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Theorem poimirlem22 37037
Description: Lemma for poimir 37048, 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 480 . . . 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 480 . . . 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 480 . . . 4 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ 𝑇 ∈ 𝑆)
8 simpr 484 . . . 4 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)))
92, 3, 5, 7, 8poimirlem15 37030 . . 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 6891 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑑 = 𝑇 β†’ (2nd β€˜π‘‘) = (2nd β€˜π‘‡))
1110breq2d 5154 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑇 β†’ (𝑦 < (2nd β€˜π‘‘) ↔ 𝑦 < (2nd β€˜π‘‡)))
1211ifbid 4547 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 = 𝑇 β†’ if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)))
1312csbeq1d 3893 . . . . . . . . . . . . . . . . . . . . 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 6896 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑇 β†’ (1st β€˜(1st β€˜π‘‘)) = (1st β€˜(1st β€˜π‘‡)))
15 2fveq3 6896 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = 𝑇 β†’ (2nd β€˜(1st β€˜π‘‘)) = (2nd β€˜(1st β€˜π‘‡)))
1615imaeq1d 6056 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = 𝑇 β†’ ((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)))
1716xpeq1d 5701 . . . . . . . . . . . . . . . . . . . . . . . 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 6056 . . . . . . . . . . . . . . . . . . . . . . . . 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 5701 . . . . . . . . . . . . . . . . . . . . . . . 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 4160 . . . . . . . . . . . . . . . . . . . . . . 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 7432 . . . . . . . . . . . . . . . . . . . . . 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 3896 . . . . . . . . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . . . . . . . . . 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 5244 . . . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . . 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 3683 . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 3674 . . . . . . . . . . . . . . . . . . . . 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 2846 . . . . . . . . . . . . . . . . . . . 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 8017 . . . . . . . . . . . . . . . . . . 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 8017 . . . . . . . . . . . . . . . . . 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 8857 . . . . . . . . . . . . . . . . 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 13650 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝐾) β†’ 𝑛 ∈ β„€)
4039ssriv 3982 . . . . . . . . . . . . . . . 16 (0..^𝐾) βŠ† β„€
41 fss 6733 . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)βŸΆβ„€)
4342adantr 480 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)βŸΆβ„€)
44 xp2nd 8018 . . . . . . . . . . . . . . . . 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 6904 . . . . . . . . . . . . . . . . 17 (2nd β€˜(1st β€˜π‘‡)) ∈ V
47 f1oeq1 6821 . . . . . . . . . . . . . . . . 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 3665 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 37016 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑑 = 𝑧 β†’ (2nd β€˜π‘‘) = (2nd β€˜π‘§))
5554breq2d 5154 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑧 β†’ (𝑦 < (2nd β€˜π‘‘) ↔ 𝑦 < (2nd β€˜π‘§)))
5655ifbid 4547 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 = 𝑧 β†’ if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd β€˜π‘§), 𝑦, (𝑦 + 1)))
5756csbeq1d 3893 . . . . . . . . . . . . . . . . . . . . 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 6896 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑧 β†’ (1st β€˜(1st β€˜π‘‘)) = (1st β€˜(1st β€˜π‘§)))
59 2fveq3 6896 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = 𝑧 β†’ (2nd β€˜(1st β€˜π‘‘)) = (2nd β€˜(1st β€˜π‘§)))
6059imaeq1d 6056 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = 𝑧 β†’ ((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) = ((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑗)))
6160xpeq1d 5701 . . . . . . . . . . . . . . . . . . . . . . . 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 6056 . . . . . . . . . . . . . . . . . . . . . . . . 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 5701 . . . . . . . . . . . . . . . . . . . . . . . 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 4160 . . . . . . . . . . . . . . . . . . . . . . 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 7432 . . . . . . . . . . . . . . . . . . . . . 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 3896 . . . . . . . . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . . . . . . . . . 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 5244 . . . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . . 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 3683 . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . 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 3674 . . . . . . . . . . . . . . . . . . . . 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 2846 . . . . . . . . . . . . . . . . . . . 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 8017 . . . . . . . . . . . . . . . . . . . 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 8017 . . . . . . . . . . . . . . . . . . 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 8857 . . . . . . . . . . . . . . . . . 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 6733 . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . 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 8018 . . . . . . . . . . . . . . . . . 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 6904 . . . . . . . . . . . . . . . . . 18 (2nd β€˜(1st β€˜π‘§)) ∈ V
87 f1oeq1 6821 . . . . . . . . . . . . . . . . . 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 3665 . . . . . . . . . . . . . . . . 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 8018 . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (2nd β€˜π‘§) ∈ (0...𝑁))
95 eldifsn 4786 . . . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . . 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 37017 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . 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 2953 . . . . . . . . . . . 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 2816 . . . . . . . . . . . . . . . 16 ((2nd β€˜π‘§) = (2nd β€˜π‘‡) β†’ ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ↔ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))))
103102biimparc 479 . . . . . . . . . . . . . . 15 (((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘‡)) β†’ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)))
104103anim2i 616 . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 5145 . . . . . . . . . . . . . . . . . 18 (𝑦 = 0 β†’ (𝑦 < (2nd β€˜π‘§) ↔ 0 < (2nd β€˜π‘§)))
108 id 22 . . . . . . . . . . . . . . . . . 18 (𝑦 = 0 β†’ 𝑦 = 0)
109107, 108ifbieq1d 4548 . . . . . . . . . . . . . . . . 17 (𝑦 = 0 β†’ if(𝑦 < (2nd β€˜π‘§), 𝑦, (𝑦 + 1)) = if(0 < (2nd β€˜π‘§), 0, (𝑦 + 1)))
110 elfznn 13548 . . . . . . . . . . . . . . . . . . . 20 ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) β†’ (2nd β€˜π‘§) ∈ β„•)
111110nngt0d 12277 . . . . . . . . . . . . . . . . . . 19 ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) β†’ 0 < (2nd β€˜π‘§))
112111iftrued 4532 . . . . . . . . . . . . . . . . . 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 2789 . . . . . . . . . . . . . . . 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 3893 . . . . . . . . . . . . . . 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 11224 . . . . . . . . . . . . . . . . . 18 0 ∈ V
117 oveq2 7422 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 β†’ (1...𝑗) = (1...0))
118 fz10 13540 . . . . . . . . . . . . . . . . . . . . . . . 24 (1...0) = βˆ…
119117, 118eqtrdi 2783 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 β†’ (1...𝑗) = βˆ…)
120119imaeq2d 6057 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 β†’ ((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑗)) = ((2nd β€˜(1st β€˜π‘§)) β€œ βˆ…))
121120xpeq1d 5701 . . . . . . . . . . . . . . . . . . . . 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 7421 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 0 β†’ (𝑗 + 1) = (0 + 1))
123 0p1e1 12350 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 + 1) = 1
124122, 123eqtrdi 2783 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 β†’ (𝑗 + 1) = 1)
125124oveq1d 7429 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 β†’ ((𝑗 + 1)...𝑁) = (1...𝑁))
126125imaeq2d 6057 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 β†’ ((2nd β€˜(1st β€˜π‘§)) β€œ ((𝑗 + 1)...𝑁)) = ((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)))
127126xpeq1d 5701 . . . . . . . . . . . . . . . . . . . . 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 4160 . . . . . . . . . . . . . . . . . . . 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 6074 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd β€˜(1st β€˜π‘§)) β€œ βˆ…) = βˆ…
130129xpeq1i 5698 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd β€˜(1st β€˜π‘§)) β€œ βˆ…) Γ— {1}) = (βˆ… Γ— {1})
131 0xp 5770 . . . . . . . . . . . . . . . . . . . . . . 23 (βˆ… Γ— {1}) = βˆ…
132130, 131eqtri 2755 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd β€˜(1st β€˜π‘§)) β€œ βˆ…) Γ— {1}) = βˆ…
133132uneq1i 4155 . . . . . . . . . . . . . . . . . . . . 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 4149 . . . . . . . . . . . . . . . . . . . . 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 4386 . . . . . . . . . . . . . . . . . . . . 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 2759 . . . . . . . . . . . . . . . . . . . 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 2783 . . . . . . . . . . . . . . . . . . 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 7430 . . . . . . . . . . . . . . . . . 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 3925 . . . . . . . . . . . . . . . . 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 6840 . . . . . . . . . . . . . . . . . . . . . 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 6810 . . . . . . . . . . . . . . . . . . . . 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 5701 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ 𝑆 β†’ (((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑁)) Γ— {0}) = ((1...𝑁) Γ— {0}))
145144oveq2d 7430 . . . . . . . . . . . . . . . . . 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 7449 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ 𝑆 β†’ (1...𝑁) ∈ V)
14780ffnd 6717 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ 𝑆 β†’ (1st β€˜(1st β€˜π‘§)) Fn (1...𝑁))
148 fnconstg 6779 . . . . . . . . . . . . . . . . . . . 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 2728 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) β†’ ((1st β€˜(1st β€˜π‘§))β€˜π‘›) = ((1st β€˜(1st β€˜π‘§))β€˜π‘›))
151116fvconst2 7210 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) β†’ (((1...𝑁) Γ— {0})β€˜π‘›) = 0)
152151adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) β†’ (((1...𝑁) Γ— {0})β€˜π‘›) = 0)
15380ffvelcdmda 7088 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) β†’ ((1st β€˜(1st β€˜π‘§))β€˜π‘›) ∈ (0..^𝐾))
154 elfzonn0 13695 . . . . . . . . . . . . . . . . . . . . . 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 12550 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) β†’ ((1st β€˜(1st β€˜π‘§))β€˜π‘›) ∈ β„‚)
157156addridd 11430 . . . . . . . . . . . . . . . . . . 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 7701 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ 𝑆 β†’ ((1st β€˜(1st β€˜π‘§)) ∘f + ((1...𝑁) Γ— {0})) = (1st β€˜(1st β€˜π‘§)))
159145, 158eqtrd 2767 . . . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . . . 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 12529 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ β„• β†’ (𝑁 βˆ’ 1) ∈ β„•0)
1641, 163syl 17 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (𝑁 βˆ’ 1) ∈ β„•0)
165 0elfz 13616 . . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (1st β€˜(1st β€˜π‘§)) ∈ V)
169106, 162, 167, 168fvmptd 7006 . . . . . . . . . . . . 13 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘§)))
170105, 169sylan 579 . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑇 β†’ (2nd β€˜π‘§) = (2nd β€˜π‘‡))
174173eleq1d 2813 . . . . . . . . . . . . . . 15 (𝑧 = 𝑇 β†’ ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ↔ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))))
175174anbi2d 628 . . . . . . . . . . . . . 14 (𝑧 = 𝑇 β†’ ((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ↔ (πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)))))
176 2fveq3 6896 . . . . . . . . . . . . . . 15 (𝑧 = 𝑇 β†’ (1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)))
177176eqeq2d 2738 . . . . . . . . . . . . . 14 (𝑧 = 𝑇 β†’ ((πΉβ€˜0) = (1st β€˜(1st β€˜π‘§)) ↔ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘‡))))
178175, 177imbi12d 344 . . . . . . . . . . . . 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 413 . . . . . . . . . . . . 13 (𝑧 ∈ 𝑆 β†’ ((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘§))))
180178, 179vtoclga 3561 . . . . . . . . . . . 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 480 . . . . . . . . . 10 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (πΉβ€˜0) = (1st β€˜(1st β€˜π‘‡)))
183172, 182eqtr3d 2769 . . . . . . . . 9 (((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) ∧ 𝑧 ∈ 𝑆) β†’ (1st β€˜(1st β€˜π‘§)) = (1st β€˜(1st β€˜π‘‡)))
184183adantr 480 . . . . . . . 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 480 . . . . . . . . . . . . . 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 8026 . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 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 8026 . . . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . . 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 2956 . . . . . . . . . 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 406 . . . . . . . . 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 37024 . . . . . . . 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 480 . . . . . . . 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 514 . . . . . . 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 412 . . . . . 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 13548 . . . . . . . . . . . . . 14 ((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) β†’ (2nd β€˜π‘‡) ∈ β„•)
207206nnred 12243 . . . . . . . . . . . . 13 ((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) β†’ (2nd β€˜π‘‡) ∈ ℝ)
208207ltp1d 12160 . . . . . . . . . . . . 13 ((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) β†’ (2nd β€˜π‘‡) < ((2nd β€˜π‘‡) + 1))
209207, 208ltned 11366 . . . . . . . . . . . 12 ((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) β†’ (2nd β€˜π‘‡) β‰  ((2nd β€˜π‘‡) + 1))
210209adantl 481 . . . . . . . . . . 11 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜π‘‡) β‰  ((2nd β€˜π‘‡) + 1))
211 fveq1 6890 . . . . . . . . . . . . 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 12070 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd β€˜π‘‡) ∈ ℝ β†’ (2nd β€˜π‘‡) < ((2nd β€˜π‘‡) + 1))
214212, 213ltned 11366 . . . . . . . . . . . . . . . . . . . . 21 ((2nd β€˜π‘‡) ∈ ℝ β†’ (2nd β€˜π‘‡) β‰  ((2nd β€˜π‘‡) + 1))
215 fvex 6904 . . . . . . . . . . . . . . . . . . . . . 22 (2nd β€˜π‘‡) ∈ V
216 ovex 7447 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd β€˜π‘‡) + 1) ∈ V
217215, 216, 216, 215fpr 7157 . . . . . . . . . . . . . . . . . . . . 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 6871 . . . . . . . . . . . . . . . . . . . . 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 6833 . . . . . . . . . . . . . . . . . . . . 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 4467 . . . . . . . . . . . . . . . . . . . . 21 ({(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)} ∩ ((1...𝑁) βˆ– {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)})) = βˆ…
223 fun 6753 . . . . . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . . . . 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 4762 . . . . . . . . . . . . . . . . . . . 20 (2nd β€˜π‘‡) ∈ {(2nd β€˜π‘‡), ((2nd β€˜π‘‡) + 1)}
227 elun1 4172 . . . . . . . . . . . . . . . . . . . 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 6991 . . . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . . . 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 6717 . . . . . . . . . . . . . . . . . . . . 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 6678 . . . . . . . . . . . . . . . . . . . . . 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 470 . . . . . . . . . . . . . . . . . . . . . 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 6983 . . . . . . . . . . . . . . . . . . . . . 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 1450 . . . . . . . . . . . . . . . . . . . . 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 7196 . . . . . . . . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 6832 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1β†’(1...𝑁))
2481nncnd 12244 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ 𝑁 ∈ β„‚)
249 npcan1 11655 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„‚ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
250248, 249syl 17 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
251164nn0zd 12600 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ (𝑁 βˆ’ 1) ∈ β„€)
252 uzid 12853 . . . . . . . . . . . . . . . . . . . 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 12901 . . . . . . . . . . . . . . . . . . 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 2829 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
257 fzss2 13559 . . . . . . . . . . . . . . . . 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 3978 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜π‘‡) ∈ (1...𝑁))
260 fzp1elp1 13572 . . . . . . . . . . . . . . . . 17 ((2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1)) β†’ ((2nd β€˜π‘‡) + 1) ∈ (1...((𝑁 βˆ’ 1) + 1)))
261260adantl 481 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜π‘‡) + 1) ∈ (1...((𝑁 βˆ’ 1) + 1)))
262250oveq2d 7430 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (1...((𝑁 βˆ’ 1) + 1)) = (1...𝑁))
263262adantr 480 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ (1...((𝑁 βˆ’ 1) + 1)) = (1...𝑁))
264261, 263eleqtrd 2830 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜π‘‡) + 1) ∈ (1...𝑁))
265 f1veqaeq 7261 . . . . . . . . . . . . . . 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 2956 . . . . . . . . . . 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 6896 . . . . . . . . . . 11 (𝑧 = 𝑇 β†’ (2nd β€˜(1st β€˜π‘§)) = (2nd β€˜(1st β€˜π‘‡)))
272271neeq1d 2995 . . . . . . . . . 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 246 . . . . . . . . 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 2958 . . . . . . . 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 480 . . . . . 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 8027 . . . . . . . 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 8027 . . . . . . . . . 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 629 . . . . . . . 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 481 . . . . 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 3141 . . 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 3721 . . 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 583 . 2 ((πœ‘ ∧ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
289 xp2nd 8018 . . . . . . 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 532 . . . . 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 12548 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑁 ∈ β„•0)
293 nn0uz 12880 . . . . . . . . . . . 12 β„•0 = (β„€β‰₯β€˜0)
294292, 293eleqtrdi 2838 . . . . . . . . . . 11 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜0))
295 fzpred 13567 . . . . . . . . . . 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 7424 . . . . . . . . . . 11 ((0 + 1)...𝑁) = (1...𝑁)
298297uneq2i 4156 . . . . . . . . . 10 ({0} βˆͺ ((0 + 1)...𝑁)) = ({0} βˆͺ (1...𝑁))
299296, 298eqtrdi 2783 . . . . . . . . 9 (πœ‘ β†’ (0...𝑁) = ({0} βˆͺ (1...𝑁)))
300299difeq1d 4117 . . . . . . . 8 (πœ‘ β†’ ((0...𝑁) βˆ– (1...(𝑁 βˆ’ 1))) = (({0} βˆͺ (1...𝑁)) βˆ– (1...(𝑁 βˆ’ 1))))
301 difundir 4276 . . . . . . . . . 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 11752 . . . . . . . . . . . . . 14 0 < 1
303 0re 11232 . . . . . . . . . . . . . . 15 0 ∈ ℝ
304 1re 11230 . . . . . . . . . . . . . . 15 1 ∈ ℝ
305303, 304ltnlei 11351 . . . . . . . . . . . . . 14 (0 < 1 ↔ Β¬ 1 ≀ 0)
306302, 305mpbi 229 . . . . . . . . . . . . 13 Β¬ 1 ≀ 0
307 elfzle1 13522 . . . . . . . . . . . . 13 (0 ∈ (1...(𝑁 βˆ’ 1)) β†’ 1 ≀ 0)
308306, 307mto 196 . . . . . . . . . . . 12 Β¬ 0 ∈ (1...(𝑁 βˆ’ 1))
309 incom 4197 . . . . . . . . . . . . . 14 ((1...(𝑁 βˆ’ 1)) ∩ {0}) = ({0} ∩ (1...(𝑁 βˆ’ 1)))
310309eqeq1i 2732 . . . . . . . . . . . . 13 (((1...(𝑁 βˆ’ 1)) ∩ {0}) = βˆ… ↔ ({0} ∩ (1...(𝑁 βˆ’ 1))) = βˆ…)
311 disjsn 4711 . . . . . . . . . . . . 13 (((1...(𝑁 βˆ’ 1)) ∩ {0}) = βˆ… ↔ Β¬ 0 ∈ (1...(𝑁 βˆ’ 1)))
312 disj3 4449 . . . . . . . . . . . . 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 4155 . . . . . . . . . 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 2758 . . . . . . . . 9 (({0} βˆͺ (1...𝑁)) βˆ– (1...(𝑁 βˆ’ 1))) = ({0} βˆͺ ((1...𝑁) βˆ– (1...(𝑁 βˆ’ 1))))
317 difundir 4276 . . . . . . . . . . . 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 4366 . . . . . . . . . . . . 13 ((1...(𝑁 βˆ’ 1)) βˆ– (1...(𝑁 βˆ’ 1))) = βˆ…
319318uneq1i 4155 . . . . . . . . . . . 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 4149 . . . . . . . . . . . . 13 (βˆ… βˆͺ ({𝑁} βˆ– (1...(𝑁 βˆ’ 1)))) = (({𝑁} βˆ– (1...(𝑁 βˆ’ 1))) βˆͺ βˆ…)
321 un0 4386 . . . . . . . . . . . . 13 (({𝑁} βˆ– (1...(𝑁 βˆ’ 1))) βˆͺ βˆ…) = ({𝑁} βˆ– (1...(𝑁 βˆ’ 1)))
322320, 321eqtri 2755 . . . . . . . . . . . 12 (βˆ… βˆͺ ({𝑁} βˆ– (1...(𝑁 βˆ’ 1)))) = ({𝑁} βˆ– (1...(𝑁 βˆ’ 1)))
323317, 319, 3223eqtri 2759 . . . . . . . . . . 11 (((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}) βˆ– (1...(𝑁 βˆ’ 1))) = ({𝑁} βˆ– (1...(𝑁 βˆ’ 1)))
324 nnuz 12881 . . . . . . . . . . . . . . . 16 β„• = (β„€β‰₯β€˜1)
3251, 324eleqtrdi 2838 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜1))
326250, 325eqeltrd 2828 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜1))
327 fzsplit2 13544 . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . 13 (πœ‘ β†’ (1...𝑁) = ((1...(𝑁 βˆ’ 1)) βˆͺ (((𝑁 βˆ’ 1) + 1)...𝑁)))
329250oveq1d 7429 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (((𝑁 βˆ’ 1) + 1)...𝑁) = (𝑁...𝑁))
3301nnzd 12601 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝑁 ∈ β„€)
331 fzsn 13561 . . . . . . . . . . . . . . . 16 (𝑁 ∈ β„€ β†’ (𝑁...𝑁) = {𝑁})
332330, 331syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (𝑁...𝑁) = {𝑁})
333329, 332eqtrd 2767 . . . . . . . . . . . . . 14 (πœ‘ β†’ (((𝑁 βˆ’ 1) + 1)...𝑁) = {𝑁})
334333uneq2d 4159 . . . . . . . . . . . . 13 (πœ‘ β†’ ((1...(𝑁 βˆ’ 1)) βˆͺ (((𝑁 βˆ’ 1) + 1)...𝑁)) = ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}))
335328, 334eqtrd 2767 . . . . . . . . . . . 12 (πœ‘ β†’ (1...𝑁) = ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}))
336335difeq1d 4117 . . . . . . . . . . 11 (πœ‘ β†’ ((1...𝑁) βˆ– (1...(𝑁 βˆ’ 1))) = (((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}) βˆ– (1...(𝑁 βˆ’ 1))))
3371nnred 12243 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝑁 ∈ ℝ)
338337ltm1d 12162 . . . . . . . . . . . . . 14 (πœ‘ β†’ (𝑁 βˆ’ 1) < 𝑁)
339164nn0red 12549 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (𝑁 βˆ’ 1) ∈ ℝ)
340339, 337ltnled 11377 . . . . . . . . . . . . . 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 13523 . . . . . . . . . . . . 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 4197 . . . . . . . . . . . . . 14 ((1...(𝑁 βˆ’ 1)) ∩ {𝑁}) = ({𝑁} ∩ (1...(𝑁 βˆ’ 1)))
345344eqeq1i 2732 . . . . . . . . . . . . 13 (((1...(𝑁 βˆ’ 1)) ∩ {𝑁}) = βˆ… ↔ ({𝑁} ∩ (1...(𝑁 βˆ’ 1))) = βˆ…)
346 disjsn 4711 . . . . . . . . . . . . 13 (((1...(𝑁 βˆ’ 1)) ∩ {𝑁}) = βˆ… ↔ Β¬ 𝑁 ∈ (1...(𝑁 βˆ’ 1)))
347 disj3 4449 . . . . . . . . . . . . 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 2793 . . . . . . . . . 10 (πœ‘ β†’ ((1...𝑁) βˆ– (1...(𝑁 βˆ’ 1))) = {𝑁})
351350uneq2d 4159 . . . . . . . . 9 (πœ‘ β†’ ({0} βˆͺ ((1...𝑁) βˆ– (1...(𝑁 βˆ’ 1)))) = ({0} βˆͺ {𝑁}))
352316, 351eqtrid 2779 . . . . . . . 8 (πœ‘ β†’ (({0} βˆͺ (1...𝑁)) βˆ– (1...(𝑁 βˆ’ 1))) = ({0} βˆͺ {𝑁}))
353300, 352eqtrd 2767 . . . . . . 7 (πœ‘ β†’ ((0...𝑁) βˆ– (1...(𝑁 βˆ’ 1))) = ({0} βˆͺ {𝑁}))
354353eleq2d 2814 . . . . . 6 (πœ‘ β†’ ((2nd β€˜π‘‡) ∈ ((0...𝑁) βˆ– (1...(𝑁 βˆ’ 1))) ↔ (2nd β€˜π‘‡) ∈ ({0} βˆͺ {𝑁})))
355 eldif 3954 . . . . . 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 4144 . . . . . . 7 ((2nd β€˜π‘‡) ∈ ({0} βˆͺ {𝑁}) ↔ ((2nd β€˜π‘‡) ∈ {0} ∨ (2nd β€˜π‘‡) ∈ {𝑁}))
357215elsn 4639 . . . . . . . 8 ((2nd β€˜π‘‡) ∈ {0} ↔ (2nd β€˜π‘‡) = 0)
358215elsn 4639 . . . . . . . 8 ((2nd β€˜π‘‡) ∈ {𝑁} ↔ (2nd β€˜π‘‡) = 𝑁)
359357, 358orbi12i 913 . . . . . . 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 476 . . 3 ((πœ‘ ∧ Β¬ (2nd β€˜π‘‡) ∈ (1...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜π‘‡) = 0 ∨ (2nd β€˜π‘‡) = 𝑁))
3641adantr 480 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 0) β†’ 𝑁 ∈ β„•)
3654adantr 480 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 0) β†’ 𝐹:(0...(𝑁 βˆ’ 1))⟢((0...𝐾) ↑m (1...𝑁)))
3666adantr 480 . . . . 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 484 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 0) β†’ (2nd β€˜π‘‡) = 0)
370364, 3, 365, 366, 368, 369poimirlem18 37033 . . . 4 ((πœ‘ ∧ (2nd β€˜π‘‡) = 0) β†’ βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
3711adantr 480 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 𝑁) β†’ 𝑁 ∈ β„•)
3724adantr 480 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 𝑁) β†’ 𝐹:(0...(𝑁 βˆ’ 1))⟢((0...𝐾) ↑m (1...𝑁)))
3736adantr 480 . . . . 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 484 . . . . 5 ((πœ‘ ∧ (2nd β€˜π‘‡) = 𝑁) β†’ (2nd β€˜π‘‡) = 𝑁)
377371, 3, 372, 373, 375, 376poimirlem21 37036 . . . 4 ((πœ‘ ∧ (2nd β€˜π‘‡) = 𝑁) β†’ βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
378370, 377jaodan 956 . . 3 ((πœ‘ ∧ ((2nd β€˜π‘‡) = 0 ∨ (2nd β€˜π‘‡) = 𝑁)) β†’ βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
379363, 378syldan 590 . 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 395   ∨ wo 846   = wceq 1534   ∈ wcel 2099  {cab 2704   β‰  wne 2935  βˆ€wral 3056  βˆƒwrex 3065  βˆƒ!wreu 3369  βˆƒ*wrmo 3370  {crab 3427  Vcvv 3469  β¦‹csb 3889   βˆ– cdif 3941   βˆͺ cun 3942   ∩ cin 3943   βŠ† wss 3944  βˆ…c0 4318  ifcif 4524  {csn 4624  {cpr 4626  βŸ¨cop 4630   class class class wbr 5142   ↦ cmpt 5225   I cid 5569   Γ— cxp 5670  ran crn 5673   β†Ύ cres 5674   β€œ cima 5675   ∘ ccom 5676   Fn wfn 6537  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€“ontoβ†’wfo 6540  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414   ∘f cof 7675  1st c1st 7983  2nd c2nd 7984   ↑m cmap 8834  β„‚cc 11122  β„cr 11123  0cc0 11124  1c1 11125   + caddc 11127   < clt 11264   ≀ cle 11265   βˆ’ cmin 11460  β„•cn 12228  β„•0cn0 12488  β„€cz 12574  β„€β‰₯cuz 12838  ...cfz 13502  ..^cfzo 13645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7677  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-oadd 8482  df-er 8716  df-map 8836  df-pm 8837  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-dju 9910  df-card 9948  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-div 11888  df-nn 12229  df-2 12291  df-3 12292  df-n0 12489  df-xnn0 12561  df-z 12575  df-uz 12839  df-fz 13503  df-fzo 13646  df-seq 13985  df-fac 14251  df-bc 14280  df-hash 14308
This theorem is referenced by:  poimirlem27  37042
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