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Mirrors > Home > MPE Home > Th. List > wlkiswwlks2lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for wlkiswwlks2 27002. (Contributed by Alexander van der Vekens, 20-Jul-2018.) |
Ref | Expression |
---|---|
wlkiswwlks2lem.f | ⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) |
Ref | Expression |
---|---|
wlkiswwlks2lem2 | ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 1))) → (𝐹‘𝐼) = (◡𝐸‘{(𝑃‘𝐼), (𝑃‘(𝐼 + 1))})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkiswwlks2lem.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) | |
2 | 1 | a1i 11 | . 2 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 1))) → 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))) |
3 | fveq2 6330 | . . . . 5 ⊢ (𝑥 = 𝐼 → (𝑃‘𝑥) = (𝑃‘𝐼)) | |
4 | fvoveq1 6814 | . . . . 5 ⊢ (𝑥 = 𝐼 → (𝑃‘(𝑥 + 1)) = (𝑃‘(𝐼 + 1))) | |
5 | 3, 4 | preq12d 4412 | . . . 4 ⊢ (𝑥 = 𝐼 → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃‘𝐼), (𝑃‘(𝐼 + 1))}) |
6 | 5 | fveq2d 6334 | . . 3 ⊢ (𝑥 = 𝐼 → (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}) = (◡𝐸‘{(𝑃‘𝐼), (𝑃‘(𝐼 + 1))})) |
7 | 6 | adantl 467 | . 2 ⊢ ((((♯‘𝑃) ∈ ℕ0 ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 1))) ∧ 𝑥 = 𝐼) → (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}) = (◡𝐸‘{(𝑃‘𝐼), (𝑃‘(𝐼 + 1))})) |
8 | simpr 471 | . 2 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 1))) → 𝐼 ∈ (0..^((♯‘𝑃) − 1))) | |
9 | fvexd 6342 | . 2 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 1))) → (◡𝐸‘{(𝑃‘𝐼), (𝑃‘(𝐼 + 1))}) ∈ V) | |
10 | 2, 7, 8, 9 | fvmptd 6428 | 1 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 1))) → (𝐹‘𝐼) = (◡𝐸‘{(𝑃‘𝐼), (𝑃‘(𝐼 + 1))})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 {cpr 4318 ↦ cmpt 4863 ◡ccnv 5248 ‘cfv 6029 (class class class)co 6791 0cc0 10136 1c1 10137 + caddc 10139 − cmin 10466 ℕ0cn0 11492 ..^cfzo 12666 ♯chash 13314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-iota 5992 df-fun 6031 df-fv 6037 df-ov 6794 |
This theorem is referenced by: wlkiswwlks2lem4 26999 |
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