆ– {1}) = ((1 + 1)...𝑁))
313 eluzp1p1 12798 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 + 1) ∈ (β„€β‰₯β€˜1) β†’ ((𝑦 + 1) + 1) ∈ (β„€β‰₯β€˜(1 + 1)))
31463, 313syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ ((𝑦 + 1) + 1) ∈ (β„€β‰₯β€˜(1 + 1)))
315 fzsplit2 13473 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 + 1) + 1) ∈ (β„€β‰₯β€˜(1 + 1)) ∧ 𝑁 ∈ (β„€β‰₯β€˜(𝑦 + 1))) β†’ ((1 + 1)...𝑁) = (((1 + 1)...(𝑦 + 1)) βˆͺ (((𝑦 + 1) + 1)...𝑁)))
316314, 75, 315syl2an2 685 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((1 + 1)...𝑁) = (((1 + 1)...(𝑦 + 1)) βˆͺ (((𝑦 + 1) + 1)...𝑁)))
317312, 316eqtrd 2777 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((1...𝑁) βˆ– {1}) = (((1 + 1)...(𝑦 + 1)) βˆͺ (((𝑦 + 1) + 1)...𝑁)))
318317imaeq2d 6018 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1...𝑁) βˆ– {1})) = ((2nd β€˜(1st β€˜π‘‡)) β€œ (((1 + 1)...(𝑦 + 1)) βˆͺ (((𝑦 + 1) + 1)...𝑁))))
319 imadif 6590 . . . . . . . . . . . . . . . . . . . . 21 (Fun β—‘(2nd β€˜(1st β€˜π‘‡)) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1...𝑁) βˆ– {1})) = (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑁)) βˆ– ((2nd β€˜(1st β€˜π‘‡)) β€œ {1})))
32045, 319syl 17 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1...𝑁) βˆ– {1})) = (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑁)) βˆ– ((2nd β€˜(1st β€˜π‘‡)) β€œ {1})))
321 eluzfz1 13455 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ (β„€β‰₯β€˜1) β†’ 1 ∈ (1...𝑁))
322184, 321syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ 1 ∈ (1...𝑁))
323 fnsnfv 6925 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd β€˜(1st β€˜π‘‡)) Fn (1...𝑁) ∧ 1 ∈ (1...𝑁)) β†’ {((2nd β€˜(1st β€˜π‘‡))β€˜1)} = ((2nd β€˜(1st β€˜π‘‡)) β€œ {1}))
32499, 322, 323syl2anc 585 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ {((2nd β€˜(1st β€˜π‘‡))β€˜1)} = ((2nd β€˜(1st β€˜π‘‡)) β€œ {1}))
325324eqcomd 2743 . . . . . . . . . . . . . . . . . . . . 21 (πœ‘ β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ {1}) = {((2nd β€˜(1st β€˜π‘‡))β€˜1)})
32681, 325difeq12d 4088 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑁)) βˆ– ((2nd β€˜(1st β€˜π‘‡)) β€œ {1})) = ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}))
327320, 326eqtrd 2777 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1...𝑁) βˆ– {1})) = ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}))
328327adantr 482 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1...𝑁) βˆ– {1})) = ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}))
329318, 328eqtr3d 2779 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((1 + 1)...(𝑦 + 1)) βˆͺ (((𝑦 + 1) + 1)...𝑁))) = ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}))
330301, 329eqtr3id 2791 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) = ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}))
331330fneq2d 6601 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) ↔ ((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)})))
332300, 331mpbid 231 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}))
333 disjdifr 4437 . . . . . . . . . . . . . . 15 (((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ∩ {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) = βˆ…
334 fnconstg 6735 . . . . . . . . . . . . . . . . . 18 (1 ∈ V β†’ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}) Fn {((2nd β€˜(1st β€˜π‘‡))β€˜1)})
33530, 334ax-mp 5 . . . . . . . . . . . . . . . . 17 ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}) Fn {((2nd β€˜(1st β€˜π‘‡))β€˜1)}
336 fvun1 6937 . . . . . . . . . . . . . . . . 17 ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ∧ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}) Fn {((2nd β€˜(1st β€˜π‘‡))β€˜1)} ∧ ((((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ∩ {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) = βˆ… ∧ 𝑛 ∈ ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}))) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}))β€˜π‘›) = (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›))
337335, 336mp3an2 1450 . . . . . . . . . . . . . . . 16 ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ∧ ((((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ∩ {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) = βˆ… ∧ 𝑛 ∈ ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}))) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}))β€˜π‘›) = (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›))
338 fnconstg 6735 . . . . . . . . . . . . . . . . . 18 (0 ∈ V β†’ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}) Fn {((2nd β€˜(1st β€˜π‘‡))β€˜1)})
33933, 338ax-mp 5 . . . . . . . . . . . . . . . . 17 ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}) Fn {((2nd β€˜(1st β€˜π‘‡))β€˜1)}
340 fvun1 6937 . . . . . . . . . . . . . . . . 17 ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ∧ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}) Fn {((2nd β€˜(1st β€˜π‘‡))β€˜1)} ∧ ((((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ∩ {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) = βˆ… ∧ 𝑛 ∈ ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}))) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}))β€˜π‘›) = (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›))
341339, 340mp3an2 1450 . . . . . . . . . . . . . . . 16 ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ∧ ((((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ∩ {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) = βˆ… ∧ 𝑛 ∈ ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}))) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}))β€˜π‘›) = (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›))
342337, 341eqtr4d 2780 . . . . . . . . . . . . . . 15 ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ∧ ((((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ∩ {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) = βˆ… ∧ 𝑛 ∈ ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}))) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}))β€˜π‘›) = ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}))β€˜π‘›))
343333, 342mpanr1 702 . . . . . . . . . . . . . 14 ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) Fn ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) ∧ 𝑛 ∈ ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)})) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}))β€˜π‘›) = ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}))β€˜π‘›))
344332, 343sylan 581 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ ((1...𝑁) βˆ– {((2nd β€˜(1st β€˜π‘‡))β€˜1)})) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}))β€˜π‘›) = ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}))β€˜π‘›))
345288, 344sylan2br 596 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ (𝑛 ∈ (1...𝑁) ∧ Β¬ 𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1))) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}))β€˜π‘›) = ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}))β€˜π‘›))
346345anassrs 469 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ Β¬ 𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1)) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}))β€˜π‘›) = ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}))β€˜π‘›))
347 fzpred 13496 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 + 1) ∈ (β„€β‰₯β€˜1) β†’ (1...(𝑦 + 1)) = ({1} βˆͺ ((1 + 1)...(𝑦 + 1))))
34863, 347syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ (1...(𝑦 + 1)) = ({1} βˆͺ ((1 + 1)...(𝑦 + 1))))
349348imaeq2d 6018 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ({1} βˆͺ ((1 + 1)...(𝑦 + 1)))))
350349adantl 483 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ({1} βˆͺ ((1 + 1)...(𝑦 + 1)))))
351324uneq1d 4127 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1)))) = (((2nd β€˜(1st β€˜π‘‡)) β€œ {1}) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1)))))
352 uncom 4118 . . . . . . . . . . . . . . . . . . . 20 (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) βˆͺ {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) = ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))))
353 imaundi 6107 . . . . . . . . . . . . . . . . . . . 20 ((2nd β€˜(1st β€˜π‘‡)) β€œ ({1} βˆͺ ((1 + 1)...(𝑦 + 1)))) = (((2nd β€˜(1st β€˜π‘‡)) β€œ {1}) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))))
354351, 352, 3533eqtr4g 2802 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) βˆͺ {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ({1} βˆͺ ((1 + 1)...(𝑦 + 1)))))
355354adantr 482 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) βˆͺ {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ({1} βˆͺ ((1 + 1)...(𝑦 + 1)))))
356350, 355eqtr4d 2780 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) = (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) βˆͺ {((2nd β€˜(1st β€˜π‘‡))β€˜1)}))
357356xpeq1d 5667 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) = ((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) βˆͺ {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) Γ— {1}))
358 xpundir 5706 . . . . . . . . . . . . . . . 16 ((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) βˆͺ {((2nd β€˜(1st β€˜π‘‡))β€˜1)}) Γ— {1}) = ((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}))
359357, 358eqtrdi 2793 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) = ((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1})))
360359uneq1d 4127 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) = (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1})) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})))
361 un23 4133 . . . . . . . . . . . . . 14 (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1})) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) = (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}))
362360, 361eqtrdi 2793 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) = (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1})))
363362fveq1d 6849 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}))β€˜π‘›))
364363ad2antrr 725 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ Β¬ 𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1)) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {1}))β€˜π‘›))
365 imaco 6208 . . . . . . . . . . . . . . . . 17 (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ (1...𝑦)))
366 df-ima 5651 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ (1...𝑦)) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β†Ύ (1...𝑦))
367 peano2uz 12833 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑁 βˆ’ 1) ∈ (β„€β‰₯β€˜π‘¦) β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜π‘¦))
36871, 367syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜π‘¦))
369368adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜π‘¦))
37070, 369eqeltrrd 2839 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘¦))
371 fzss2 13488 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ (β„€β‰₯β€˜π‘¦) β†’ (1...𝑦) βŠ† (1...𝑁))
372370, 371syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (1...𝑦) βŠ† (1...𝑁))
373372resmptd 5999 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β†Ύ (1...𝑦)) = (𝑛 ∈ (1...𝑦) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
374 fzss2 13488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑁 βˆ’ 1) ∈ (β„€β‰₯β€˜π‘¦) β†’ (1...𝑦) βŠ† (1...(𝑁 βˆ’ 1)))
37571, 374syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ (1...𝑦) βŠ† (1...(𝑁 βˆ’ 1)))
376375adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (1...𝑦) βŠ† (1...(𝑁 βˆ’ 1)))
377168adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ Β¬ 𝑁 ∈ (1...(𝑁 βˆ’ 1)))
378376, 377ssneldd 3952 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ Β¬ 𝑁 ∈ (1...𝑦))
379 eleq1 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 𝑁 β†’ (𝑛 ∈ (1...𝑦) ↔ 𝑁 ∈ (1...𝑦)))
380379notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑁 β†’ (Β¬ 𝑛 ∈ (1...𝑦) ↔ Β¬ 𝑁 ∈ (1...𝑦)))
381378, 380syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (𝑛 = 𝑁 β†’ Β¬ 𝑛 ∈ (1...𝑦)))
382381necon2ad 2959 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (𝑛 ∈ (1...𝑦) β†’ 𝑛 β‰  𝑁))
383382imp 408 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑦)) β†’ 𝑛 β‰  𝑁)
384383, 206syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑦)) β†’ if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
385384mpteq2dva 5210 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (𝑛 ∈ (1...𝑦) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
386373, 385eqtrd 2777 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β†Ύ (1...𝑦)) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
387386rneqd 5898 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β†Ύ (1...𝑦)) = ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
388366, 387eqtrid 2789 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ (1...𝑦)) = ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
389 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1))
390389elrnmpt 5916 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ V β†’ (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ βˆƒπ‘› ∈ (1...𝑦)𝑗 = (𝑛 + 1)))
391390elv 3454 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ βˆƒπ‘› ∈ (1...𝑦)𝑗 = (𝑛 + 1))
392 elfzelz 13448 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ 𝑦 ∈ β„€)
393392adantl 483 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ 𝑦 ∈ β„€)
394 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ β„€ ∧ 𝑛 ∈ (1...𝑦)) β†’ 𝑛 ∈ (1...𝑦))
395118jctl 525 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ β„€ β†’ (1 ∈ β„€ ∧ 𝑦 ∈ β„€))
396 elfzelz 13448 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (1...𝑦) β†’ 𝑛 ∈ β„€)
397396, 118jctir 522 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 ∈ (1...𝑦) β†’ (𝑛 ∈ β„€ ∧ 1 ∈ β„€))
398 fzaddel 13482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1 ∈ β„€ ∧ 𝑦 ∈ β„€) ∧ (𝑛 ∈ β„€ ∧ 1 ∈ β„€)) β†’ (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
399395, 397, 398syl2an 597 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ β„€ ∧ 𝑛 ∈ (1...𝑦)) β†’ (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
400394, 399mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ β„€ ∧ 𝑛 ∈ (1...𝑦)) β†’ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1)))
401 eleq1 2826 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = (𝑛 + 1) β†’ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
402400, 401syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ β„€ ∧ 𝑛 ∈ (1...𝑦)) β†’ (𝑗 = (𝑛 + 1) β†’ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
403402rexlimdva 3153 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ β„€ β†’ (βˆƒπ‘› ∈ (1...𝑦)𝑗 = (𝑛 + 1) β†’ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
404 elfzelz 13448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) β†’ 𝑗 ∈ β„€)
405404zcnd 12615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) β†’ 𝑗 ∈ β„‚)
406 npcan1 11587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ β„‚ β†’ ((𝑗 βˆ’ 1) + 1) = 𝑗)
407405, 406syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) β†’ ((𝑗 βˆ’ 1) + 1) = 𝑗)
408407eleq1d 2823 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) β†’ (((𝑗 βˆ’ 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
409408ibir 268 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) β†’ ((𝑗 βˆ’ 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))
410409adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ β„€ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) β†’ ((𝑗 βˆ’ 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))
411 peano2zm 12553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ β„€ β†’ (𝑗 βˆ’ 1) ∈ β„€)
412404, 411syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) β†’ (𝑗 βˆ’ 1) ∈ β„€)
413412, 118jctir 522 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) β†’ ((𝑗 βˆ’ 1) ∈ β„€ ∧ 1 ∈ β„€))
414 fzaddel 13482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1 ∈ β„€ ∧ 𝑦 ∈ β„€) ∧ ((𝑗 βˆ’ 1) ∈ β„€ ∧ 1 ∈ β„€)) β†’ ((𝑗 βˆ’ 1) ∈ (1...𝑦) ↔ ((𝑗 βˆ’ 1) + 1) ∈ ((1 + 1)...(𝑦 + 1))))
415395, 413, 414syl2an 597 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ β„€ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) β†’ ((𝑗 βˆ’ 1) ∈ (1...𝑦) ↔ ((𝑗 βˆ’ 1) + 1) ∈ ((1 + 1)...(𝑦 + 1))))
416410, 415mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ β„€ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) β†’ (𝑗 βˆ’ 1) ∈ (1...𝑦))
417405adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ β„€ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) β†’ 𝑗 ∈ β„‚)
418406eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ β„‚ β†’ 𝑗 = ((𝑗 βˆ’ 1) + 1))
419417, 418syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ β„€ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) β†’ 𝑗 = ((𝑗 βˆ’ 1) + 1))
420 oveq1 7369 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = (𝑗 βˆ’ 1) β†’ (𝑛 + 1) = ((𝑗 βˆ’ 1) + 1))
421420rspceeqv 3600 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑗 βˆ’ 1) ∈ (1...𝑦) ∧ 𝑗 = ((𝑗 βˆ’ 1) + 1)) β†’ βˆƒπ‘› ∈ (1...𝑦)𝑗 = (𝑛 + 1))
422416, 419, 421syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ β„€ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) β†’ βˆƒπ‘› ∈ (1...𝑦)𝑗 = (𝑛 + 1))
423422ex 414 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ β„€ β†’ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) β†’ βˆƒπ‘› ∈ (1...𝑦)𝑗 = (𝑛 + 1)))
424403, 423impbid 211 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ β„€ β†’ (βˆƒπ‘› ∈ (1...𝑦)𝑗 = (𝑛 + 1) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
425393, 424syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (βˆƒπ‘› ∈ (1...𝑦)𝑗 = (𝑛 + 1) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
426391, 425bitrid 283 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
427426eqrdv 2735 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = ((1 + 1)...(𝑦 + 1)))
428388, 427eqtrd 2777 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ (1...𝑦)) = ((1 + 1)...(𝑦 + 1)))
429428imaeq2d 6018 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ (1...𝑦))) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))))
430365, 429eqtrid 2789 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))))
431430xpeq1d 5667 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) = (((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}))
432 imaundi 6107 . . . . . . . . . . . . . . . . . . 19 (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ({𝑁} βˆͺ ((𝑦 + 1)...(𝑁 βˆ’ 1)))) = ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ {𝑁}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...(𝑁 βˆ’ 1))))
433 imaco 6208 . . . . . . . . . . . . . . . . . . . 20 (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ {𝑁}) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ {𝑁}))
434 imaco 6208 . . . . . . . . . . . . . . . . . . . 20 (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...(𝑁 βˆ’ 1))) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...(𝑁 βˆ’ 1))))
435433, 434uneq12i 4126 . . . . . . . . . . . . . . . . . . 19 ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ {𝑁}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...(𝑁 βˆ’ 1)))) = (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ {𝑁})) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...(𝑁 βˆ’ 1)))))
436432, 435eqtri 2765 . . . . . . . . . . . . . . . . . 18 (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ({𝑁} βˆͺ ((𝑦 + 1)...(𝑁 βˆ’ 1)))) = (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ {𝑁})) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...(𝑁 βˆ’ 1)))))
437189adantr 482 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ 𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
438 fzsplit2 13473 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜(𝑦 + 1)) ∧ 𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1))) β†’ ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 βˆ’ 1)) βˆͺ (((𝑁 βˆ’ 1) + 1)...𝑁)))
43973, 437, 438syl2an2 685 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 βˆ’ 1)) βˆͺ (((𝑁 βˆ’ 1) + 1)...𝑁)))
440195uneq2d 4128 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ (((𝑦 + 1)...(𝑁 βˆ’ 1)) βˆͺ (((𝑁 βˆ’ 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 βˆ’ 1)) βˆͺ {𝑁}))
441440adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((𝑦 + 1)...(𝑁 βˆ’ 1)) βˆͺ (((𝑁 βˆ’ 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 βˆ’ 1)) βˆͺ {𝑁}))
442439, 441eqtrd 2777 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 βˆ’ 1)) βˆͺ {𝑁}))
443 uncom 4118 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 + 1)...(𝑁 βˆ’ 1)) βˆͺ {𝑁}) = ({𝑁} βˆͺ ((𝑦 + 1)...(𝑁 βˆ’ 1)))
444442, 443eqtrdi 2793 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑦 + 1)...𝑁) = ({𝑁} βˆͺ ((𝑦 + 1)...(𝑁 βˆ’ 1))))
445444imaeq2d 6018 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) = (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ({𝑁} βˆͺ ((𝑦 + 1)...(𝑁 βˆ’ 1)))))
446249sneqd 4603 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))β€˜π‘)} = {1})
447 fnsnfv 6925 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) β†’ {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))β€˜π‘)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ {𝑁}))
448252, 246, 447syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))β€˜π‘)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ {𝑁}))
449446, 448eqtr3d 2779 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ {1} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ {𝑁}))
450449imaeq2d 6018 . . . . . . . . . . . . . . . . . . . . 21 (πœ‘ β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ {1}) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ {𝑁})))
451324, 450eqtrd 2777 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ {((2nd β€˜(1st β€˜π‘‡))β€˜1)} = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ {𝑁})))
452451adantr 482 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ {((2nd β€˜(1st β€˜π‘‡))β€˜1)} = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ {𝑁})))
453 df-ima 5651 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...(𝑁 βˆ’ 1))) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β†Ύ ((𝑦 + 1)...(𝑁 βˆ’ 1)))
454 fzss1 13487 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 + 1) ∈ (β„€β‰₯β€˜1) β†’ ((𝑦 + 1)...(𝑁 βˆ’ 1)) βŠ† (1...(𝑁 βˆ’ 1)))
45563, 454syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ ((𝑦 + 1)...(𝑁 βˆ’ 1)) βŠ† (1...(𝑁 βˆ’ 1)))
456 fzss2 13488 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)) β†’ (1...(𝑁 βˆ’ 1)) βŠ† (1...𝑁))
457189, 456syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (πœ‘ β†’ (1...(𝑁 βˆ’ 1)) βŠ† (1...𝑁))
458455, 457sylan9ssr 3963 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑦 + 1)...(𝑁 βˆ’ 1)) βŠ† (1...𝑁))
459458resmptd 5999 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β†Ύ ((𝑦 + 1)...(𝑁 βˆ’ 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
460 elfzle2 13452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) β†’ 𝑁 ≀ (𝑁 βˆ’ 1))
461166, 460nsyl 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (πœ‘ β†’ Β¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)))
462 eleq1 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 = 𝑁 β†’ (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↔ 𝑁 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))))
463462notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = 𝑁 β†’ (Β¬ 𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↔ Β¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))))
464461, 463syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (πœ‘ β†’ (𝑛 = 𝑁 β†’ Β¬ 𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))))
465464con2d 134 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (πœ‘ β†’ (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) β†’ Β¬ 𝑛 = 𝑁))
466465imp 408 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((πœ‘ ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))) β†’ Β¬ 𝑛 = 𝑁)
467466iffalsed 4502 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((πœ‘ ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))) β†’ if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
468467mpteq2dva 5210 . . . . . . . . . . . . . . . . . . . . . . . . 25 (πœ‘ β†’ (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)))
469468adantr 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)))
470459, 469eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β†Ύ ((𝑦 + 1)...(𝑁 βˆ’ 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)))
471470rneqd 5898 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β†Ύ ((𝑦 + 1)...(𝑁 βˆ’ 1))) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)))
472453, 471eqtrid 2789 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...(𝑁 βˆ’ 1))) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)))
473 elfzelz 13448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) β†’ 𝑗 ∈ β„€)
474473zcnd 12615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) β†’ 𝑗 ∈ β„‚)
475474, 406syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) β†’ ((𝑗 βˆ’ 1) + 1) = 𝑗)
476475eleq1d 2823 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) β†’ (((𝑗 βˆ’ 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) ↔ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1))))
477476ibir 268 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) β†’ ((𝑗 βˆ’ 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)))
478477adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1))) β†’ ((𝑗 βˆ’ 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)))
47950nnzd 12533 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ (𝑦 + 1) ∈ β„€)
480117, 479anim12ci 615 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑦 + 1) ∈ β„€ ∧ (𝑁 βˆ’ 1) ∈ β„€))
481473, 411syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) β†’ (𝑗 βˆ’ 1) ∈ β„€)
482481, 118jctir 522 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) β†’ ((𝑗 βˆ’ 1) ∈ β„€ ∧ 1 ∈ β„€))
483 fzaddel 13482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑦 + 1) ∈ β„€ ∧ (𝑁 βˆ’ 1) ∈ β„€) ∧ ((𝑗 βˆ’ 1) ∈ β„€ ∧ 1 ∈ β„€)) β†’ ((𝑗 βˆ’ 1) ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↔ ((𝑗 βˆ’ 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1))))
484480, 482, 483syl2an 597 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1))) β†’ ((𝑗 βˆ’ 1) ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↔ ((𝑗 βˆ’ 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1))))
485478, 484mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1))) β†’ (𝑗 βˆ’ 1) ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)))
486474, 418syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) β†’ 𝑗 = ((𝑗 βˆ’ 1) + 1))
487486adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1))) β†’ 𝑗 = ((𝑗 βˆ’ 1) + 1))
488420rspceeqv 3600 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑗 βˆ’ 1) ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ∧ 𝑗 = ((𝑗 βˆ’ 1) + 1)) β†’ βˆƒπ‘› ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))𝑗 = (𝑛 + 1))
489485, 487, 488syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1))) β†’ βˆƒπ‘› ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))𝑗 = (𝑛 + 1))
490489ex 414 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) β†’ βˆƒπ‘› ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))𝑗 = (𝑛 + 1)))
491 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))) β†’ 𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)))
492 elfzelz 13448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) β†’ 𝑛 ∈ β„€)
493492, 118jctir 522 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) β†’ (𝑛 ∈ β„€ ∧ 1 ∈ β„€))
494 fzaddel 13482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑦 + 1) ∈ β„€ ∧ (𝑁 βˆ’ 1) ∈ β„€) ∧ (𝑛 ∈ β„€ ∧ 1 ∈ β„€)) β†’ (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1))))
495480, 493, 494syl2an 597 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))) β†’ (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1))))
496491, 495mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))) β†’ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)))
497 eleq1 2826 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = (𝑛 + 1) β†’ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1))))
498496, 497syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))) β†’ (𝑗 = (𝑛 + 1) β†’ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1))))
499498rexlimdva 3153 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (βˆƒπ‘› ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))𝑗 = (𝑛 + 1) β†’ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1))))
500490, 499impbid 211 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) ↔ βˆƒπ‘› ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))𝑗 = (𝑛 + 1)))
501 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1))
502501elrnmpt 5916 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ V β†’ (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)) ↔ βˆƒπ‘› ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))𝑗 = (𝑛 + 1)))
503502elv 3454 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)) ↔ βˆƒπ‘› ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1))𝑗 = (𝑛 + 1))
504500, 503bitr4di 289 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) ↔ 𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1))))
505504eqrdv 2735 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 βˆ’ 1)) ↦ (𝑛 + 1)))
50669oveq2d 7378 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) = (((𝑦 + 1) + 1)...𝑁))
507506adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((𝑦 + 1) + 1)...((𝑁 βˆ’ 1) + 1)) = (((𝑦 + 1) + 1)...𝑁))
508472, 505, 5073eqtr2rd 2784 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((𝑦 + 1) + 1)...𝑁) = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...(𝑁 βˆ’ 1))))
509508imaeq2d 6018 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...(𝑁 βˆ’ 1)))))
510452, 509uneq12d 4129 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) = (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ {𝑁})) βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) β€œ ((𝑦 + 1)...(𝑁 βˆ’ 1))))))
511436, 445, 5103eqtr4a 2803 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) = ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))))
512511xpeq1d 5667 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}) = (({((2nd β€˜(1st β€˜π‘‡))β€˜1)} βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) Γ— {0}))
513 xpundir 5706 . . . . . . . . . . . . . . . 16 (({((2nd β€˜(1st β€˜π‘‡))β€˜1)} βˆͺ ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁))) Γ— {0}) = (({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))
514512, 513eqtrdi 2793 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}) = (({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})))
515431, 514uneq12d 4129 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})) = ((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))))
516 unass 4131 . . . . . . . . . . . . . . 15 (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0})) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) = ((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})))
517 un23 4133 . . . . . . . . . . . . . . 15 (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0})) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) = (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}))
518516, 517eqtr3i 2767 . . . . . . . . . . . . . 14 ((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))) = (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}))
519515, 518eqtrdi 2793 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})) = (((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0})))
520519fveq1d 6849 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›) = ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}))β€˜π‘›))
521520ad2antrr 725 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ Β¬ 𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1)) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›) = ((((((2nd β€˜(1st β€˜π‘‡)) β€œ ((1 + 1)...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})) βˆͺ ({((2nd β€˜(1st β€˜π‘‡))β€˜1)} Γ— {0}))β€˜π‘›))
522346, 364, 5213eqtr4d 2787 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ Β¬ 𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1)) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›))
523 snssi 4773 . . . . . . . . . . . . . . 15 (1 ∈ β„‚ β†’ {1} βŠ† β„‚)
524138, 523ax-mp 5 . . . . . . . . . . . . . 14 {1} βŠ† β„‚
525 0cn 11154 . . . . . . . . . . . . . . 15 0 ∈ β„‚
526 snssi 4773 . . . . . . . . . . . . . . 15 (0 ∈ β„‚ β†’ {0} βŠ† β„‚)
527525, 526ax-mp 5 . . . . . . . . . . . . . 14 {0} βŠ† β„‚
528524, 527unssi 4150 . . . . . . . . . . . . 13 ({1} βˆͺ {0}) βŠ† β„‚
52930fconst 6733 . . . . . . . . . . . . . . . . 17 ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}):(((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦))⟢{1}
53033fconst 6733 . . . . . . . . . . . . . . . . 17 ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}):(((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁))⟢{0}
531529, 530pm3.2i 472 . . . . . . . . . . . . . . . 16 (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}):(((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦))⟢{1} ∧ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}):(((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁))⟢{0})
532 fun 6709 . . . . . . . . . . . . . . . 16 (((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}):(((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦))⟢{1} ∧ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}):(((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁))⟢{0}) ∧ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) ∩ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁))) = βˆ…) β†’ (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})):((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)))⟢({1} βˆͺ {0}))
533531, 238, 532sylancr 588 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})):((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)))⟢({1} βˆͺ {0}))
534 imaundi 6107 . . . . . . . . . . . . . . . . 17 (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((1...𝑦) βˆͺ ((𝑦 + 1)...𝑁))) = ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)))
535 fzsplit2 13473 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 + 1) ∈ (β„€β‰₯β€˜1) ∧ 𝑁 ∈ (β„€β‰₯β€˜π‘¦)) β†’ (1...𝑁) = ((1...𝑦) βˆͺ ((𝑦 + 1)...𝑁)))
53663, 370, 535syl2an2 685 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (1...𝑁) = ((1...𝑦) βˆͺ ((𝑦 + 1)...𝑁)))
537536imaeq2d 6018 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑁)) = (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((1...𝑦) βˆͺ ((𝑦 + 1)...𝑁))))
538 f1ofo 6796 . . . . . . . . . . . . . . . . . . . 20 (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-ontoβ†’(1...𝑁) β†’ ((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–ontoβ†’(1...𝑁))
539 foima 6766 . . . . . . . . . . . . . . . . . . . 20 (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–ontoβ†’(1...𝑁) β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑁)) = (1...𝑁))
540225, 538, 5393syl 18 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑁)) = (1...𝑁))
541540adantr 482 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑁)) = (1...𝑁))
542537, 541eqtr3d 2779 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((1...𝑦) βˆͺ ((𝑦 + 1)...𝑁))) = (1...𝑁))
543534, 542eqtr3id 2791 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁))) = (1...𝑁))
544543feq2d 6659 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})):((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)))⟢({1} βˆͺ {0}) ↔ (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})):(1...𝑁)⟢({1} βˆͺ {0})))
545533, 544mpbid 231 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})):(1...𝑁)⟢({1} βˆͺ {0}))
546545ffvelcdmda 7040 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›) ∈ ({1} βˆͺ {0}))
547528, 546sselid 3947 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›) ∈ β„‚)
548547addid2d 11363 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) β†’ (0 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)) = ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›))
549548adantr 482 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ Β¬ 𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1)) β†’ (0 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)) = ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›))
550522, 549eqtr4d 2780 . . . . . . . . 9 ((((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ Β¬ 𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1)) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = (0 + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)))
55193, 95, 284, 550ifbothda 4529 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) β†’ (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›) = (if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0) + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)))
552551oveq2d 7378 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) β†’ (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›)) = (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + (if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0) + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›))))
553 elmapi 8794 . . . . . . . . . . . . 13 ((1st β€˜(1st β€˜π‘‡)) ∈ ((0..^𝐾) ↑m (1...𝑁)) β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)⟢(0..^𝐾))
55426, 553syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)⟢(0..^𝐾))
555554ffvelcdmda 7040 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ (1...𝑁)) β†’ ((1st β€˜(1st β€˜π‘‡))β€˜π‘›) ∈ (0..^𝐾))
556 elfzonn0 13624 . . . . . . . . . . 11 (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) ∈ (0..^𝐾) β†’ ((1st β€˜(1st β€˜π‘‡))β€˜π‘›) ∈ β„•0)
557555, 556syl 17 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (1...𝑁)) β†’ ((1st β€˜(1st β€˜π‘‡))β€˜π‘›) ∈ β„•0)
558557nn0cnd 12482 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (1...𝑁)) β†’ ((1st β€˜(1st β€˜π‘‡))β€˜π‘›) ∈ β„‚)
559558adantlr 714 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) β†’ ((1st β€˜(1st β€˜π‘‡))β€˜π‘›) ∈ β„‚)
560138, 525ifcli 4538 . . . . . . . . 9 if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0) ∈ β„‚
561560a1i 11 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) β†’ if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0) ∈ β„‚)
562559, 561, 547addassd 11184 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) β†’ ((((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0)) + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)) = (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + (if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0) + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›))))
563552, 562eqtr4d 2780 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) β†’ (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›)) = ((((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0)) + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)))
564563mpteq2dva 5210 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (𝑛 ∈ (1...𝑁) ↦ (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + (((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))β€˜π‘›))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0)) + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›))))
56591, 564eqtrd 2777 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0)) + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›))))
566 poimirlem18.4 . . . . . . . . . 10 (πœ‘ β†’ (2nd β€˜π‘‡) = 0)
567566adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (2nd β€˜π‘‡) = 0)
568 elfzle1 13451 . . . . . . . . . 10 (𝑦 ∈ (0...(𝑁 βˆ’ 1)) β†’ 0 ≀ 𝑦)
569568adantl 483 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ 0 ≀ 𝑦)
570567, 569eqbrtrd 5132 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (2nd β€˜π‘‡) ≀ 𝑦)
571 0re 11164 . . . . . . . . . 10 0 ∈ ℝ
572566, 571eqeltrdi 2846 . . . . . . . . 9 (πœ‘ β†’ (2nd β€˜π‘‡) ∈ ℝ)
573 lenlt 11240 . . . . . . . . 9 (((2nd β€˜π‘‡) ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ ((2nd β€˜π‘‡) ≀ 𝑦 ↔ Β¬ 𝑦 < (2nd β€˜π‘‡)))
574572, 231, 573syl2an 597 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜π‘‡) ≀ 𝑦 ↔ Β¬ 𝑦 < (2nd β€˜π‘‡)))
575570, 574mpbid 231 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ Β¬ 𝑦 < (2nd β€˜π‘‡))
576575iffalsed 4502 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)) = (𝑦 + 1))
577576csbeq1d 3864 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ⦋if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) = ⦋(𝑦 + 1) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))
578 ovex 7395 . . . . . 6 (𝑦 + 1) ∈ V
579 oveq2 7370 . . . . . . . . . 10 (𝑗 = (𝑦 + 1) β†’ (1...𝑗) = (1...(𝑦 + 1)))
580579imaeq2d 6018 . . . . . . . . 9 (𝑗 = (𝑦 + 1) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))))
581580xpeq1d 5667 . . . . . . . 8 (𝑗 = (𝑦 + 1) β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) = (((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}))
582 oveq1 7369 . . . . . . . . . . 11 (𝑗 = (𝑦 + 1) β†’ (𝑗 + 1) = ((𝑦 + 1) + 1))
583582oveq1d 7377 . . . . . . . . . 10 (𝑗 = (𝑦 + 1) β†’ ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
584583imaeq2d 6018 . . . . . . . . 9 (𝑗 = (𝑦 + 1) β†’ ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)))
585584xpeq1d 5667 . . . . . . . 8 (𝑗 = (𝑦 + 1) β†’ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}) = (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))
586581, 585uneq12d 4129 . . . . . . 7 (𝑗 = (𝑦 + 1) β†’ ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})) = ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})))
587586oveq2d 7378 . . . . . 6 (𝑗 = (𝑦 + 1) β†’ ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) = ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))))
588578, 587csbie 3896 . . . . 5 ⦋(𝑦 + 1) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) = ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0})))
589577, 588eqtrdi 2793 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ⦋if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) = ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...(𝑦 + 1))) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ (((𝑦 + 1) + 1)...𝑁)) Γ— {0}))))
590 ovexd 7397 . . . . 5 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) β†’ (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0)) ∈ V)
591 fvexd 6862 . . . . 5 (((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) ∧ 𝑛 ∈ (1...𝑁)) β†’ ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›) ∈ V)
592 eqidd 2738 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (𝑛 ∈ (1...𝑁) ↦ (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0))) = (𝑛 ∈ (1...𝑁) ↦ (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0))))
593545ffnd 6674 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})) Fn (1...𝑁))
594 nfcv 2908 . . . . . . . . . . 11 Ⅎ𝑛(2nd β€˜(1st β€˜π‘‡))
595 nfmpt1 5218 . . . . . . . . . . 11 Ⅎ𝑛(𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
596594, 595nfco 5826 . . . . . . . . . 10 Ⅎ𝑛((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
597 nfcv 2908 . . . . . . . . . 10 Ⅎ𝑛(1...𝑦)
598596, 597nfima 6026 . . . . . . . . 9 Ⅎ𝑛(((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦))
599 nfcv 2908 . . . . . . . . 9 Ⅎ𝑛{1}
600598, 599nfxp 5671 . . . . . . . 8 Ⅎ𝑛((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1})
601 nfcv 2908 . . . . . . . . . 10 Ⅎ𝑛((𝑦 + 1)...𝑁)
602596, 601nfima 6026 . . . . . . . . 9 Ⅎ𝑛(((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁))
603 nfcv 2908 . . . . . . . . 9 Ⅎ𝑛{0}
604602, 603nfxp 5671 . . . . . . . 8 Ⅎ𝑛((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})
605600, 604nfun 4130 . . . . . . 7 Ⅎ𝑛(((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))
606605dffn5f 6918 . . . . . 6 ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})) Fn (1...𝑁) ↔ (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)))
607593, 606sylib 217 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›)))
60887, 590, 591, 592, 607offval2 7642 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑛 ∈ (1...𝑁) ↦ (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0))) ∘f + (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0)) + ((((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))β€˜π‘›))))
609565, 589, 6083eqtr4rd 2788 . . 3 ((πœ‘ ∧ 𝑦 ∈ (0...(𝑁 βˆ’ 1))) β†’ ((𝑛 ∈ (1...𝑁) ↦ (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0))) ∘f + (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0}))) = ⦋if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))
610609mpteq2dva 5210 . 2 (πœ‘ β†’ (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0))) ∘f + (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})))) = (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ⦋if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‡)) ∘f + ((((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))))
61119, 610eqtr4d 2780 1 (πœ‘ β†’ 𝐹 = (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st β€˜(1st β€˜π‘‡))β€˜π‘›) + if(𝑛 = ((2nd β€˜(1st β€˜π‘‡))β€˜1), 1, 0))) ∘f + (((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ (1...𝑦)) Γ— {1}) βˆͺ ((((2nd β€˜(1st β€˜π‘‡)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) β€œ ((𝑦 + 1)...𝑁)) Γ— {0})))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2714   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  βˆƒ!wreu 3354  {crab 3410  Vcvv 3448  β¦‹csb 3860   βˆ– cdif 3912   βˆͺ cun 3913   ∩ cin 3914   βŠ† wss 3915  βˆ…c0 4287  ifcif 4491  {csn 4591  βŸ¨cop 4597   class class class wbr 5110   ↦ cmpt 5193   Γ— cxp 5636  β—‘ccnv 5637  ran crn 5639   β†Ύ cres 5640   β€œ cima 5641   ∘ ccom 5642  Fun wfun 6495   Fn wfn 6496  βŸΆwf 6497  β€“ontoβ†’wfo 6499  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  (class class class)co 7362   ∘f cof 7620  1st c1st 7924  2nd c2nd 7925   ↑m cmap 8772  β„‚cc 11056  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   < clt 11196   ≀ cle 11197   βˆ’ cmin 11392  β„•cn 12160  β„•0cn0 12420  β„€cz 12506  β„€β‰₯cuz 12770  ...cfz 13431  ..^cfzo 13574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575
This theorem is referenced by:  poimirlem17  36124
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