Theorem crctisclwlk 29814

Description: A circuit is a closed walk. (Contributed by AV, 17-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)

Assertion

Ref	Expression
crctisclwlk	⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(ClWalks‘𝐺)𝑃)

Proof of Theorem crctisclwlk

Step	Hyp	Ref	Expression
1		crctprop 29812	. 2 ⊢ (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2		trliswlk 29715	. . 3 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3		isclwlk 29793	. . . 4 ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
4	3	biimpri 228	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → 𝐹(ClWalks‘𝐺)𝑃)
5	2, 4	sylan 580	. 2 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → 𝐹(ClWalks‘𝐺)𝑃)
6	1, 5	syl 17	1 ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(ClWalks‘𝐺)𝑃)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   class class class wbr 5143  ‘cfv 6561  0cc0 11155  ♯chash 14369  Walkscwlks 29614  Trailsctrls 29708  ClWalkscclwlks 29790  Circuitsccrcts 29804

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

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-wlks 29617  df-trls 29710  df-clwlks 29791  df-crcts 29806

This theorem is referenced by: (None)