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Theorem clwlkcompbp 28677
Description: Basic properties of the components of a closed walk. (Contributed by AV, 23-May-2022.)
Hypotheses
Ref Expression
clwlkcompbp.1 𝐹 = (1st𝑊)
clwlkcompbp.2 𝑃 = (2nd𝑊)
Assertion
Ref Expression
clwlkcompbp (𝑊 ∈ (ClWalks‘𝐺) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Proof of Theorem clwlkcompbp
StepHypRef Expression
1 clwlkwlk 28670 . . 3 (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺))
2 wlkop 28523 . . 3 (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
31, 2syl 17 . 2 (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
4 eleq1 2825 . . . 4 (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) ↔ ⟨(1st𝑊), (2nd𝑊)⟩ ∈ (ClWalks‘𝐺)))
5 df-br 5106 . . . 4 ((1st𝑊)(ClWalks‘𝐺)(2nd𝑊) ↔ ⟨(1st𝑊), (2nd𝑊)⟩ ∈ (ClWalks‘𝐺))
64, 5bitr4di 288 . . 3 (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) ↔ (1st𝑊)(ClWalks‘𝐺)(2nd𝑊)))
7 isclwlk 28668 . . . 4 ((1st𝑊)(ClWalks‘𝐺)(2nd𝑊) ↔ ((1st𝑊)(Walks‘𝐺)(2nd𝑊) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(♯‘(1st𝑊)))))
8 clwlkcompbp.1 . . . . . 6 𝐹 = (1st𝑊)
9 clwlkcompbp.2 . . . . . 6 𝑃 = (2nd𝑊)
108, 9breq12i 5114 . . . . 5 (𝐹(Walks‘𝐺)𝑃 ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊))
119fveq1i 6843 . . . . . 6 (𝑃‘0) = ((2nd𝑊)‘0)
128fveq2i 6845 . . . . . . 7 (♯‘𝐹) = (♯‘(1st𝑊))
139, 12fveq12i 6848 . . . . . 6 (𝑃‘(♯‘𝐹)) = ((2nd𝑊)‘(♯‘(1st𝑊)))
1411, 13eqeq12i 2754 . . . . 5 ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ ((2nd𝑊)‘0) = ((2nd𝑊)‘(♯‘(1st𝑊))))
1510, 14anbi12i 627 . . . 4 ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) ↔ ((1st𝑊)(Walks‘𝐺)(2nd𝑊) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(♯‘(1st𝑊)))))
167, 15sylbb2 237 . . 3 ((1st𝑊)(ClWalks‘𝐺)(2nd𝑊) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
176, 16syl6bi 252 . 2 (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))))
183, 17mpcom 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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