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Mirrors > Home > MPE Home > Th. List > clwlkcompbp | Structured version Visualization version GIF version |
Description: Basic properties of the components of a closed walk. (Contributed by AV, 23-May-2022.) |
Ref | Expression |
---|---|
clwlkcompbp.1 | ⊢ 𝐹 = (1st ‘𝑊) |
clwlkcompbp.2 | ⊢ 𝑃 = (2nd ‘𝑊) |
Ref | Expression |
---|---|
clwlkcompbp | ⊢ (𝑊 ∈ (ClWalks‘𝐺) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwlkwlk 28670 | . . 3 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺)) | |
2 | wlkop 28523 | . . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) |
4 | eleq1 2825 | . . . 4 ⊢ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 → (𝑊 ∈ (ClWalks‘𝐺) ↔ 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∈ (ClWalks‘𝐺))) | |
5 | df-br 5106 | . . . 4 ⊢ ((1st ‘𝑊)(ClWalks‘𝐺)(2nd ‘𝑊) ↔ 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∈ (ClWalks‘𝐺)) | |
6 | 4, 5 | bitr4di 288 | . . 3 ⊢ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 → (𝑊 ∈ (ClWalks‘𝐺) ↔ (1st ‘𝑊)(ClWalks‘𝐺)(2nd ‘𝑊))) |
7 | isclwlk 28668 | . . . 4 ⊢ ((1st ‘𝑊)(ClWalks‘𝐺)(2nd ‘𝑊) ↔ ((1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊) ∧ ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘(♯‘(1st ‘𝑊))))) | |
8 | clwlkcompbp.1 | . . . . . 6 ⊢ 𝐹 = (1st ‘𝑊) | |
9 | clwlkcompbp.2 | . . . . . 6 ⊢ 𝑃 = (2nd ‘𝑊) | |
10 | 8, 9 | breq12i 5114 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 ↔ (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊)) |
11 | 9 | fveq1i 6843 | . . . . . 6 ⊢ (𝑃‘0) = ((2nd ‘𝑊)‘0) |
12 | 8 | fveq2i 6845 | . . . . . . 7 ⊢ (♯‘𝐹) = (♯‘(1st ‘𝑊)) |
13 | 9, 12 | fveq12i 6848 | . . . . . 6 ⊢ (𝑃‘(♯‘𝐹)) = ((2nd ‘𝑊)‘(♯‘(1st ‘𝑊))) |
14 | 11, 13 | eqeq12i 2754 | . . . . 5 ⊢ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘(♯‘(1st ‘𝑊)))) |
15 | 10, 14 | anbi12i 627 | . . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) ↔ ((1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊) ∧ ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘(♯‘(1st ‘𝑊))))) |
16 | 7, 15 | sylbb2 237 | . . 3 ⊢ ((1st ‘𝑊)(ClWalks‘𝐺)(2nd ‘𝑊) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) |
17 | 6, 16 | syl6bi 252 | . 2 ⊢ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 → (𝑊 ∈ (ClWalks‘𝐺) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))) |
18 | 3, 17 | mpcom 38 | 1 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 〈cop 4592 class class class wbr 5105 ‘cfv 6496 1st c1st 7918 2nd c2nd 7919 0cc0 11050 ♯chash 14229 Walkscwlks 28491 ClWalkscclwlks 28665 |
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-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-ral 3065 df-rex 3074 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-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 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-iota 6448 df-fun 6498 df-fv 6504 df-1st 7920 df-2nd 7921 df-wlks 28494 df-clwlks 28666 |
This theorem is referenced by: (None) |
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