Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > clwlksfclwwlk1hashn | Structured version Visualization version GIF version |
Description: The size of the first component of a closed walk. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 2-May-2021.) |
Ref | Expression |
---|---|
clwlksfclwwlk.1 | ⊢ 𝐴 = (1st ‘𝑐) |
clwlksfclwwlk.2 | ⊢ 𝐵 = (2nd ‘𝑐) |
clwlksfclwwlk.c | ⊢ 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁} |
clwlksfclwwlk.f | ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr 〈0, (#‘𝐴)〉)) |
Ref | Expression |
---|---|
clwlksfclwwlk1hashn | ⊢ (𝑊 ∈ 𝐶 → (#‘(1st ‘𝑊)) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwlksfclwwlk.1 | . . . . . 6 ⊢ 𝐴 = (1st ‘𝑐) | |
2 | 1 | fveq2i 6106 | . . . . 5 ⊢ (#‘𝐴) = (#‘(1st ‘𝑐)) |
3 | 2 | eqeq1i 2615 | . . . 4 ⊢ ((#‘𝐴) = 𝑁 ↔ (#‘(1st ‘𝑐)) = 𝑁) |
4 | fveq2 6103 | . . . . . 6 ⊢ (𝑐 = 𝑊 → (1st ‘𝑐) = (1st ‘𝑊)) | |
5 | 4 | fveq2d 6107 | . . . . 5 ⊢ (𝑐 = 𝑊 → (#‘(1st ‘𝑐)) = (#‘(1st ‘𝑊))) |
6 | 5 | eqeq1d 2612 | . . . 4 ⊢ (𝑐 = 𝑊 → ((#‘(1st ‘𝑐)) = 𝑁 ↔ (#‘(1st ‘𝑊)) = 𝑁)) |
7 | 3, 6 | syl5bb 271 | . . 3 ⊢ (𝑐 = 𝑊 → ((#‘𝐴) = 𝑁 ↔ (#‘(1st ‘𝑊)) = 𝑁)) |
8 | clwlksfclwwlk.c | . . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁} | |
9 | 7, 8 | elrab2 3333 | . 2 ⊢ (𝑊 ∈ 𝐶 ↔ (𝑊 ∈ (ClWalkS‘𝐺) ∧ (#‘(1st ‘𝑊)) = 𝑁)) |
10 | 9 | simprbi 479 | 1 ⊢ (𝑊 ∈ 𝐶 → (#‘(1st ‘𝑊)) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 〈cop 4131 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 0cc0 9815 #chash 12979 substr csubstr 13150 ClWalkScclwlks 40976 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 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-iota 5768 df-fv 5812 |
This theorem is referenced by: clwlksf1clwwlklem 41275 clwlksf1clwwlk 41276 |
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