Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > av-numclwlk2lem2fv | Structured version Visualization version GIF version |
Description: Value of the function R. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) |
Ref | Expression |
---|---|
av-numclwwlk.v | ⊢ 𝑉 = (Vtx‘𝐺) |
av-numclwwlk.q | ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) |
av-numclwwlk.f | ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣}) |
av-numclwwlk.h | ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) |
av-numclwwlk.r | ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr 〈0, (𝑁 + 1)〉)) |
Ref | Expression |
---|---|
av-numclwlk2lem2fv | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑊) = (𝑊 substr 〈0, (𝑁 + 1)〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | av-numclwwlk.r | . . . 4 ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr 〈0, (𝑁 + 1)〉)) | |
2 | 1 | a1i 11 | . . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr 〈0, (𝑁 + 1)〉))) |
3 | oveq1 6556 | . . . 4 ⊢ (𝑥 = 𝑊 → (𝑥 substr 〈0, (𝑁 + 1)〉) = (𝑊 substr 〈0, (𝑁 + 1)〉)) | |
4 | 3 | adantl 481 | . . 3 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) ∧ 𝑥 = 𝑊) → (𝑥 substr 〈0, (𝑁 + 1)〉) = (𝑊 substr 〈0, (𝑁 + 1)〉)) |
5 | simpr 476 | . . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) | |
6 | ovex 6577 | . . . 4 ⊢ (𝑊 substr 〈0, (𝑁 + 1)〉) ∈ V | |
7 | 6 | a1i 11 | . . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑊 substr 〈0, (𝑁 + 1)〉) ∈ V) |
8 | 2, 4, 5, 7 | fvmptd 6197 | . 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑅‘𝑊) = (𝑊 substr 〈0, (𝑁 + 1)〉)) |
9 | 8 | ex 449 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑊) = (𝑊 substr 〈0, (𝑁 + 1)〉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 〈cop 4131 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 0cc0 9815 1c1 9816 + caddc 9818 − cmin 10145 ℕcn 10897 2c2 10947 lastS clsw 13147 substr csubstr 13150 Vtxcvtx 25673 WWalkSN cwwlksn 41029 ClWWalkSN cclwwlksn 41184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 |
This theorem is referenced by: av-numclwlk2lem2f1o 41535 |
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