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Theorem clwlkcompbp 30168

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 30161	. . 3 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺))
2		wlkop 30014	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
3	1, 2	syl 18	. 2 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
4		eleq1 2854	. . . 4 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ (ClWalks‘𝐺)))
5		df-br 5115	. . . 4 ⊢ ((1^st ‘𝑊)(ClWalks‘𝐺)(2^nd ‘𝑊) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ (ClWalks‘𝐺))
6	4, 5	bitr4di 292	. . 3 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) ↔ (1^st ‘𝑊)(ClWalks‘𝐺)(2^nd ‘𝑊)))
7		isclwlk 30159	. . . 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 5123	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
11	9	fveq1i 6889	. . . . . 6 ⊢ (𝑃‘0) = ((2^nd ‘𝑊)‘0)
12	8	fveq2i 6891	. . . . . . 7 ⊢ (♯‘𝐹) = (♯‘(1^st ‘𝑊))
13	9, 12	fveq12i 6894	. . . . . 6 ⊢ (𝑃‘(♯‘𝐹)) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊)))
14	11, 13	eqeq12i 2784	. . . . 5 ⊢ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊))))
15	10, 14	anbi12i 640	. . . 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 241	. . 3 ⊢ ((1^st ‘𝑊)(ClWalks‘𝐺)(2^nd ‘𝑊) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
17	6, 16	biimtrdi 256	. 2 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))))
18	3, 17	mpcom 39	1 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ⟨cop 4600 class class class wbr 5114 ‘cfv 6543 1^st c1st 7993 2^nd c2nd 7994 0cc0 11118 ♯chash 14386 Walkscwlks 29983 ClWalkscclwlks 30156

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pr 5409 ax-un 7745

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6499 df-fun 6545 df-fv 6551 df-1st 7995 df-2nd 7996 df-wlks 29986 df-clwlks 30157

This theorem is used by: (None)

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