![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > wlkon2n0 | Structured version Visualization version GIF version |
Description: The length of a walk between two different vertices is not 0 (i.e. is at least 1). (Contributed by AV, 3-Apr-2021.) |
Ref | Expression |
---|---|
wlkon2n0 | ⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐴 ≠ 𝐵) → (♯‘𝐹) ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | 1 | wlkonprop 29348 | . . . 4 ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵))) |
3 | fveqeq2 6900 | . . . . . . . . 9 ⊢ ((♯‘𝐹) = 0 → ((𝑃‘(♯‘𝐹)) = 𝐵 ↔ (𝑃‘0) = 𝐵)) | |
4 | 3 | anbi2d 628 | . . . . . . . 8 ⊢ ((♯‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ↔ ((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵))) |
5 | eqtr2 2755 | . . . . . . . . 9 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵) → 𝐴 = 𝐵) | |
6 | nne 2943 | . . . . . . . . 9 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵) | |
7 | 5, 6 | sylibr 233 | . . . . . . . 8 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵) → ¬ 𝐴 ≠ 𝐵) |
8 | 4, 7 | syl6bi 253 | . . . . . . 7 ⊢ ((♯‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) → ¬ 𝐴 ≠ 𝐵)) |
9 | 8 | com12 32 | . . . . . 6 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) → ((♯‘𝐹) = 0 → ¬ 𝐴 ≠ 𝐵)) |
10 | 9 | 3adant1 1129 | . . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) → ((♯‘𝐹) = 0 → ¬ 𝐴 ≠ 𝐵)) |
11 | 10 | 3ad2ant3 1134 | . . . 4 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)) → ((♯‘𝐹) = 0 → ¬ 𝐴 ≠ 𝐵)) |
12 | 2, 11 | syl 17 | . . 3 ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((♯‘𝐹) = 0 → ¬ 𝐴 ≠ 𝐵)) |
13 | 12 | necon2ad 2954 | . 2 ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → (𝐴 ≠ 𝐵 → (♯‘𝐹) ≠ 0)) |
14 | 13 | imp 406 | 1 ⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐴 ≠ 𝐵) → (♯‘𝐹) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 Vcvv 3473 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 0cc0 11116 ♯chash 14297 Vtxcvtx 28689 Walkscwlks 29286 WalksOncwlkson 29287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-wlkson 29290 |
This theorem is referenced by: conngrv2edg 29881 |
Copyright terms: Public domain | W3C validator |