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Theorem poimirlem18 36096
Description: Lemma for poimir 36111 stating that, given a face not on a front face of the main cube and a simplex in which it's opposite the first vertex on the walk, there exists exactly one other simplex containing it. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (πœ‘ β†’ 𝑁 ∈ β„•)
poimirlem22.s 𝑆 = {𝑑 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 βˆ’ 1)) ↦ ⦋if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) / π‘—β¦Œ((1st β€˜(1st β€˜π‘‘)) ∘f + ((((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))}
poimirlem22.1 (πœ‘ β†’ 𝐹:(0...(𝑁 βˆ’ 1))⟢((0...𝐾) ↑m (1...𝑁)))
poimirlem22.2 (πœ‘ β†’ 𝑇 ∈ 𝑆)
poimirlem18.3 ((πœ‘ ∧ 𝑛 ∈ (1...𝑁)) β†’ βˆƒπ‘ ∈ ran 𝐹(π‘β€˜π‘›) β‰  𝐾)
poimirlem18.4 (πœ‘ β†’ (2nd β€˜π‘‡) = 0)
Assertion
Ref Expression
poimirlem18 (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 poimirlem18
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . 3 (πœ‘ β†’ 𝑁 ∈ β„•)
2 poimirlem22.s . . 3 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}))))}
3 poimirlem22.1 . . 3 (πœ‘ β†’ 𝐹:(0...(𝑁 βˆ’ 1))⟢((0...𝐾) ↑m (1...𝑁)))
4 poimirlem22.2 . . 3 (πœ‘ β†’ 𝑇 ∈ 𝑆)
5 poimirlem18.3 . . 3 ((πœ‘ ∧ 𝑛 ∈ (1...𝑁)) β†’ βˆƒπ‘ ∈ ran 𝐹(π‘β€˜π‘›) β‰  𝐾)
6 poimirlem18.4 . . 3 (πœ‘ β†’ (2nd β€˜π‘‡) = 0)
71, 2, 3, 4, 5, 6poimirlem17 36095 . 2 (πœ‘ β†’ βˆƒπ‘§ ∈ 𝑆 𝑧 β‰  𝑇)
86adantr 481 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ (2nd β€˜π‘‡) = 0)
9 0nnn 12189 . . . . . . . . . . . . 13 Β¬ 0 ∈ β„•
10 elfznn 13470 . . . . . . . . . . . . 13 (0 ∈ (1...(𝑁 βˆ’ 1)) β†’ 0 ∈ β„•)
119, 10mto 196 . . . . . . . . . . . 12 Β¬ 0 ∈ (1...(𝑁 βˆ’ 1))
12 eleq1 2825 . . . . . . . . . . . 12 ((2nd β€˜π‘§) = 0 β†’ ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ↔ 0 ∈ (1...(𝑁 βˆ’ 1))))
1311, 12mtbiri 326 . . . . . . . . . . 11 ((2nd β€˜π‘§) = 0 β†’ Β¬ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)))
1413necon2ai 2973 . . . . . . . . . 10 ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) β†’ (2nd β€˜π‘§) β‰  0)
151ad2antrr 724 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) β†’ 𝑁 ∈ β„•)
16 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑧 β†’ (2nd β€˜π‘‘) = (2nd β€˜π‘§))
1716breq2d 5117 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 = 𝑧 β†’ (𝑦 < (2nd β€˜π‘‘) ↔ 𝑦 < (2nd β€˜π‘§)))
1817ifbid 4509 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑧 β†’ if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd β€˜π‘§), 𝑦, (𝑦 + 1)))
1918csbeq1d 3859 . . . . . . . . . . . . . . . . . . . 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}))))
20 2fveq3 6847 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 = 𝑧 β†’ (1st β€˜(1st β€˜π‘‘)) = (1st β€˜(1st β€˜π‘§)))
21 2fveq3 6847 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = 𝑧 β†’ (2nd β€˜(1st β€˜π‘‘)) = (2nd β€˜(1st β€˜π‘§)))
2221imaeq1d 6012 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑑 = 𝑧 β†’ ((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) = ((2nd β€˜(1st β€˜π‘§)) β€œ (1...𝑗)))
2322xpeq1d 5662 . . . . . . . . . . . . . . . . . . . . . . 23 (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}))
2421imaeq1d 6012 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑑 = 𝑧 β†’ ((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) = ((2nd β€˜(1st β€˜π‘§)) β€œ ((𝑗 + 1)...𝑁)))
2524xpeq1d 5662 . . . . . . . . . . . . . . . . . . . . . . 23 (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}))
2623, 25uneq12d 4124 . . . . . . . . . . . . . . . . . . . . . 22 (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})))
2720, 26oveq12d 7375 . . . . . . . . . . . . . . . . . . . . 21 (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}))))
2827csbeq2dv 3862 . . . . . . . . . . . . . . . . . . . 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}))))
2919, 28eqtrd 2776 . . . . . . . . . . . . . . . . . . 19 (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}))))
3029mpteq2dv 5207 . . . . . . . . . . . . . . . . . 18 (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})))))
3130eqeq2d 2747 . . . . . . . . . . . . . . . . 17 (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}))))))
3231, 2elrab2 3648 . . . . . . . . . . . . . . . 16 (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}))))))
3332simprbi 497 . . . . . . . . . . . . . . 15 (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})))))
3433ad2antlr 725 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑧 ∈ 𝑆) ∧ (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})))))
35 elrabi 3639 . . . . . . . . . . . . . . . . . . . 20 (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)))
3635, 2eleq2s 2855 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ 𝑆 β†’ 𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)))
37 xp1st 7953 . . . . . . . . . . . . . . . . . . 19 (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)}))
3836, 37syl 17 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ 𝑆 β†’ (1st β€˜π‘§) ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}))
39 xp1st 7953 . . . . . . . . . . . . . . . . . 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)))
4038, 39syl 17 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ 𝑆 β†’ (1st β€˜(1st β€˜π‘§)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
41 elmapi 8787 . . . . . . . . . . . . . . . . 17 ((1st β€˜(1st β€˜π‘§)) ∈ ((0..^𝐾) ↑m (1...𝑁)) β†’ (1st β€˜(1st β€˜π‘§)):(1...𝑁)⟢(0..^𝐾))
4240, 41syl 17 . . . . . . . . . . . . . . . 16 (𝑧 ∈ 𝑆 β†’ (1st β€˜(1st β€˜π‘§)):(1...𝑁)⟢(0..^𝐾))
43 elfzoelz 13572 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝐾) β†’ 𝑛 ∈ β„€)
4443ssriv 3948 . . . . . . . . . . . . . . . 16 (0..^𝐾) βŠ† β„€
45 fss 6685 . . . . . . . . . . . . . . . 16 (((1st β€˜(1st β€˜π‘§)):(1...𝑁)⟢(0..^𝐾) ∧ (0..^𝐾) βŠ† β„€) β†’ (1st β€˜(1st β€˜π‘§)):(1...𝑁)βŸΆβ„€)
4642, 44, 45sylancl 586 . . . . . . . . . . . . . . 15 (𝑧 ∈ 𝑆 β†’ (1st β€˜(1st β€˜π‘§)):(1...𝑁)βŸΆβ„€)
4746ad2antlr 725 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) β†’ (1st β€˜(1st β€˜π‘§)):(1...𝑁)βŸΆβ„€)
48 xp2nd 7954 . . . . . . . . . . . . . . . . 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)})
4938, 48syl 17 . . . . . . . . . . . . . . . 16 (𝑧 ∈ 𝑆 β†’ (2nd β€˜(1st β€˜π‘§)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})
50 fvex 6855 . . . . . . . . . . . . . . . . 17 (2nd β€˜(1st β€˜π‘§)) ∈ V
51 f1oeq1 6772 . . . . . . . . . . . . . . . . 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)))
5250, 51elab 3630 . . . . . . . . . . . . . . . 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))
5349, 52sylib 217 . . . . . . . . . . . . . . 15 (𝑧 ∈ 𝑆 β†’ (2nd β€˜(1st β€˜π‘§)):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
5453ad2antlr 725 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜(1st β€˜π‘§)):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
55 simpr 485 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)))
5615, 34, 47, 54, 55poimirlem1 36079 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑧 ∈ 𝑆) ∧ (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‘›))
571ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ (2nd β€˜π‘‡) β‰  (2nd β€˜π‘§)) β†’ 𝑁 ∈ β„•)
58 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑 = 𝑇 β†’ (2nd β€˜π‘‘) = (2nd β€˜π‘‡))
5958breq2d 5117 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = 𝑇 β†’ (𝑦 < (2nd β€˜π‘‘) ↔ 𝑦 < (2nd β€˜π‘‡)))
6059ifbid 4509 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = 𝑇 β†’ if(𝑦 < (2nd β€˜π‘‘), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd β€˜π‘‡), 𝑦, (𝑦 + 1)))
6160csbeq1d 3859 . . . . . . . . . . . . . . . . . . . . . . . 24 (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}))))
62 2fveq3 6847 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = 𝑇 β†’ (1st β€˜(1st β€˜π‘‘)) = (1st β€˜(1st β€˜π‘‡)))
63 2fveq3 6847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑑 = 𝑇 β†’ (2nd β€˜(1st β€˜π‘‘)) = (2nd β€˜(1st β€˜π‘‡)))
6463imaeq1d 6012 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑑 = 𝑇 β†’ ((2nd β€˜(1st β€˜π‘‘)) β€œ (1...𝑗)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ (1...𝑗)))
6564xpeq1d 5662 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (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}))
6663imaeq1d 6012 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑑 = 𝑇 β†’ ((2nd β€˜(1st β€˜π‘‘)) β€œ ((𝑗 + 1)...𝑁)) = ((2nd β€˜(1st β€˜π‘‡)) β€œ ((𝑗 + 1)...𝑁)))
6766xpeq1d 5662 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (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}))
6865, 67uneq12d 4124 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (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})))
6962, 68oveq12d 7375 . . . . . . . . . . . . . . . . . . . . . . . . 25 (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}))))
7069csbeq2dv 3862 . . . . . . . . . . . . . . . . . . . . . . . 24 (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}))))
7161, 70eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . 23 (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}))))
7271mpteq2dv 5207 . . . . . . . . . . . . . . . . . . . . . 22 (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})))))
7372eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . 21 (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}))))))
7473, 2elrab2 3648 . . . . . . . . . . . . . . . . . . . 20 (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}))))))
7574simprbi 497 . . . . . . . . . . . . . . . . . . 19 (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})))))
764, 75syl 17 . . . . . . . . . . . . . . . . . 18 (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})))))
7776ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ (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})))))
78 elrabi 3639 . . . . . . . . . . . . . . . . . . . . . . . 24 (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)))
7978, 2eleq2s 2855 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑇 ∈ 𝑆 β†’ 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)))
804, 79syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) Γ— (0...𝑁)))
81 xp1st 7953 . . . . . . . . . . . . . . . . . . . . . 22 (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)}))
8280, 81syl 17 . . . . . . . . . . . . . . . . . . . . 21 (πœ‘ β†’ (1st β€˜π‘‡) ∈ (((0..^𝐾) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}))
83 xp1st 7953 . . . . . . . . . . . . . . . . . . . . 21 ((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)))
8482, 83syl 17 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ (1st β€˜(1st β€˜π‘‡)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
85 elmapi 8787 . . . . . . . . . . . . . . . . . . . 20 ((1st β€˜(1st β€˜π‘‡)) ∈ ((0..^𝐾) ↑m (1...𝑁)) β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)⟢(0..^𝐾))
8684, 85syl 17 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)⟢(0..^𝐾))
87 fss 6685 . . . . . . . . . . . . . . . . . . 19 (((1st β€˜(1st β€˜π‘‡)):(1...𝑁)⟢(0..^𝐾) ∧ (0..^𝐾) βŠ† β„€) β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)βŸΆβ„€)
8886, 44, 87sylancl 586 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)βŸΆβ„€)
8988ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ (2nd β€˜π‘‡) β‰  (2nd β€˜π‘§)) β†’ (1st β€˜(1st β€˜π‘‡)):(1...𝑁)βŸΆβ„€)
90 xp2nd 7954 . . . . . . . . . . . . . . . . . . . 20 ((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)})
9182, 90syl 17 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ (2nd β€˜(1st β€˜π‘‡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})
92 fvex 6855 . . . . . . . . . . . . . . . . . . . 20 (2nd β€˜(1st β€˜π‘‡)) ∈ V
93 f1oeq1 6772 . . . . . . . . . . . . . . . . . . . 20 (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)))
9492, 93elab 3630 . . . . . . . . . . . . . . . . . . 19 ((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))
9591, 94sylib 217 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ (2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
9695ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ (2nd β€˜π‘‡) β‰  (2nd β€˜π‘§)) β†’ (2nd β€˜(1st β€˜π‘‡)):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
97 simplr 767 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ (2nd β€˜π‘‡) β‰  (2nd β€˜π‘§)) β†’ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)))
98 xp2nd 7954 . . . . . . . . . . . . . . . . . . . 20 (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))
9980, 98syl 17 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ (2nd β€˜π‘‡) ∈ (0...𝑁))
10099adantr 481 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜π‘‡) ∈ (0...𝑁))
101 eldifsn 4747 . . . . . . . . . . . . . . . . . . 19 ((2nd β€˜π‘‡) ∈ ((0...𝑁) βˆ– {(2nd β€˜π‘§)}) ↔ ((2nd β€˜π‘‡) ∈ (0...𝑁) ∧ (2nd β€˜π‘‡) β‰  (2nd β€˜π‘§)))
102101biimpri 227 . . . . . . . . . . . . . . . . . 18 (((2nd β€˜π‘‡) ∈ (0...𝑁) ∧ (2nd β€˜π‘‡) β‰  (2nd β€˜π‘§)) β†’ (2nd β€˜π‘‡) ∈ ((0...𝑁) βˆ– {(2nd β€˜π‘§)}))
103100, 102sylan 580 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ (2nd β€˜π‘‡) β‰  (2nd β€˜π‘§)) β†’ (2nd β€˜π‘‡) ∈ ((0...𝑁) βˆ– {(2nd β€˜π‘§)}))
10457, 77, 89, 96, 97, 103poimirlem2 36080 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) ∧ (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‘›))
105104ex 413 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) 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‘›)))
106105necon1bd 2961 . . . . . . . . . . . . . 14 ((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‘›) β†’ (2nd β€˜π‘‡) = (2nd β€˜π‘§)))
107106adantlr 713 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑧 ∈ 𝑆) ∧ (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‘›) β†’ (2nd β€˜π‘‡) = (2nd β€˜π‘§)))
10856, 107mpd 15 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) β†’ (2nd β€˜π‘‡) = (2nd β€˜π‘§))
109108neeq1d 3003 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ 𝑆) ∧ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))) β†’ ((2nd β€˜π‘‡) β‰  0 ↔ (2nd β€˜π‘§) β‰  0))
110109exbiri 809 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) β†’ ((2nd β€˜π‘§) β‰  0 β†’ (2nd β€˜π‘‡) β‰  0)))
11114, 110mpdi 45 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) β†’ (2nd β€˜π‘‡) β‰  0))
112111necon2bd 2959 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ ((2nd β€˜π‘‡) = 0 β†’ Β¬ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1))))
1138, 112mpd 15 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ Β¬ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)))
114 xp2nd 7954 . . . . . . . . 9 (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))
11536, 114syl 17 . . . . . . . 8 (𝑧 ∈ 𝑆 β†’ (2nd β€˜π‘§) ∈ (0...𝑁))
1161nncnd 12169 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ 𝑁 ∈ β„‚)
117 npcan1 11580 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„‚ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
118116, 117syl 17 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
119 nnuz 12806 . . . . . . . . . . . . . . . . . . 19 β„• = (β„€β‰₯β€˜1)
1201, 119eleqtrdi 2847 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜1))
121118, 120eqeltrd 2837 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜1))
1221nnzd 12526 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ 𝑁 ∈ β„€)
123 peano2zm 12546 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„€ β†’ (𝑁 βˆ’ 1) ∈ β„€)
124122, 123syl 17 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ (𝑁 βˆ’ 1) ∈ β„€)
125 uzid 12778 . . . . . . . . . . . . . . . . . . 19 ((𝑁 βˆ’ 1) ∈ β„€ β†’ (𝑁 βˆ’ 1) ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
126 peano2uz 12826 . . . . . . . . . . . . . . . . . . 19 ((𝑁 βˆ’ 1) ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)) β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
127124, 125, 1263syl 18 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ ((𝑁 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
128118, 127eqeltrrd 2838 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
129 fzsplit2 13466 . . . . . . . . . . . . . . . . 17 ((((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)))
130121, 128, 129syl2anc 584 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (1...𝑁) = ((1...(𝑁 βˆ’ 1)) βˆͺ (((𝑁 βˆ’ 1) + 1)...𝑁)))
131118oveq1d 7372 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ (((𝑁 βˆ’ 1) + 1)...𝑁) = (𝑁...𝑁))
132 fzsn 13483 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„€ β†’ (𝑁...𝑁) = {𝑁})
133122, 132syl 17 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ (𝑁...𝑁) = {𝑁})
134131, 133eqtrd 2776 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (((𝑁 βˆ’ 1) + 1)...𝑁) = {𝑁})
135134uneq2d 4123 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ ((1...(𝑁 βˆ’ 1)) βˆͺ (((𝑁 βˆ’ 1) + 1)...𝑁)) = ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}))
136130, 135eqtrd 2776 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (1...𝑁) = ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}))
137136eleq2d 2823 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((2nd β€˜π‘§) ∈ (1...𝑁) ↔ (2nd β€˜π‘§) ∈ ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁})))
138137notbid 317 . . . . . . . . . . . . 13 (πœ‘ β†’ (Β¬ (2nd β€˜π‘§) ∈ (1...𝑁) ↔ Β¬ (2nd β€˜π‘§) ∈ ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁})))
139 ioran 982 . . . . . . . . . . . . . 14 (Β¬ ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ∨ (2nd β€˜π‘§) = 𝑁) ↔ (Β¬ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ∧ Β¬ (2nd β€˜π‘§) = 𝑁))
140 elun 4108 . . . . . . . . . . . . . . 15 ((2nd β€˜π‘§) ∈ ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}) ↔ ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ∨ (2nd β€˜π‘§) ∈ {𝑁}))
141 fvex 6855 . . . . . . . . . . . . . . . . 17 (2nd β€˜π‘§) ∈ V
142141elsn 4601 . . . . . . . . . . . . . . . 16 ((2nd β€˜π‘§) ∈ {𝑁} ↔ (2nd β€˜π‘§) = 𝑁)
143142orbi2i 911 . . . . . . . . . . . . . . 15 (((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ∨ (2nd β€˜π‘§) ∈ {𝑁}) ↔ ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ∨ (2nd β€˜π‘§) = 𝑁))
144140, 143bitri 274 . . . . . . . . . . . . . 14 ((2nd β€˜π‘§) ∈ ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}) ↔ ((2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ∨ (2nd β€˜π‘§) = 𝑁))
145139, 144xchnxbir 332 . . . . . . . . . . . . 13 (Β¬ (2nd β€˜π‘§) ∈ ((1...(𝑁 βˆ’ 1)) βˆͺ {𝑁}) ↔ (Β¬ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ∧ Β¬ (2nd β€˜π‘§) = 𝑁))
146138, 145bitrdi 286 . . . . . . . . . . . 12 (πœ‘ β†’ (Β¬ (2nd β€˜π‘§) ∈ (1...𝑁) ↔ (Β¬ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ∧ Β¬ (2nd β€˜π‘§) = 𝑁)))
147146anbi2d 629 . . . . . . . . . . 11 (πœ‘ β†’ (((2nd β€˜π‘§) ∈ (0...𝑁) ∧ Β¬ (2nd β€˜π‘§) ∈ (1...𝑁)) ↔ ((2nd β€˜π‘§) ∈ (0...𝑁) ∧ (Β¬ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ∧ Β¬ (2nd β€˜π‘§) = 𝑁))))
1481nnnn0d 12473 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝑁 ∈ β„•0)
149 nn0uz 12805 . . . . . . . . . . . . . . . . 17 β„•0 = (β„€β‰₯β€˜0)
150148, 149eleqtrdi 2847 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜0))
151 fzpred 13489 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (β„€β‰₯β€˜0) β†’ (0...𝑁) = ({0} βˆͺ ((0 + 1)...𝑁)))
152150, 151syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (0...𝑁) = ({0} βˆͺ ((0 + 1)...𝑁)))
153152difeq1d 4081 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((0...𝑁) βˆ– (1...𝑁)) = (({0} βˆͺ ((0 + 1)...𝑁)) βˆ– (1...𝑁)))
154 difun2 4440 . . . . . . . . . . . . . . 15 (({0} βˆͺ (1...𝑁)) βˆ– (1...𝑁)) = ({0} βˆ– (1...𝑁))
155 0p1e1 12275 . . . . . . . . . . . . . . . . . 18 (0 + 1) = 1
156155oveq1i 7367 . . . . . . . . . . . . . . . . 17 ((0 + 1)...𝑁) = (1...𝑁)
157156uneq2i 4120 . . . . . . . . . . . . . . . 16 ({0} βˆͺ ((0 + 1)...𝑁)) = ({0} βˆͺ (1...𝑁))
158157difeq1i 4078 . . . . . . . . . . . . . . 15 (({0} βˆͺ ((0 + 1)...𝑁)) βˆ– (1...𝑁)) = (({0} βˆͺ (1...𝑁)) βˆ– (1...𝑁))
159 incom 4161 . . . . . . . . . . . . . . . . 17 ({0} ∩ (1...𝑁)) = ((1...𝑁) ∩ {0})
160 elfznn 13470 . . . . . . . . . . . . . . . . . . 19 (0 ∈ (1...𝑁) β†’ 0 ∈ β„•)
1619, 160mto 196 . . . . . . . . . . . . . . . . . 18 Β¬ 0 ∈ (1...𝑁)
162 disjsn 4672 . . . . . . . . . . . . . . . . . 18 (((1...𝑁) ∩ {0}) = βˆ… ↔ Β¬ 0 ∈ (1...𝑁))
163161, 162mpbir 230 . . . . . . . . . . . . . . . . 17 ((1...𝑁) ∩ {0}) = βˆ…
164159, 163eqtri 2764 . . . . . . . . . . . . . . . 16 ({0} ∩ (1...𝑁)) = βˆ…
165 disj3 4413 . . . . . . . . . . . . . . . 16 (({0} ∩ (1...𝑁)) = βˆ… ↔ {0} = ({0} βˆ– (1...𝑁)))
166164, 165mpbi 229 . . . . . . . . . . . . . . 15 {0} = ({0} βˆ– (1...𝑁))
167154, 158, 1663eqtr4i 2774 . . . . . . . . . . . . . 14 (({0} βˆͺ ((0 + 1)...𝑁)) βˆ– (1...𝑁)) = {0}
168153, 167eqtrdi 2792 . . . . . . . . . . . . 13 (πœ‘ β†’ ((0...𝑁) βˆ– (1...𝑁)) = {0})
169168eleq2d 2823 . . . . . . . . . . . 12 (πœ‘ β†’ ((2nd β€˜π‘§) ∈ ((0...𝑁) βˆ– (1...𝑁)) ↔ (2nd β€˜π‘§) ∈ {0}))
170 eldif 3920 . . . . . . . . . . . 12 ((2nd β€˜π‘§) ∈ ((0...𝑁) βˆ– (1...𝑁)) ↔ ((2nd β€˜π‘§) ∈ (0...𝑁) ∧ Β¬ (2nd β€˜π‘§) ∈ (1...𝑁)))
171141elsn 4601 . . . . . . . . . . . 12 ((2nd β€˜π‘§) ∈ {0} ↔ (2nd β€˜π‘§) = 0)
172169, 170, 1713bitr3g 312 . . . . . . . . . . 11 (πœ‘ β†’ (((2nd β€˜π‘§) ∈ (0...𝑁) ∧ Β¬ (2nd β€˜π‘§) ∈ (1...𝑁)) ↔ (2nd β€˜π‘§) = 0))
173147, 172bitr3d 280 . . . . . . . . . 10 (πœ‘ β†’ (((2nd β€˜π‘§) ∈ (0...𝑁) ∧ (Β¬ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ∧ Β¬ (2nd β€˜π‘§) = 𝑁)) ↔ (2nd β€˜π‘§) = 0))
174173biimpd 228 . . . . . . . . 9 (πœ‘ β†’ (((2nd β€˜π‘§) ∈ (0...𝑁) ∧ (Β¬ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ∧ Β¬ (2nd β€˜π‘§) = 𝑁)) β†’ (2nd β€˜π‘§) = 0))
175174expdimp 453 . . . . . . . 8 ((πœ‘ ∧ (2nd β€˜π‘§) ∈ (0...𝑁)) β†’ ((Β¬ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ∧ Β¬ (2nd β€˜π‘§) = 𝑁) β†’ (2nd β€˜π‘§) = 0))
176115, 175sylan2 593 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ ((Β¬ (2nd β€˜π‘§) ∈ (1...(𝑁 βˆ’ 1)) ∧ Β¬ (2nd β€˜π‘§) = 𝑁) β†’ (2nd β€˜π‘§) = 0))
177113, 176mpand 693 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ (Β¬ (2nd β€˜π‘§) = 𝑁 β†’ (2nd β€˜π‘§) = 0))
1781, 2, 3poimirlem13 36091 . . . . . . . . . 10 (πœ‘ β†’ βˆƒ*𝑧 ∈ 𝑆 (2nd β€˜π‘§) = 0)
179 fveqeq2 6851 . . . . . . . . . . 11 (𝑧 = 𝑠 β†’ ((2nd β€˜π‘§) = 0 ↔ (2nd β€˜π‘ ) = 0))
180179rmo4 3688 . . . . . . . . . 10 (βˆƒ*𝑧 ∈ 𝑆 (2nd β€˜π‘§) = 0 ↔ βˆ€π‘§ ∈ 𝑆 βˆ€π‘  ∈ 𝑆 (((2nd β€˜π‘§) = 0 ∧ (2nd β€˜π‘ ) = 0) β†’ 𝑧 = 𝑠))
181178, 180sylib 217 . . . . . . . . 9 (πœ‘ β†’ βˆ€π‘§ ∈ 𝑆 βˆ€π‘  ∈ 𝑆 (((2nd β€˜π‘§) = 0 ∧ (2nd β€˜π‘ ) = 0) β†’ 𝑧 = 𝑠))
182181r19.21bi 3234 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ βˆ€π‘  ∈ 𝑆 (((2nd β€˜π‘§) = 0 ∧ (2nd β€˜π‘ ) = 0) β†’ 𝑧 = 𝑠))
1834adantr 481 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ 𝑇 ∈ 𝑆)
184 fveqeq2 6851 . . . . . . . . . . 11 (𝑠 = 𝑇 β†’ ((2nd β€˜π‘ ) = 0 ↔ (2nd β€˜π‘‡) = 0))
185184anbi2d 629 . . . . . . . . . 10 (𝑠 = 𝑇 β†’ (((2nd β€˜π‘§) = 0 ∧ (2nd β€˜π‘ ) = 0) ↔ ((2nd β€˜π‘§) = 0 ∧ (2nd β€˜π‘‡) = 0)))
186 eqeq2 2748 . . . . . . . . . 10 (𝑠 = 𝑇 β†’ (𝑧 = 𝑠 ↔ 𝑧 = 𝑇))
187185, 186imbi12d 344 . . . . . . . . 9 (𝑠 = 𝑇 β†’ ((((2nd β€˜π‘§) = 0 ∧ (2nd β€˜π‘ ) = 0) β†’ 𝑧 = 𝑠) ↔ (((2nd β€˜π‘§) = 0 ∧ (2nd β€˜π‘‡) = 0) β†’ 𝑧 = 𝑇)))
188187rspccv 3578 . . . . . . . 8 (βˆ€π‘  ∈ 𝑆 (((2nd β€˜π‘§) = 0 ∧ (2nd β€˜π‘ ) = 0) β†’ 𝑧 = 𝑠) β†’ (𝑇 ∈ 𝑆 β†’ (((2nd β€˜π‘§) = 0 ∧ (2nd β€˜π‘‡) = 0) β†’ 𝑧 = 𝑇)))
189182, 183, 188sylc 65 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ (((2nd β€˜π‘§) = 0 ∧ (2nd β€˜π‘‡) = 0) β†’ 𝑧 = 𝑇))
1908, 189mpan2d 692 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ ((2nd β€˜π‘§) = 0 β†’ 𝑧 = 𝑇))
191177, 190syld 47 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ (Β¬ (2nd β€˜π‘§) = 𝑁 β†’ 𝑧 = 𝑇))
192191necon1ad 2960 . . . 4 ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ (𝑧 β‰  𝑇 β†’ (2nd β€˜π‘§) = 𝑁))
193192ralrimiva 3143 . . 3 (πœ‘ β†’ βˆ€π‘§ ∈ 𝑆 (𝑧 β‰  𝑇 β†’ (2nd β€˜π‘§) = 𝑁))
1941, 2, 3poimirlem14 36092 . . 3 (πœ‘ β†’ βˆƒ*𝑧 ∈ 𝑆 (2nd β€˜π‘§) = 𝑁)
195 rmoim 3698 . . 3 (βˆ€π‘§ ∈ 𝑆 (𝑧 β‰  𝑇 β†’ (2nd β€˜π‘§) = 𝑁) β†’ (βˆƒ*𝑧 ∈ 𝑆 (2nd β€˜π‘§) = 𝑁 β†’ βˆƒ*𝑧 ∈ 𝑆 𝑧 β‰  𝑇))
196193, 194, 195sylc 65 . 2 (πœ‘ β†’ βˆƒ*𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
197 reu5 3355 . 2 (βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇 ↔ (βˆƒπ‘§ ∈ 𝑆 𝑧 β‰  𝑇 ∧ βˆƒ*𝑧 ∈ 𝑆 𝑧 β‰  𝑇))
1987, 196, 197sylanbrc 583 1 (πœ‘ β†’ βˆƒ!𝑧 ∈ 𝑆 𝑧 β‰  𝑇)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  {cab 2713   β‰  wne 2943  βˆ€wral 3064  βˆƒwrex 3073  βˆƒ!wreu 3351  βˆƒ*wrmo 3352  {crab 3407  β¦‹csb 3855   βˆ– cdif 3907   βˆͺ cun 3908   ∩ cin 3909   βŠ† wss 3910  βˆ…c0 4282  ifcif 4486  {csn 4586   class class class wbr 5105   ↦ cmpt 5188   Γ— cxp 5631  ran crn 5634   β€œ cima 5636  βŸΆwf 6492  β€“1-1-ontoβ†’wf1o 6495  β€˜cfv 6496  (class class class)co 7357   ∘f cof 7615  1st c1st 7919  2nd c2nd 7920   ↑m cmap 8765  β„‚cc 11049  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189   βˆ’ cmin 11385  β„•cn 12153  β„•0cn0 12413  β„€cz 12499  β„€β‰₯cuz 12763  ...cfz 13424  ..^cfzo 13567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568
This theorem is referenced by:  poimirlem22  36100
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