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Theorem clwlkcompbp 29875

Description: Basic properties of the components of a closed walk. (Contributed by AV, 23-May-2022.)

**Hypotheses**
Ref	Expression
clwlkcompbp.1	⊢ 𝐹 = (1^st ‘𝑊)
clwlkcompbp.2	⊢ 𝑃 = (2^nd ‘𝑊)

**Assertion**
Ref	Expression
clwlkcompbp	⊢ (𝑊 ∈ (ClWalks‘𝐺) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

**Proof of Theorem clwlkcompbp**
Step	Hyp	Ref	Expression
1		clwlkwlk 29868	. . 3 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺))
2		wlkop 29721	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
3	1, 2	syl 17	. 2 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
4		eleq1 2828	. . . 4 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ (ClWalks‘𝐺)))
5		df-br 5080	. . . 4 ⊢ ((1^st ‘𝑊)(ClWalks‘𝐺)(2^nd ‘𝑊) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ (ClWalks‘𝐺))
6	4, 5	bitr4di 290	. . 3 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) ↔ (1^st ‘𝑊)(ClWalks‘𝐺)(2^nd ‘𝑊)))
7		isclwlk 29866	. . . 4 ⊢ ((1^st ‘𝑊)(ClWalks‘𝐺)(2^nd ‘𝑊) ↔ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) ∧ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊)))))
8		clwlkcompbp.1	. . . . . 6 ⊢ 𝐹 = (1^st ‘𝑊)
9		clwlkcompbp.2	. . . . . 6 ⊢ 𝑃 = (2^nd ‘𝑊)
10	8, 9	breq12i 5088	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
11	9	fveq1i 6835	. . . . . 6 ⊢ (𝑃‘0) = ((2^nd ‘𝑊)‘0)
12	8	fveq2i 6837	. . . . . . 7 ⊢ (♯‘𝐹) = (♯‘(1^st ‘𝑊))
13	9, 12	fveq12i 6840	. . . . . 6 ⊢ (𝑃‘(♯‘𝐹)) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊)))
14	11, 13	eqeq12i 2758	. . . . 5 ⊢ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊))))
15	10, 14	anbi12i 634	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) ↔ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) ∧ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊)))))
16	7, 15	sylbb2 239	. . 3 ⊢ ((1^st ‘𝑊)(ClWalks‘𝐺)(2^nd ‘𝑊) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
17	6, 16	biimtrdi 254	. 2 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘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 1547 ∈ wcel 2119 ⟨cop 4568 class class class wbr 5079 ‘cfv 6492 1^st c1st 7936 2^nd c2nd 7937 0cc0 11036 ♯chash 14290 Walkscwlks 29690 ClWalkscclwlks 29863

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pr 5369 ax-un 7685

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6448 df-fun 6494 df-fv 6500 df-1st 7938 df-2nd 7939 df-wlks 29693 df-clwlks 29864

This theorem is referenced by: (None)

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