Theorem crctisclwlk 29862

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 29860	. 2 ⊢ (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2		trliswlk 29764	. . 3 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3		isclwlk 29841	. . . 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 581	. 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 1542   class class class wbr 5085  ‘cfv 6498  0cc0 11038  ♯chash 14292  Walkscwlks 29665  Trailsctrls 29757  ClWalkscclwlks 29838  Circuitsccrcts 29852

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fv 6506  df-wlks 29668  df-trls 29759  df-clwlks 29839  df-crcts 29854

This theorem is referenced by: (None)