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Theorem poimirlem22 36129
Description: Lemma for poimir 36140, that a given face belongs to exactly two simplices, provided it's not on the boundary of the cube. (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)
poimirlem22.3 ((πœ‘ ∧ 𝑛 ∈ (1...𝑁)) β†’ βˆƒπ‘ ∈ ran 𝐹(π‘β€˜π‘›) β‰  0)
poimirlem22.4 ((πœ‘ ∧ 𝑛 ∈ (1...𝑁)) β†’ βˆƒπ‘ ∈ ran 𝐹(π‘β€˜π‘›) β‰  𝐾)
Assertion
Ref Expression
poimirlem22 (πœ‘ β†’ βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
Distinct variable groups:   𝑓,𝑗,𝑛,𝑝,𝑑,𝑦,𝑧   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   πœ‘,𝑧   𝑓,𝐹,𝑝,𝑑,𝑧   𝑧,𝐾   𝑧,𝑁   𝑑,𝑇,𝑧   𝑆,𝑗,𝑛,𝑝,𝑑,𝑦,𝑧
Allowed substitution hints:   πœ‘(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem22
StepHypRef Expression
1 poimir.0 . . . . 5 (πœ‘ β†’ 𝑁 ∈ β„•)
21adantr 482 . . . 4 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ 𝑁 ∈ β„•)
3 poimirlem22.s . . . 4 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}))))}
4 poimirlem22.1 . . . . 5 (πœ‘ β†’ 𝐹:(0...(𝑁 βˆ’ 1))⟢((0...𝐾) ↑m (1...𝑁)))
54adantr 482 . . . 4 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ 𝐹:(0...(𝑁 βˆ’ 1))⟢((0...𝐾) ↑m (1...𝑁)))
6 poimirlem22.2 . . . . 5 (πœ‘ β†’ 𝑇 ∈ 𝑆)
76adantr 482 . . . 4 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ 𝑇 ∈ 𝑆)
8 simpr 486 . . . 4 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)))
92, 3, 5, 7, 8poimirlem15 36122 . . 3 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ ⟨⟨(1st β€˜(1st β€˜π‘‡)), ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))⟩, (2nd β€˜π‘‡)⟩ ∈ 𝑆)
10 fveq2 6847 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑑 = 𝑇 β†’ (2nd β€˜π‘‘) = (2nd β€˜π‘‡))
1110breq2d 5122 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑇 β†’ (𝑦 < (2nd β€˜π‘‘) ↔ 𝑦 < (2nd β€˜π‘‡)))
1211ifbid 4514 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 = 𝑇 β†’ if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)))
1312csbeq1d 3864 . . . . . . . . . . . . . . . . . . . . 21 (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}))))
14 2fveq3 6852 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑇 β†’ (1st β€˜(1st β€˜π‘‘)) = (1st β€˜(1st β€˜π‘‡)))
15 2fveq3 6852 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = 𝑇 β†’ (2nd β€˜(1st β€˜π‘‘)) = (2nd β€˜(1st β€˜π‘‡)))
1615imaeq1d 6017 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = 𝑇 β†’ ((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)))
1716xpeq1d 5667 . . . . . . . . . . . . . . . . . . . . . . . 24 (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}))
1815imaeq1d 6017 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = 𝑇 β†’ ((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)))
1918xpeq1d 5667 . . . . . . . . . . . . . . . . . . . . . . . 24 (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}))
2017, 19uneq12d 4129 . . . . . . . . . . . . . . . . . . . . . . 23 (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})))
2114, 20oveq12d 7380 . . . . . . . . . . . . . . . . . . . . . 22 (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}))))
2221csbeq2dv 3867 . . . . . . . . . . . . . . . . . . . . 21 (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}))))
2313, 22eqtrd 2777 . . . . . . . . . . . . . . . . . . . 20 (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}))))
2423mpteq2dv 5212 . . . . . . . . . . . . . . . . . . 19 (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})))))
2524eqeq2d 2748 . . . . . . . . . . . . . . . . . 18 (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}))))))
2625, 3elrab2 3653 . . . . . . . . . . . . . . . . 17 (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}))))))
2726simprbi 498 . . . . . . . . . . . . . . . 16 (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})))))
286, 27syl 17 . . . . . . . . . . . . . . 15 (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})))))
2928adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) 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})))))
30 elrabi 3644 . . . . . . . . . . . . . . . . . . . . 21 (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)))
3130, 3eleq2s 2856 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ 𝑆 β†’ 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)))
326, 31syl 17 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)))
33 xp1st 7958 . . . . . . . . . . . . . . . . . . 19 (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)}))
3432, 33syl 17 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ (1st β€˜π‘‡) ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}))
35 xp1st 7958 . . . . . . . . . . . . . . . . . 18 ((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)))
3634, 35syl 17 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (1st β€˜(1st β€˜π‘‡)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
37 elmapi 8794 . . . . . . . . . . . . . . . . 17 ((1st β€˜(1st β€˜π‘‡)) ∈ ((0..^𝐾) ↑m (1...𝑁)) β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)⟢(0..^𝐾))
3836, 37syl 17 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)⟢(0..^𝐾))
39 elfzoelz 13579 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝐾) β†’ 𝑛 ∈ β„€)
4039ssriv 3953 . . . . . . . . . . . . . . . 16 (0..^𝐾) βŠ† β„€
41 fss 6690 . . . . . . . . . . . . . . . 16 (((1st β€˜(1st β€˜π‘‡)):(1...𝑁)⟢(0..^𝐾) ∧ (0..^𝐾) βŠ† β„€) β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)βŸΆβ„€)
4238, 40, 41sylancl 587 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)βŸΆβ„€)
4342adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)βŸΆβ„€)
44 xp2nd 7959 . . . . . . . . . . . . . . . . 17 ((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)})
4534, 44syl 17 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (2nd β€˜(1st β€˜π‘‡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})
46 fvex 6860 . . . . . . . . . . . . . . . . 17 (2nd β€˜(1st β€˜π‘‡)) ∈ V
47 f1oeq1 6777 . . . . . . . . . . . . . . . . 17 (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)))
4846, 47elab 3635 . . . . . . . . . . . . . . . 16 ((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))
4945, 48sylib 217 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
5049adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
512, 29, 43, 50, 8poimirlem1 36108 . . . . . . . . . . . . 13 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ Β¬ βˆƒ*𝑛 ∈ (1...𝑁)((πΉβ€˜((2nd β€˜π‘‡) βˆ’ 1))β€˜π‘›) β‰  ((πΉβ€˜(2nd β€˜π‘‡))β€˜π‘›))
5251adantr 482 . . . . . . . . . . . 12 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ Β¬ βˆƒ*𝑛 ∈ (1...𝑁)((πΉβ€˜((2nd β€˜π‘‡) βˆ’ 1))β€˜π‘›) β‰  ((πΉβ€˜(2nd β€˜π‘‡))β€˜π‘›))
531ad3antrrr 729 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) β‰  (2nd β€˜π‘‡)) β†’ 𝑁 ∈ β„•)
54 fveq2 6847 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑑 = 𝑧 β†’ (2nd β€˜π‘‘) = (2nd β€˜π‘§))
5554breq2d 5122 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑧 β†’ (𝑦 < (2nd β€˜π‘‘) ↔ 𝑦 < (2nd β€˜π‘§)))
5655ifbid 4514 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 = 𝑧 β†’ if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd β€˜π‘§), 𝑦, (𝑦 + 1)))
5756csbeq1d 3864 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑧 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}))))
58 2fveq3 6852 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑧 β†’ (1st β€˜(1st β€˜π‘‘)) = (1st β€˜(1st β€˜π‘§)))
59 2fveq3 6852 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = 𝑧 β†’ (2nd β€˜(1st β€˜π‘‘)) = (2nd β€˜(1st β€˜π‘§)))
6059imaeq1d 6017 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = 𝑧 β†’ ((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) = ((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑗)))
6160xpeq1d 5667 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑑 = 𝑧 β†’ (((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) Γ— {1}) = (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑗)) Γ— {1}))
6259imaeq1d 6017 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = 𝑧 β†’ ((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) = ((2nd β€˜(1st β€˜π‘§)) β€œ ((𝑗 + 1)...𝑁)))
6362xpeq1d 5667 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑑 = 𝑧 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}))
6461, 63uneq12d 4129 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑧 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})))
6558, 64oveq12d 7380 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 = 𝑧 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}))))
6665csbeq2dv 3867 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑧 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}))))
6757, 66eqtrd 2777 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = 𝑧 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}))))
6867mpteq2dv 5212 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑧 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})))))
6968eqeq2d 2748 . . . . . . . . . . . . . . . . . 18 (𝑑 = 𝑧 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}))))))
7069, 3elrab2 3653 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ 𝑆 ↔ (𝑧 ∈ ((((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}))))))
7170simprbi 498 . . . . . . . . . . . . . . . 16 (𝑧 ∈ 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})))))
7271ad2antlr 726 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) β‰  (2nd β€˜π‘‡)) 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})))))
73 elrabi 3644 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ {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†’ 𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)))
7473, 3eleq2s 2856 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ 𝑆 β†’ 𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)))
75 xp1st 7958 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ((((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)}))
7674, 75syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ 𝑆 β†’ (1st β€˜π‘§) ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}))
77 xp1st 7958 . . . . . . . . . . . . . . . . . . 19 ((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)))
7876, 77syl 17 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ 𝑆 β†’ (1st β€˜(1st β€˜π‘§)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
79 elmapi 8794 . . . . . . . . . . . . . . . . . 18 ((1st β€˜(1st β€˜π‘§)) ∈ ((0..^𝐾) ↑m (1...𝑁)) β†’ (1st β€˜(1st β€˜π‘§)):(1...𝑁)⟢(0..^𝐾))
8078, 79syl 17 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ 𝑆 β†’ (1st β€˜(1st β€˜π‘§)):(1...𝑁)⟢(0..^𝐾))
81 fss 6690 . . . . . . . . . . . . . . . . 17 (((1st β€˜(1st β€˜π‘§)):(1...𝑁)⟢(0..^𝐾) ∧ (0..^𝐾) βŠ† β„€) β†’ (1st β€˜(1st β€˜π‘§)):(1...𝑁)βŸΆβ„€)
8280, 40, 81sylancl 587 . . . . . . . . . . . . . . . 16 (𝑧 ∈ 𝑆 β†’ (1st β€˜(1st β€˜π‘§)):(1...𝑁)βŸΆβ„€)
8382ad2antlr 726 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) β‰  (2nd β€˜π‘‡)) β†’ (1st β€˜(1st β€˜π‘§)):(1...𝑁)βŸΆβ„€)
84 xp2nd 7959 . . . . . . . . . . . . . . . . . 18 ((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)})
8576, 84syl 17 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ 𝑆 β†’ (2nd β€˜(1st β€˜π‘§)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})
86 fvex 6860 . . . . . . . . . . . . . . . . . 18 (2nd β€˜(1st β€˜π‘§)) ∈ V
87 f1oeq1 6777 . . . . . . . . . . . . . . . . . 18 (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)))
8886, 87elab 3635 . . . . . . . . . . . . . . . . 17 ((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))
8985, 88sylib 217 . . . . . . . . . . . . . . . 16 (𝑧 ∈ 𝑆 β†’ (2nd β€˜(1st β€˜π‘§)):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
9089ad2antlr 726 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) β‰  (2nd β€˜π‘‡)) β†’ (2nd β€˜(1st β€˜π‘§)):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
91 simpllr 775 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) β‰  (2nd β€˜π‘‡)) β†’ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)))
92 xp2nd 7959 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) β†’ (2nd β€˜π‘§) ∈ (0...𝑁))
9374, 92syl 17 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ 𝑆 β†’ (2nd β€˜π‘§) ∈ (0...𝑁))
9493adantl 483 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (2nd β€˜π‘§) ∈ (0...𝑁))
95 eldifsn 4752 . . . . . . . . . . . . . . . . 17 ((2nd β€˜π‘§) ∈ ((0...𝑁) βˆ– {(2nd β€˜π‘‡)}) ↔ ((2nd β€˜π‘§) ∈ (0...𝑁) ∧ (2nd β€˜π‘§) β‰  (2nd β€˜π‘‡)))
9695biimpri 227 . . . . . . . . . . . . . . . 16 (((2nd β€˜π‘§) ∈ (0...𝑁) ∧ (2nd β€˜π‘§) β‰  (2nd β€˜π‘‡)) β†’ (2nd β€˜π‘§) ∈ ((0...𝑁) βˆ– {(2nd β€˜π‘‡)}))
9794, 96sylan 581 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) β‰  (2nd β€˜π‘‡)) β†’ (2nd β€˜π‘§) ∈ ((0...𝑁) βˆ– {(2nd β€˜π‘‡)}))
9853, 72, 83, 90, 91, 97poimirlem2 36109 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) β‰  (2nd β€˜π‘‡)) β†’ βˆƒ*𝑛 ∈ (1...𝑁)((πΉβ€˜((2nd β€˜π‘‡) βˆ’ 1))β€˜π‘›) β‰  ((πΉβ€˜(2nd β€˜π‘‡))β€˜π‘›))
9998ex 414 . . . . . . . . . . . . 13 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ ((2nd β€˜π‘§) β‰  (2nd β€˜π‘‡) β†’ βˆƒ*𝑛 ∈ (1...𝑁)((πΉβ€˜((2nd β€˜π‘‡) βˆ’ 1))β€˜π‘›) β‰  ((πΉβ€˜(2nd β€˜π‘‡))β€˜π‘›)))
10099necon1bd 2962 . . . . . . . . . . . 12 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (Β¬ βˆƒ*𝑛 ∈ (1...𝑁)((πΉβ€˜((2nd β€˜π‘‡) βˆ’ 1))β€˜π‘›) β‰  ((πΉβ€˜(2nd β€˜π‘‡))β€˜π‘›) β†’ (2nd β€˜π‘§) = (2nd β€˜π‘‡)))
10152, 100mpd 15 . . . . . . . . . . 11 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (2nd β€˜π‘§) = (2nd β€˜π‘‡))
102 eleq1 2826 . . . . . . . . . . . . . . . 16 ((2nd β€˜π‘§) = (2nd β€˜π‘‡) β†’ ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ↔ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))))
103102biimparc 481 . . . . . . . . . . . . . . 15 (((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡)) β†’ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)))
104103anim2i 618 . . . . . . . . . . . . . 14 ((πœ‘ ∧ ((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡))) β†’ (πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))))
105104anassrs 469 . . . . . . . . . . . . 13 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡)) β†’ (πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))))
10671adantl 483 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 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})))))
107 breq1 5113 . . . . . . . . . . . . . . . . . 18 (𝑦 = 0 β†’ (𝑦 < (2nd β€˜π‘§) ↔ 0 < (2nd β€˜π‘§)))
108 id 22 . . . . . . . . . . . . . . . . . 18 (𝑦 = 0 β†’ 𝑦 = 0)
109107, 108ifbieq1d 4515 . . . . . . . . . . . . . . . . 17 (𝑦 = 0 β†’ if(𝑦 < (2nd β€˜π‘§), 𝑦, (𝑦 + 1)) = if(0 < (2nd β€˜π‘§), 0, (𝑦 + 1)))
110 elfznn 13477 . . . . . . . . . . . . . . . . . . . 20 ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) β†’ (2nd β€˜π‘§) ∈ β„•)
111110nngt0d 12209 . . . . . . . . . . . . . . . . . . 19 ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) β†’ 0 < (2nd β€˜π‘§))
112111iftrued 4499 . . . . . . . . . . . . . . . . . 18 ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) β†’ if(0 < (2nd β€˜π‘§), 0, (𝑦 + 1)) = 0)
113112ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ if(0 < (2nd β€˜π‘§), 0, (𝑦 + 1)) = 0)
114109, 113sylan9eqr 2799 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑦 = 0) β†’ if(𝑦 < (2nd β€˜π‘§), 𝑦, (𝑦 + 1)) = 0)
115114csbeq1d 3864 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑦 = 0) 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}))) = ⦋0 / 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}))))
116 c0ex 11156 . . . . . . . . . . . . . . . . . 18 0 ∈ V
117 oveq2 7370 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 β†’ (1...𝑗) = (1...0))
118 fz10 13469 . . . . . . . . . . . . . . . . . . . . . . . 24 (1...0) = βˆ…
119117, 118eqtrdi 2793 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 β†’ (1...𝑗) = βˆ…)
120119imaeq2d 6018 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 β†’ ((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑗)) = ((2nd β€˜(1st β€˜π‘§)) β€œ βˆ…))
121120xpeq1d 5667 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 0 β†’ (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑗)) Γ— {1}) = (((2nd β€˜(1st β€˜π‘§)) β€œ βˆ…) Γ— {1}))
122 oveq1 7369 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 0 β†’ (𝑗 + 1) = (0 + 1))
123 0p1e1 12282 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 + 1) = 1
124122, 123eqtrdi 2793 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 β†’ (𝑗 + 1) = 1)
125124oveq1d 7377 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 β†’ ((𝑗 + 1)...𝑁) = (1...𝑁))
126125imaeq2d 6018 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 β†’ ((2nd β€˜(1st β€˜π‘§)) β€œ ((𝑗 + 1)...𝑁)) = ((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)))
127126xpeq1d 5667 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 0 β†’ (((2nd β€˜(1st β€˜π‘§)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}) = (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0}))
128121, 127uneq12d 4129 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 0 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 βˆ…) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0})))
129 ima0 6034 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd β€˜(1st β€˜π‘§)) β€œ βˆ…) = βˆ…
130129xpeq1i 5664 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd β€˜(1st β€˜π‘§)) β€œ βˆ…) Γ— {1}) = (βˆ… Γ— {1})
131 0xp 5735 . . . . . . . . . . . . . . . . . . . . . . 23 (βˆ… Γ— {1}) = βˆ…
132130, 131eqtri 2765 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd β€˜(1st β€˜π‘§)) β€œ βˆ…) Γ— {1}) = βˆ…
133132uneq1i 4124 . . . . . . . . . . . . . . . . . . . . 21 ((((2nd β€˜(1st β€˜π‘§)) β€œ βˆ…) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0})) = (βˆ… βˆͺ (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0}))
134 uncom 4118 . . . . . . . . . . . . . . . . . . . . 21 (βˆ… βˆͺ (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0})) = ((((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0}) βˆͺ βˆ…)
135 un0 4355 . . . . . . . . . . . . . . . . . . . . 21 ((((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0}) βˆͺ βˆ…) = (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0})
136133, 134, 1353eqtri 2769 . . . . . . . . . . . . . . . . . . . 20 ((((2nd β€˜(1st β€˜π‘§)) β€œ βˆ…) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0})) = (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0})
137128, 136eqtrdi 2793 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 0 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...𝑁)) Γ— {0}))
138137oveq2d 7378 . . . . . . . . . . . . . . . . . 18 (𝑗 = 0 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...𝑁)) Γ— {0})))
139116, 138csbie 3896 . . . . . . . . . . . . . . . . 17 ⦋0 / 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...𝑁)) Γ— {0}))
140 f1ofo 6796 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd β€˜(1st β€˜π‘§)):(1...𝑁)–1-1-ontoβ†’(1...𝑁) β†’ (2nd β€˜(1st β€˜π‘§)):(1...𝑁)–ontoβ†’(1...𝑁))
14189, 140syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ 𝑆 β†’ (2nd β€˜(1st β€˜π‘§)):(1...𝑁)–ontoβ†’(1...𝑁))
142 foima 6766 . . . . . . . . . . . . . . . . . . . . 21 ((2nd β€˜(1st β€˜π‘§)):(1...𝑁)–ontoβ†’(1...𝑁) β†’ ((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) = (1...𝑁))
143141, 142syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ 𝑆 β†’ ((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) = (1...𝑁))
144143xpeq1d 5667 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ 𝑆 β†’ (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0}) = ((1...𝑁) Γ— {0}))
145144oveq2d 7378 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ 𝑆 β†’ ((1st β€˜(1st β€˜π‘§)) ∘f + (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0})) = ((1st β€˜(1st β€˜π‘§)) ∘f + ((1...𝑁) Γ— {0})))
146 ovexd 7397 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ 𝑆 β†’ (1...𝑁) ∈ V)
14780ffnd 6674 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ 𝑆 β†’ (1st β€˜(1st β€˜π‘§)) Fn (1...𝑁))
148 fnconstg 6735 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ V β†’ ((1...𝑁) Γ— {0}) Fn (1...𝑁))
149116, 148mp1i 13 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ 𝑆 β†’ ((1...𝑁) Γ— {0}) Fn (1...𝑁))
150 eqidd 2738 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) β†’ ((1st β€˜(1st β€˜π‘§))β€˜π‘›) = ((1st β€˜(1st β€˜π‘§))β€˜π‘›))
151116fvconst2 7158 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) β†’ (((1...𝑁) Γ— {0})β€˜π‘›) = 0)
152151adantl 483 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) β†’ (((1...𝑁) Γ— {0})β€˜π‘›) = 0)
15380ffvelcdmda 7040 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) β†’ ((1st β€˜(1st β€˜π‘§))β€˜π‘›) ∈ (0..^𝐾))
154 elfzonn0 13624 . . . . . . . . . . . . . . . . . . . . . 22 (((1st β€˜(1st β€˜π‘§))β€˜π‘›) ∈ (0..^𝐾) β†’ ((1st β€˜(1st β€˜π‘§))β€˜π‘›) ∈ β„•0)
155153, 154syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) β†’ ((1st β€˜(1st β€˜π‘§))β€˜π‘›) ∈ β„•0)
156155nn0cnd 12482 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) β†’ ((1st β€˜(1st β€˜π‘§))β€˜π‘›) ∈ β„‚)
157156addid1d 11362 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) β†’ (((1st β€˜(1st β€˜π‘§))β€˜π‘›) + 0) = ((1st β€˜(1st β€˜π‘§))β€˜π‘›))
158146, 147, 149, 147, 150, 152, 157offveq 7646 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ 𝑆 β†’ ((1st β€˜(1st β€˜π‘§)) ∘f + ((1...𝑁) Γ— {0})) = (1st β€˜(1st β€˜π‘§)))
159145, 158eqtrd 2777 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ 𝑆 β†’ ((1st β€˜(1st β€˜π‘§)) ∘f + (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0})) = (1st β€˜(1st β€˜π‘§)))
160139, 159eqtrid 2789 . . . . . . . . . . . . . . . 16 (𝑧 ∈ 𝑆 β†’ ⦋0 / 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‘§)))
161160ad2antlr 726 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑦 = 0) β†’ ⦋0 / 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‘§)))
162115, 161eqtrd 2777 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑦 = 0) 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‘§)))
163 nnm1nn0 12461 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ β„• β†’ (𝑁 βˆ’ 1) ∈ β„•0)
1641, 163syl 17 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (𝑁 βˆ’ 1) ∈ β„•0)
165 0elfz 13545 . . . . . . . . . . . . . . . 16 ((𝑁 βˆ’ 1) ∈ β„•0 β†’ 0 ∈ (0...(𝑁 βˆ’ 1)))
166164, 165syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 0 ∈ (0...(𝑁 βˆ’ 1)))
167166ad2antrr 725 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ 0 ∈ (0...(𝑁 βˆ’ 1)))
168 fvexd 6862 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (1st β€˜(1st β€˜π‘§)) ∈ V)
169106, 162, 167, 168fvmptd 6960 . . . . . . . . . . . . 13 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘§)))
170105, 169sylan 581 . . . . . . . . . . . 12 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡)) ∧ 𝑧 ∈ 𝑆) β†’ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘§)))
171170an32s 651 . . . . . . . . . . 11 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡)) β†’ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘§)))
172101, 171mpdan 686 . . . . . . . . . 10 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘§)))
173 fveq2 6847 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑇 β†’ (2nd β€˜π‘§) = (2nd β€˜π‘‡))
174173eleq1d 2823 . . . . . . . . . . . . . . 15 (𝑧 = 𝑇 β†’ ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ↔ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))))
175174anbi2d 630 . . . . . . . . . . . . . 14 (𝑧 = 𝑇 β†’ ((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ↔ (πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)))))
176 2fveq3 6852 . . . . . . . . . . . . . . 15 (𝑧 = 𝑇 β†’ (1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)))
177176eqeq2d 2748 . . . . . . . . . . . . . 14 (𝑧 = 𝑇 β†’ ((πΉβ€˜0) = (1st β€˜(1st β€˜π‘§)) ↔ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘‡))))
178175, 177imbi12d 345 . . . . . . . . . . . . 13 (𝑧 = 𝑇 β†’ (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘§))) ↔ ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘‡)))))
179169expcom 415 . . . . . . . . . . . . 13 (𝑧 ∈ 𝑆 β†’ ((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘§))))
180178, 179vtoclga 3537 . . . . . . . . . . . 12 (𝑇 ∈ 𝑆 β†’ ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘‡))))
1817, 180mpcom 38 . . . . . . . . . . 11 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘‡)))
182181adantr 482 . . . . . . . . . 10 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘‡)))
183172, 182eqtr3d 2779 . . . . . . . . 9 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)))
184183adantr 482 . . . . . . . 8 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 β‰  𝑇) β†’ (1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)))
1851ad3antrrr 729 . . . . . . . . 9 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 β‰  𝑇) β†’ 𝑁 ∈ β„•)
1866ad3antrrr 729 . . . . . . . . 9 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 β‰  𝑇) β†’ 𝑇 ∈ 𝑆)
187 simpllr 775 . . . . . . . . 9 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 β‰  𝑇) β†’ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)))
188 simplr 768 . . . . . . . . 9 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 β‰  𝑇) β†’ 𝑧 ∈ 𝑆)
18934adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (1st β€˜π‘‡) ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}))
190 xpopth 7967 . . . . . . . . . . . . . 14 (((1st β€˜π‘§) ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∧ (1st β€˜π‘‡) ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})) β†’ (((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = (2nd β€˜(1st β€˜π‘‡))) ↔ (1st β€˜π‘§) = (1st β€˜π‘‡)))
19176, 189, 190syl2anr 598 . . . . . . . . . . . . 13 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = (2nd β€˜(1st β€˜π‘‡))) ↔ (1st β€˜π‘§) = (1st β€˜π‘‡)))
19232adantr 482 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)))
193 xpopth 7967 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) ∧ 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁))) β†’ (((1st β€˜π‘§) = (1st β€˜π‘‡) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡)) ↔ 𝑧 = 𝑇))
194193biimpd 228 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) ∧ 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁))) β†’ (((1st β€˜π‘§) = (1st β€˜π‘‡) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡)) β†’ 𝑧 = 𝑇))
19574, 192, 194syl2anr 598 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (((1st β€˜π‘§) = (1st β€˜π‘‡) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡)) β†’ 𝑧 = 𝑇))
196101, 195mpan2d 693 . . . . . . . . . . . . 13 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ ((1st β€˜π‘§) = (1st β€˜π‘‡) β†’ 𝑧 = 𝑇))
197191, 196sylbid 239 . . . . . . . . . . . 12 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = (2nd β€˜(1st β€˜π‘‡))) β†’ 𝑧 = 𝑇))
198183, 197mpand 694 . . . . . . . . . . 11 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ ((2nd β€˜(1st β€˜π‘§)) = (2nd β€˜(1st β€˜π‘‡)) β†’ 𝑧 = 𝑇))
199198necon3d 2965 . . . . . . . . . 10 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (𝑧 β‰  𝑇 β†’ (2nd β€˜(1st β€˜π‘§)) β‰  (2nd β€˜(1st β€˜π‘‡))))
200199imp 408 . . . . . . . . 9 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 β‰  𝑇) β†’ (2nd β€˜(1st β€˜π‘§)) β‰  (2nd β€˜(1st β€˜π‘‡)))
201185, 3, 186, 187, 188, 200poimirlem9 36116 . . . . . . . 8 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 β‰  𝑇) β†’ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))))
202101adantr 482 . . . . . . . 8 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 β‰  𝑇) β†’ (2nd β€˜π‘§) = (2nd β€˜π‘‡))
203184, 201, 202jca31 516 . . . . . . 7 ((((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 β‰  𝑇) β†’ (((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡)))
204203ex 414 . . . . . 6 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (𝑧 β‰  𝑇 β†’ (((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡))))
205 simplr 768 . . . . . . . 8 ((((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡)) β†’ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))))
206 elfznn 13477 . . . . . . . . . . . . . 14 ((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) β†’ (2nd β€˜π‘‡) ∈ β„•)
207206nnred 12175 . . . . . . . . . . . . 13 ((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) β†’ (2nd β€˜π‘‡) ∈ ℝ)
208207ltp1d 12092 . . . . . . . . . . . . 13 ((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) β†’ (2nd β€˜π‘‡) < ((2nd β€˜π‘‡) + 1))
209207, 208ltned 11298 . . . . . . . . . . . 12 ((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) β†’ (2nd β€˜π‘‡) β‰  ((2nd β€˜π‘‡) + 1))
210209adantl 483 . . . . . . . . . . 11 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜π‘‡) β‰  ((2nd β€˜π‘‡) + 1))
211 fveq1 6846 . . . . . . . . . . . . 13 ((2nd β€˜(1st β€˜π‘‡)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))) β†’ ((2nd β€˜(1st β€˜π‘‡))β€˜(2nd β€˜π‘‡)) = (((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))β€˜(2nd β€˜π‘‡)))
212 id 22 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd β€˜π‘‡) ∈ ℝ β†’ (2nd β€˜π‘‡) ∈ ℝ)
213 ltp1 12002 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd β€˜π‘‡) ∈ ℝ β†’ (2nd β€˜π‘‡) < ((2nd β€˜π‘‡) + 1))
214212, 213ltned 11298 . . . . . . . . . . . . . . . . . . . . 21 ((2nd β€˜π‘‡) ∈ ℝ β†’ (2nd β€˜π‘‡) β‰  ((2nd β€˜π‘‡) + 1))
215 fvex 6860 . . . . . . . . . . . . . . . . . . . . . 22 (2nd β€˜π‘‡) ∈ V
216 ovex 7395 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd β€˜π‘‡) + 1) ∈ V
217215, 216, 216, 215fpr 7105 . . . . . . . . . . . . . . . . . . . . 21 ((2nd β€˜π‘‡) β‰  ((2nd β€˜π‘‡) + 1) β†’ {⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩}:{(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}⟢{((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)})
218214, 217syl 17 . . . . . . . . . . . . . . . . . . . 20 ((2nd β€˜π‘‡) ∈ ℝ β†’ {⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩}:{(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}⟢{((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)})
219 f1oi 6827 . . . . . . . . . . . . . . . . . . . . 21 ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})):((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})–1-1-ontoβ†’((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})
220 f1of 6789 . . . . . . . . . . . . . . . . . . . . 21 (( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})):((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})–1-1-ontoβ†’((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}) β†’ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})):((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})⟢((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))
221219, 220ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})):((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})⟢((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})
222 disjdif 4436 . . . . . . . . . . . . . . . . . . . . 21 ({(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} ∩ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})) = βˆ…
223 fun 6709 . . . . . . . . . . . . . . . . . . . . 21 ((({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩}:{(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}⟢{((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)} ∧ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})):((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})⟢((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})) ∧ ({(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} ∩ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})) = βˆ…) β†’ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))):({(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} βˆͺ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))⟢({((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)} βˆͺ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))
224222, 223mpan2 690 . . . . . . . . . . . . . . . . . . . 20 (({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩}:{(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}⟢{((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)} ∧ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})):((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})⟢((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})) β†’ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))):({(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} βˆͺ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))⟢({((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)} βˆͺ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))
225218, 221, 224sylancl 587 . . . . . . . . . . . . . . . . . . 19 ((2nd β€˜π‘‡) ∈ ℝ β†’ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))):({(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} βˆͺ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))⟢({((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)} βˆͺ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))
226215prid1 4728 . . . . . . . . . . . . . . . . . . . 20 (2nd β€˜π‘‡) ∈ {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}
227 elun1 4141 . . . . . . . . . . . . . . . . . . . 20 ((2nd β€˜π‘‡) ∈ {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} β†’ (2nd β€˜π‘‡) ∈ ({(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} βˆͺ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))
228226, 227ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (2nd β€˜π‘‡) ∈ ({(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} βˆͺ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))
229 fvco3 6945 . . . . . . . . . . . . . . . . . . 19 ((({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))):({(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} βˆͺ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))⟢({((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)} βˆͺ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})) ∧ (2nd β€˜π‘‡) ∈ ({(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} βˆͺ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))) β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))β€˜(2nd β€˜π‘‡)) = ((2nd β€˜(1st β€˜π‘‡))β€˜(({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))β€˜(2nd β€˜π‘‡))))
230225, 228, 229sylancl 587 . . . . . . . . . . . . . . . . . 18 ((2nd β€˜π‘‡) ∈ ℝ β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))β€˜(2nd β€˜π‘‡)) = ((2nd β€˜(1st β€˜π‘‡))β€˜(({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))β€˜(2nd β€˜π‘‡))))
231218ffnd 6674 . . . . . . . . . . . . . . . . . . . . 21 ((2nd β€˜π‘‡) ∈ ℝ β†’ {⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} Fn {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})
232 fnresi 6635 . . . . . . . . . . . . . . . . . . . . . 22 ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})) Fn ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})
233222, 226pm3.2i 472 . . . . . . . . . . . . . . . . . . . . . 22 (({(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} ∩ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})) = βˆ… ∧ (2nd β€˜π‘‡) ∈ {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})
234 fvun1 6937 . . . . . . . . . . . . . . . . . . . . . 22 (({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} Fn {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} ∧ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})) Fn ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}) ∧ (({(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} ∩ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})) = βˆ… ∧ (2nd β€˜π‘‡) ∈ {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})) β†’ (({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))β€˜(2nd β€˜π‘‡)) = ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩}β€˜(2nd β€˜π‘‡)))
235232, 233, 234mp3an23 1454 . . . . . . . . . . . . . . . . . . . . 21 ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} Fn {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} β†’ (({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))β€˜(2nd β€˜π‘‡)) = ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩}β€˜(2nd β€˜π‘‡)))
236231, 235syl 17 . . . . . . . . . . . . . . . . . . . 20 ((2nd β€˜π‘‡) ∈ ℝ β†’ (({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))β€˜(2nd β€˜π‘‡)) = ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩}β€˜(2nd β€˜π‘‡)))
237215, 216fvpr1 7144 . . . . . . . . . . . . . . . . . . . . 21 ((2nd β€˜π‘‡) β‰  ((2nd β€˜π‘‡) + 1) β†’ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩}β€˜(2nd β€˜π‘‡)) = ((2nd β€˜π‘‡) + 1))
238214, 237syl 17 . . . . . . . . . . . . . . . . . . . 20 ((2nd β€˜π‘‡) ∈ ℝ β†’ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩}β€˜(2nd β€˜π‘‡)) = ((2nd β€˜π‘‡) + 1))
239236, 238eqtrd 2777 . . . . . . . . . . . . . . . . . . 19 ((2nd β€˜π‘‡) ∈ ℝ β†’ (({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))β€˜(2nd β€˜π‘‡)) = ((2nd β€˜π‘‡) + 1))
240239fveq2d 6851 . . . . . . . . . . . . . . . . . 18 ((2nd β€˜π‘‡) ∈ ℝ β†’ ((2nd β€˜(1st β€˜π‘‡))β€˜(({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))β€˜(2nd β€˜π‘‡))) = ((2nd β€˜(1st β€˜π‘‡))β€˜((2nd β€˜π‘‡) + 1)))
241230, 240eqtrd 2777 . . . . . . . . . . . . . . . . 17 ((2nd β€˜π‘‡) ∈ ℝ β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))β€˜(2nd β€˜π‘‡)) = ((2nd β€˜(1st β€˜π‘‡))β€˜((2nd β€˜π‘‡) + 1)))
242207, 241syl 17 . . . . . . . . . . . . . . . 16 ((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) β†’ (((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))β€˜(2nd β€˜π‘‡)) = ((2nd β€˜(1st β€˜π‘‡))β€˜((2nd β€˜π‘‡) + 1)))
243242eqeq2d 2748 . . . . . . . . . . . . . . 15 ((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) β†’ (((2nd β€˜(1st β€˜π‘‡))β€˜(2nd β€˜π‘‡)) = (((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))β€˜(2nd β€˜π‘‡)) ↔ ((2nd β€˜(1st β€˜π‘‡))β€˜(2nd β€˜π‘‡)) = ((2nd β€˜(1st β€˜π‘‡))β€˜((2nd β€˜π‘‡) + 1))))
244243adantl 483 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡))β€˜(2nd β€˜π‘‡)) = (((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))β€˜(2nd β€˜π‘‡)) ↔ ((2nd β€˜(1st β€˜π‘‡))β€˜(2nd β€˜π‘‡)) = ((2nd β€˜(1st β€˜π‘‡))β€˜((2nd β€˜π‘‡) + 1))))
245 f1of1 6788 . . . . . . . . . . . . . . . . 17 ((2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1-ontoβ†’(1...𝑁) β†’ (2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1β†’(1...𝑁))
24649, 245syl 17 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1β†’(1...𝑁))
247246adantr 482 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1β†’(1...𝑁))
2481nncnd 12176 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ 𝑁 ∈ β„‚)
249 npcan1 11587 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„‚ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
250248, 249syl 17 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
251164nn0zd 12532 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ (𝑁 βˆ’ 1) ∈ β„€)
252 uzid 12785 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 βˆ’ 1) ∈ β„€ β†’ (𝑁 βˆ’ 1) ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
253251, 252syl 17 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ (𝑁 βˆ’ 1) ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
254 peano2uz 12833 . . . . . . . . . . . . . . . . . . 19 ((𝑁 βˆ’ 1) ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)) β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
255253, 254syl 17 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
256250, 255eqeltrrd 2839 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
257 fzss2 13488 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)) β†’ (1...(𝑁 βˆ’ 1)) βŠ† (1...𝑁))
258256, 257syl 17 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (1...(𝑁 βˆ’ 1)) βŠ† (1...𝑁))
259258sselda 3949 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜π‘‡) ∈ (1...𝑁))
260 fzp1elp1 13501 . . . . . . . . . . . . . . . . 17 ((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) β†’ ((2nd β€˜π‘‡) + 1) ∈ (1...((𝑁 βˆ’ 1) + 1)))
261260adantl 483 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜π‘‡) + 1) ∈ (1...((𝑁 βˆ’ 1) + 1)))
262250oveq2d 7378 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (1...((𝑁 βˆ’ 1) + 1)) = (1...𝑁))
263262adantr 482 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (1...((𝑁 βˆ’ 1) + 1)) = (1...𝑁))
264261, 263eleqtrd 2840 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜π‘‡) + 1) ∈ (1...𝑁))
265 f1veqaeq 7209 . . . . . . . . . . . . . . 15 (((2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1β†’(1...𝑁) ∧ ((2nd β€˜π‘‡) ∈ (1...𝑁) ∧ ((2nd β€˜π‘‡) + 1) ∈ (1...𝑁))) β†’ (((2nd β€˜(1st β€˜π‘‡))β€˜(2nd β€˜π‘‡)) = ((2nd β€˜(1st β€˜π‘‡))β€˜((2nd β€˜π‘‡) + 1)) β†’ (2nd β€˜π‘‡) = ((2nd β€˜π‘‡) + 1)))
266247, 259, 264, 265syl12anc 836 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡))β€˜(2nd β€˜π‘‡)) = ((2nd β€˜(1st β€˜π‘‡))β€˜((2nd β€˜π‘‡) + 1)) β†’ (2nd β€˜π‘‡) = ((2nd β€˜π‘‡) + 1)))
267244, 266sylbid 239 . . . . . . . . . . . . 13 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (((2nd β€˜(1st β€˜π‘‡))β€˜(2nd β€˜π‘‡)) = (((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))β€˜(2nd β€˜π‘‡)) β†’ (2nd β€˜π‘‡) = ((2nd β€˜π‘‡) + 1)))
268211, 267syl5 34 . . . . . . . . . . . 12 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘‡)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))) β†’ (2nd β€˜π‘‡) = ((2nd β€˜π‘‡) + 1)))
269268necon3d 2965 . . . . . . . . . . 11 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜π‘‡) β‰  ((2nd β€˜π‘‡) + 1) β†’ (2nd β€˜(1st β€˜π‘‡)) β‰  ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))))
270210, 269mpd 15 . . . . . . . . . 10 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜(1st β€˜π‘‡)) β‰  ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))))
271 2fveq3 6852 . . . . . . . . . . 11 (𝑧 = 𝑇 β†’ (2nd β€˜(1st β€˜π‘§)) = (2nd β€˜(1st β€˜π‘‡)))
272271neeq1d 3004 . . . . . . . . . 10 (𝑧 = 𝑇 β†’ ((2nd β€˜(1st β€˜π‘§)) β‰  ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))) ↔ (2nd β€˜(1st β€˜π‘‡)) β‰  ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))))
273270, 272syl5ibrcom 247 . . . . . . . . 9 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (𝑧 = 𝑇 β†’ (2nd β€˜(1st β€˜π‘§)) β‰  ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))))
274273necon2d 2967 . . . . . . . 8 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))) β†’ 𝑧 β‰  𝑇))
275205, 274syl5 34 . . . . . . 7 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ ((((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡)) β†’ 𝑧 β‰  𝑇))
276275adantr 482 . . . . . 6 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ ((((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡)) β†’ 𝑧 β‰  𝑇))
277204, 276impbid 211 . . . . 5 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (𝑧 β‰  𝑇 ↔ (((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡))))
278 eqop 7968 . . . . . . . 8 (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) β†’ (𝑧 = ⟨⟨(1st β€˜(1st β€˜π‘‡)), ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))⟩, (2nd β€˜π‘‡)⟩ ↔ ((1st β€˜π‘§) = ⟨(1st β€˜(1st β€˜π‘‡)), ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))⟩ ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡))))
279 eqop 7968 . . . . . . . . . 10 ((1st β€˜π‘§) ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) β†’ ((1st β€˜π‘§) = ⟨(1st β€˜(1st β€˜π‘‡)), ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))⟩ ↔ ((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))))))
28075, 279syl 17 . . . . . . . . 9 (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) β†’ ((1st β€˜π‘§) = ⟨(1st β€˜(1st β€˜π‘‡)), ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))⟩ ↔ ((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})))))))
281280anbi1d 631 . . . . . . . 8 (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) β†’ (((1st β€˜π‘§) = ⟨(1st β€˜(1st β€˜π‘‡)), ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))⟩ ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡)) ↔ (((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡))))
282278, 281bitrd 279 . . . . . . 7 (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) β†’ (𝑧 = ⟨⟨(1st β€˜(1st β€˜π‘‡)), ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))⟩, (2nd β€˜π‘‡)⟩ ↔ (((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡))))
28374, 282syl 17 . . . . . 6 (𝑧 ∈ 𝑆 β†’ (𝑧 = ⟨⟨(1st β€˜(1st β€˜π‘‡)), ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))⟩, (2nd β€˜π‘‡)⟩ ↔ (((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡))))
284283adantl 483 . . . . 5 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (𝑧 = ⟨⟨(1st β€˜(1st β€˜π‘‡)), ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))⟩, (2nd β€˜π‘‡)⟩ ↔ (((1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)) ∧ (2nd β€˜(1st β€˜π‘§)) = ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡))))
285277, 284bitr4d 282 . . . 4 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (𝑧 β‰  𝑇 ↔ 𝑧 = ⟨⟨(1st β€˜(1st β€˜π‘‡)), ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))⟩, (2nd β€˜π‘‡)⟩))
286285ralrimiva 3144 . . 3 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ βˆ€π‘§ ∈ 𝑆 (𝑧 β‰  𝑇 ↔ 𝑧 = ⟨⟨(1st β€˜(1st β€˜π‘‡)), ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))⟩, (2nd β€˜π‘‡)⟩))
287 reu6i 3691 . . 3 ((⟨⟨(1st β€˜(1st β€˜π‘‡)), ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))⟩, (2nd β€˜π‘‡)⟩ ∈ 𝑆 ∧ βˆ€π‘§ ∈ 𝑆 (𝑧 β‰  𝑇 ↔ 𝑧 = ⟨⟨(1st β€˜(1st β€˜π‘‡)), ((2nd β€˜(1st β€˜π‘‡)) ∘ ({⟨(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)⟩, ⟨((2nd β€˜π‘‡) + 1), (2nd β€˜π‘‡)⟩} βˆͺ ( I β†Ύ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}))))⟩, (2nd β€˜π‘‡)⟩)) β†’ βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
2889, 286, 287syl2anc 585 . 2 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
289 xp2nd 7959 . . . . . . 7 (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) β†’ (2nd β€˜π‘‡) ∈ (0...𝑁))
29032, 289syl 17 . . . . . 6 (πœ‘ β†’ (2nd β€˜π‘‡) ∈ (0...𝑁))
291290biantrurd 534 . . . . 5 (πœ‘ β†’ (Β¬ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) ↔ ((2nd β€˜π‘‡) ∈ (0...𝑁) ∧ Β¬ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)))))
2921nnnn0d 12480 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑁 ∈ β„•0)
293 nn0uz 12812 . . . . . . . . . . . 12 β„•0 = (β„€β‰₯β€˜0)
294292, 293eleqtrdi 2848 . . . . . . . . . . 11 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜0))
295 fzpred 13496 . . . . . . . . . . 11 (𝑁 ∈ (β„€β‰₯β€˜0) β†’ (0...𝑁) = ({0} βˆͺ ((0 + 1)...𝑁)))
296294, 295syl 17 . . . . . . . . . 10 (πœ‘ β†’ (0...𝑁) = ({0} βˆͺ ((0 + 1)...𝑁)))
297123oveq1i 7372 . . . . . . . . . . 11 ((0 + 1)...𝑁) = (1...𝑁)
298297uneq2i 4125 . . . . . . . . . 10 ({0} βˆͺ ((0 + 1)...𝑁)) = ({0} βˆͺ (1...𝑁))
299296, 298eqtrdi 2793 . . . . . . . . 9 (πœ‘ β†’ (0...𝑁) = ({0} βˆͺ (1...𝑁)))
300299difeq1d 4086 . . . . . . . 8 (πœ‘ β†’ ((0...𝑁) βˆ– (1...(𝑁 βˆ’ 1))) = (({0} βˆͺ (1...𝑁)) βˆ– (1...(𝑁 βˆ’ 1))))
301 difundir 4245 . . . . . . . . . 10 (({0} βˆͺ (1...𝑁)) βˆ– (1...(𝑁 βˆ’ 1))) = (({0} βˆ– (1...(𝑁 βˆ’ 1))) βˆͺ ((1...𝑁) βˆ– (1...(𝑁 βˆ’ 1))))
302 0lt1 11684 . . . . . . . . . . . . . 14 0 < 1
303 0re 11164 . . . . . . . . . . . . . . 15 0 ∈ ℝ
304 1re 11162 . . . . . . . . . . . . . . 15 1 ∈ ℝ
305303, 304ltnlei 11283 . . . . . . . . . . . . . 14 (0 < 1 ↔ Β¬ 1 ≀ 0)
306302, 305mpbi 229 . . . . . . . . . . . . 13 Β¬ 1 ≀ 0
307 elfzle1 13451 . . . . . . . . . . . . 13 (0 ∈ (1...(𝑁 βˆ’ 1)) β†’ 1 ≀ 0)
308306, 307mto 196 . . . . . . . . . . . 12 Β¬ 0 ∈ (1...(𝑁 βˆ’ 1))
309 incom 4166 . . . . . . . . . . . . . 14 ((1...(𝑁 βˆ’ 1)) ∩ {0}) = ({0} ∩ (1...(𝑁 βˆ’ 1)))
310309eqeq1i 2742 . . . . . . . . . . . . 13 (((1...(𝑁 βˆ’ 1)) ∩ {0}) = βˆ… ↔ ({0} ∩ (1...(𝑁 βˆ’ 1))) = βˆ…)
311 disjsn 4677 . . . . . . . . . . . . 13 (((1...(𝑁 βˆ’ 1)) ∩ {0}) = βˆ… ↔ Β¬ 0 ∈ (1...(𝑁 βˆ’ 1)))
312 disj3 4418 . . . . . . . . . . . . 13 (({0} ∩ (1...(𝑁 βˆ’ 1))) = βˆ… ↔ {0} = ({0} βˆ– (1...(𝑁 βˆ’ 1))))
313310, 311, 3123bitr3i 301 . . . . . . . . . . . 12 (Β¬ 0 ∈ (1...(𝑁 βˆ’ 1)) ↔ {0} = ({0} βˆ– (1...(𝑁 βˆ’ 1))))
314308, 313mpbi 229 . . . . . . . . . . 11 {0} = ({0} βˆ– (1...(𝑁 βˆ’ 1)))
315314uneq1i 4124 . . . . . . . . . 10 ({0} βˆͺ ((1...𝑁) βˆ– (1...(𝑁 βˆ’ 1)))) = (({0} βˆ– (1...(𝑁 βˆ’ 1))) βˆͺ ((1...𝑁) βˆ– (1...(𝑁 βˆ’ 1))))
316301, 315eqtr4i 2768 . . . . . . . . 9 (({0} βˆͺ (1...𝑁)) βˆ– (1...(𝑁 βˆ’ 1))) = ({0} βˆͺ ((1...𝑁) βˆ– (1...(𝑁 βˆ’ 1))))
317 difundir 4245 . . . . . . . . . . . 12 (((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}) βˆ– (1...(𝑁 βˆ’ 1))) = (((1...(𝑁 βˆ’ 1)) βˆ– (1...(𝑁 βˆ’ 1))) βˆͺ ({𝑁} βˆ– (1...(𝑁 βˆ’ 1))))
318 difid 4335 . . . . . . . . . . . . 13 ((1...(𝑁 βˆ’ 1)) βˆ– (1...(𝑁 βˆ’ 1))) = βˆ…
319318uneq1i 4124 . . . . . . . . . . . 12 (((1...(𝑁 βˆ’ 1)) βˆ– (1...(𝑁 βˆ’ 1))) βˆͺ ({𝑁} βˆ– (1...(𝑁 βˆ’ 1)))) = (βˆ… βˆͺ ({𝑁} βˆ– (1...(𝑁 βˆ’ 1))))
320 uncom 4118 . . . . . . . . . . . . 13 (βˆ… βˆͺ ({𝑁} βˆ– (1...(𝑁 βˆ’ 1)))) = (({𝑁} βˆ– (1...(𝑁 βˆ’ 1))) βˆͺ βˆ…)
321 un0 4355 . . . . . . . . . . . . 13 (({𝑁} βˆ– (1...(𝑁 βˆ’ 1))) βˆͺ βˆ…) = ({𝑁} βˆ– (1...(𝑁 βˆ’ 1)))
322320, 321eqtri 2765 . . . . . . . . . . . 12 (βˆ… βˆͺ ({𝑁} βˆ– (1...(𝑁 βˆ’ 1)))) = ({𝑁} βˆ– (1...(𝑁 βˆ’ 1)))
323317, 319, 3223eqtri 2769 . . . . . . . . . . 11 (((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}) βˆ– (1...(𝑁 βˆ’ 1))) = ({𝑁} βˆ– (1...(𝑁 βˆ’ 1)))
324 nnuz 12813 . . . . . . . . . . . . . . . 16 β„• = (β„€β‰₯β€˜1)
3251, 324eleqtrdi 2848 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜1))
326250, 325eqeltrd 2838 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜1))
327 fzsplit2 13473 . . . . . . . . . . . . . 14 ((((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)))
328326, 256, 327syl2anc 585 . . . . . . . . . . . . 13 (πœ‘ β†’ (1...𝑁) = ((1...(𝑁 βˆ’ 1)) βˆͺ (((𝑁 βˆ’ 1) + 1)...𝑁)))
329250oveq1d 7377 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (((𝑁 βˆ’ 1) + 1)...𝑁) = (𝑁...𝑁))
3301nnzd 12533 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝑁 ∈ β„€)
331 fzsn 13490 . . . . . . . . . . . . . . . 16 (𝑁 ∈ β„€ β†’ (𝑁...𝑁) = {𝑁})
332330, 331syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (𝑁...𝑁) = {𝑁})
333329, 332eqtrd 2777 . . . . . . . . . . . . . 14 (πœ‘ β†’ (((𝑁 βˆ’ 1) + 1)...𝑁) = {𝑁})
334333uneq2d 4128 . . . . . . . . . . . . 13 (πœ‘ β†’ ((1...(𝑁 βˆ’ 1)) βˆͺ (((𝑁 βˆ’ 1) + 1)...𝑁)) = ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}))
335328, 334eqtrd 2777 . . . . . . . . . . . 12 (πœ‘ β†’ (1...𝑁) = ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}))
336335difeq1d 4086 . . . . . . . . . . 11 (πœ‘ β†’ ((1...𝑁) βˆ– (1...(𝑁 βˆ’ 1))) = (((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}) βˆ– (1...(𝑁 βˆ’ 1))))
3371nnred 12175 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝑁 ∈ ℝ)
338337ltm1d 12094 . . . . . . . . . . . . . 14 (πœ‘ β†’ (𝑁 βˆ’ 1) < 𝑁)
339164nn0red 12481 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (𝑁 βˆ’ 1) ∈ ℝ)
340339, 337ltnled 11309 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((𝑁 βˆ’ 1) < 𝑁 ↔ Β¬ 𝑁 ≀ (𝑁 βˆ’ 1)))
341338, 340mpbid 231 . . . . . . . . . . . . 13 (πœ‘ β†’ Β¬ 𝑁 ≀ (𝑁 βˆ’ 1))
342 elfzle2 13452 . . . . . . . . . . . . 13 (𝑁 ∈ (1...(𝑁 βˆ’ 1)) β†’ 𝑁 ≀ (𝑁 βˆ’ 1))
343341, 342nsyl 140 . . . . . . . . . . . 12 (πœ‘ β†’ Β¬ 𝑁 ∈ (1...(𝑁 βˆ’ 1)))
344 incom 4166 . . . . . . . . . . . . . 14 ((1...(𝑁 βˆ’ 1)) ∩ {𝑁}) = ({𝑁} ∩ (1...(𝑁 βˆ’ 1)))
345344eqeq1i 2742 . . . . . . . . . . . . 13 (((1...(𝑁 βˆ’ 1)) ∩ {𝑁}) = βˆ… ↔ ({𝑁} ∩ (1...(𝑁 βˆ’ 1))) = βˆ…)
346 disjsn 4677 . . . . . . . . . . . . 13 (((1...(𝑁 βˆ’ 1)) ∩ {𝑁}) = βˆ… ↔ Β¬ 𝑁 ∈ (1...(𝑁 βˆ’ 1)))
347 disj3 4418 . . . . . . . . . . . . 13 (({𝑁} ∩ (1...(𝑁 βˆ’ 1))) = βˆ… ↔ {𝑁} = ({𝑁} βˆ– (1...(𝑁 βˆ’ 1))))
348345, 346, 3473bitr3i 301 . . . . . . . . . . . 12 (Β¬ 𝑁 ∈ (1...(𝑁 βˆ’ 1)) ↔ {𝑁} = ({𝑁} βˆ– (1...(𝑁 βˆ’ 1))))
349343, 348sylib 217 . . . . . . . . . . 11 (πœ‘ β†’ {𝑁} = ({𝑁} βˆ– (1...(𝑁 βˆ’ 1))))
350323, 336, 3493eqtr4a 2803 . . . . . . . . . 10 (πœ‘ β†’ ((1...𝑁) βˆ– (1...(𝑁 βˆ’ 1))) = {𝑁})
351350uneq2d 4128 . . . . . . . . 9 (πœ‘ β†’ ({0} βˆͺ ((1...𝑁) βˆ– (1...(𝑁 βˆ’ 1)))) = ({0} βˆͺ {𝑁}))
352316, 351eqtrid 2789 . . . . . . . 8 (πœ‘ β†’ (({0} βˆͺ (1...𝑁)) βˆ– (1...(𝑁 βˆ’ 1))) = ({0} βˆͺ {𝑁}))
353300, 352eqtrd 2777 . . . . . . 7 (πœ‘ β†’ ((0...𝑁) βˆ– (1...(𝑁 βˆ’ 1))) = ({0} βˆͺ {𝑁}))
354353eleq2d 2824 . . . . . 6 (πœ‘ β†’ ((2nd β€˜π‘‡) ∈ ((0...𝑁) βˆ– (1...(𝑁 βˆ’ 1))) ↔ (2nd β€˜π‘‡) ∈ ({0} βˆͺ {𝑁})))
355 eldif 3925 . . . . . 6 ((2nd β€˜π‘‡) ∈ ((0...𝑁) βˆ– (1...(𝑁 βˆ’ 1))) ↔ ((2nd β€˜π‘‡) ∈ (0...𝑁) ∧ Β¬ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))))
356 elun 4113 . . . . . . 7 ((2nd β€˜π‘‡) ∈ ({0} βˆͺ {𝑁}) ↔ ((2nd β€˜π‘‡) ∈ {0} ∨ (2nd β€˜π‘‡) ∈ {𝑁}))
357215elsn 4606 . . . . . . . 8 ((2nd β€˜π‘‡) ∈ {0} ↔ (2nd β€˜π‘‡) = 0)
358215elsn 4606 . . . . . . . 8 ((2nd β€˜π‘‡) ∈ {𝑁} ↔ (2nd β€˜π‘‡) = 𝑁)
359357, 358orbi12i 914 . . . . . . 7 (((2nd β€˜π‘‡) ∈ {0} ∨ (2nd β€˜π‘‡) ∈ {𝑁}) ↔ ((2nd β€˜π‘‡) = 0 ∨ (2nd β€˜π‘‡) = 𝑁))
360356, 359bitri 275 . . . . . 6 ((2nd β€˜π‘‡) ∈ ({0} βˆͺ {𝑁}) ↔ ((2nd β€˜π‘‡) = 0 ∨ (2nd β€˜π‘‡) = 𝑁))
361354, 355, 3603bitr3g 313 . . . . 5 (πœ‘ β†’ (((2nd β€˜π‘‡) ∈ (0...𝑁) ∧ Β¬ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ↔ ((2nd β€˜π‘‡) = 0 ∨ (2nd β€˜π‘‡) = 𝑁)))
362291, 361bitrd 279 . . . 4 (πœ‘ β†’ (Β¬ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) ↔ ((2nd β€˜π‘‡) = 0 ∨ (2nd β€˜π‘‡) = 𝑁)))
363362biimpa 478 . . 3 ((πœ‘ ∧ Β¬ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜π‘‡) = 0 ∨ (2nd β€˜π‘‡) = 𝑁))
3641adantr 482 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 0) β†’ 𝑁 ∈ β„•)
3654adantr 482 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 0) β†’ 𝐹:(0...(𝑁 βˆ’ 1))⟢((0...𝐾) ↑m (1...𝑁)))
3666adantr 482 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 0) β†’ 𝑇 ∈ 𝑆)
367 poimirlem22.4 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (1...𝑁)) β†’ βˆƒπ‘ ∈ ran 𝐹(π‘β€˜π‘›) β‰  𝐾)
368367adantlr 714 . . . . 5 (((πœ‘ ∧ (2nd β€˜π‘‡) = 0) ∧ 𝑛 ∈ (1...𝑁)) β†’ βˆƒπ‘ ∈ ran 𝐹(π‘β€˜π‘›) β‰  𝐾)
369 simpr 486 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 0) β†’ (2nd β€˜π‘‡) = 0)
370364, 3, 365, 366, 368, 369poimirlem18 36125 . . . 4 ((πœ‘ ∧ (2nd β€˜π‘‡) = 0) β†’ βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
3711adantr 482 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 𝑁) β†’ 𝑁 ∈ β„•)
3724adantr 482 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 𝑁) β†’ 𝐹:(0...(𝑁 βˆ’ 1))⟢((0...𝐾) ↑m (1...𝑁)))
3736adantr 482 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 𝑁) β†’ 𝑇 ∈ 𝑆)
374 poimirlem22.3 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (1...𝑁)) β†’ βˆƒπ‘ ∈ ran 𝐹(π‘β€˜π‘›) β‰  0)
375374adantlr 714 . . . . 5 (((πœ‘ ∧ (2nd β€˜π‘‡) = 𝑁) ∧ 𝑛 ∈ (1...𝑁)) β†’ βˆƒπ‘ ∈ ran 𝐹(π‘β€˜π‘›) β‰  0)
376 simpr 486 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 𝑁) β†’ (2nd β€˜π‘‡) = 𝑁)
377371, 3, 372, 373, 375, 376poimirlem21 36128 . . . 4 ((πœ‘ ∧ (2nd β€˜π‘‡) = 𝑁) β†’ βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
378370, 377jaodan 957 . . 3 ((πœ‘ ∧ ((2nd β€˜π‘‡) = 0 ∨ (2nd β€˜π‘‡) = 𝑁)) β†’ βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
379363, 378syldan 592 . 2 ((πœ‘ ∧ Β¬ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
380288, 379pm2.61dan 812 1 (πœ‘ β†’ βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  {cab 2714   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  βˆƒ!wreu 3354  βˆƒ*wrmo 3355  {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  {cpr 4593  βŸ¨cop 4597   class class class wbr 5110   ↦ cmpt 5193   I cid 5535   Γ— cxp 5636  ran crn 5639   β†Ύ cres 5640   β€œ cima 5641   ∘ ccom 5642   Fn wfn 6496  βŸΆwf 6497  β€“1-1β†’wf1 6498  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-rmo 3356  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-tp 4596  df-op 4598  df-uni 4871  df-int 4913  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-1o 8417  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-seq 13914  df-fac 14181  df-bc 14210  df-hash 14238
This theorem is referenced by:  poimirlem27  36134
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