Theorem crctisclwlk 30180

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 30178	. 2 ⊢ (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2		trliswlk 30082	. . 3 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3		isclwlk 30159	. . . 4 ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
4	3	biimpri 231	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → 𝐹(ClWalks‘𝐺)𝑃)
5	2, 4	sylan 592	. 2 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → 𝐹(ClWalks‘𝐺)𝑃)
6	1, 5	syl 18	1 ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(ClWalks‘𝐺)𝑃)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   class class class wbr 5114  ‘cfv 6543  0cc0 11118  ♯chash 14386  Walkscwlks 29983  Trailsctrls 30075  ClWalkscclwlks 30156  Circuitsccrcts 30170

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fv 6551  df-wlks 29986  df-trls 30077  df-clwlks 30157  df-crcts 30172

This theorem is used by: (None)