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| Mirrors > Home > MPE Home > Th. List > crctisclwlk | Structured version Visualization version GIF version | ||
| Description: A circuit is a closed walk. (Contributed by AV, 17-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| Ref | Expression |
|---|---|
| crctisclwlk | ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(ClWalks‘𝐺)𝑃) |
| 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) |
| Colors of variables: wff setvar class |
| 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) |
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