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Theorem clwlkcompbp 30374

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 30367	. . 3 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺))
2		wlkop 30213	. . 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 2848	. . . 4 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ (ClWalks‘𝐺)))
5		df-br 5103	. . . 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 30365	. . . 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 5111	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
11	9	fveq1i 6886	. . . . . 6 ⊢ (𝑃‘0) = ((2^nd ‘𝑊)‘0)
12	8	fveq2i 6888	. . . . . . 7 ⊢ (♯‘𝐹) = (♯‘(1^st ‘𝑊))
13	9, 12	fveq12i 6891	. . . . . 6 ⊢ (𝑃‘(♯‘𝐹)) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊)))
14	11, 13	eqeq12i 2778	. . . . 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 1569 ∈ wcel 2144 ⟨cop 4834 class class class wbr 5102 ‘cfv 6538 1^st c1st 7999 2^nd c2nd 8000 0cc0 11203 ♯chash 14477 Walkscwlks 30182 ClWalkscclwlks 30362

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fv 6546 df-1st 8001 df-2nd 8002 df-wlks 30185 df-clwlks 30363

This theorem is used by: (None)

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