clwlkcompbp - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > clwlkcompbp		Structured version Visualization version GIF version

Theorem clwlkcompbp 29802

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 29795	. . 3 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺))
2		wlkop 29646	. . 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 2829	. . . 4 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ (ClWalks‘𝐺)))
5		df-br 5144	. . . 4 ⊢ ((1^st ‘𝑊)(ClWalks‘𝐺)(2^nd ‘𝑊) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ (ClWalks‘𝐺))
6	4, 5	bitr4di 289	. . 3 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) ↔ (1^st ‘𝑊)(ClWalks‘𝐺)(2^nd ‘𝑊)))
7		isclwlk 29793	. . . 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 5152	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
11	9	fveq1i 6907	. . . . . 6 ⊢ (𝑃‘0) = ((2^nd ‘𝑊)‘0)
12	8	fveq2i 6909	. . . . . . 7 ⊢ (♯‘𝐹) = (♯‘(1^st ‘𝑊))
13	9, 12	fveq12i 6912	. . . . . 6 ⊢ (𝑃‘(♯‘𝐹)) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊)))
14	11, 13	eqeq12i 2755	. . . . 5 ⊢ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊))))
15	10, 14	anbi12i 628	. . . 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 238	. . 3 ⊢ ((1^st ‘𝑊)(ClWalks‘𝐺)(2^nd ‘𝑊) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
17	6, 16	biimtrdi 253	. 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 395 = wceq 1540 ∈ wcel 2108 ⟨cop 4632 class class class wbr 5143 ‘cfv 6561 1^st c1st 8012 2^nd c2nd 8013 0cc0 11155 ♯chash 14369 Walkscwlks 29614 ClWalkscclwlks 29790

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015 df-wlks 29617 df-clwlks 29791

This theorem is referenced by: (None)

W3C validator