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Mirrors > Home > MPE Home > Th. List > 2wlkond | Structured version Visualization version GIF version |
Description: A walk of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 30-Jan-2021.) (Revised by AV, 24-Mar-2021.) |
Ref | Expression |
---|---|
2wlkd.p | ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 |
2wlkd.f | ⊢ 𝐹 = 〈“𝐽𝐾”〉 |
2wlkd.s | ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
2wlkd.n | ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) |
2wlkd.e | ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) |
2wlkd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
2wlkd.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
2wlkond | ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2wlkd.p | . . 3 ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 | |
2 | 2wlkd.f | . . 3 ⊢ 𝐹 = 〈“𝐽𝐾”〉 | |
3 | 2wlkd.s | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
4 | 2wlkd.n | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) | |
5 | 2wlkd.e | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) | |
6 | 2wlkd.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
7 | 2wlkd.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
8 | 1, 2, 3, 4, 5, 6, 7 | 2wlkd 29819 | . 2 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
9 | 3 | simp1d 1139 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
10 | 1 | fveq1i 6897 | . . . 4 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶”〉‘0) |
11 | s3fv0 14878 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
12 | 10, 11 | eqtrid 2777 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑃‘0) = 𝐴) |
13 | 9, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝑃‘0) = 𝐴) |
14 | 2 | fveq2i 6899 | . . . . 5 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾”〉) |
15 | s2len 14876 | . . . . 5 ⊢ (♯‘〈“𝐽𝐾”〉) = 2 | |
16 | 14, 15 | eqtri 2753 | . . . 4 ⊢ (♯‘𝐹) = 2 |
17 | 1, 16 | fveq12i 6902 | . . 3 ⊢ (𝑃‘(♯‘𝐹)) = (〈“𝐴𝐵𝐶”〉‘2) |
18 | 3 | simp3d 1141 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
19 | s3fv2 14880 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) |
21 | 17, 20 | eqtrid 2777 | . 2 ⊢ (𝜑 → (𝑃‘(♯‘𝐹)) = 𝐶) |
22 | 3simpb 1146 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
23 | 3, 22 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
24 | s2cli 14867 | . . . . 5 ⊢ 〈“𝐽𝐾”〉 ∈ Word V | |
25 | 2, 24 | eqeltri 2821 | . . . 4 ⊢ 𝐹 ∈ Word V |
26 | s3cli 14868 | . . . . 5 ⊢ 〈“𝐴𝐵𝐶”〉 ∈ Word V | |
27 | 1, 26 | eqeltri 2821 | . . . 4 ⊢ 𝑃 ∈ Word V |
28 | 25, 27 | pm3.2i 469 | . . 3 ⊢ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V) |
29 | 6 | iswlkon 29543 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)) → (𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶))) |
30 | 23, 28, 29 | sylancl 584 | . 2 ⊢ (𝜑 → (𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶))) |
31 | 8, 13, 21, 30 | mpbir3and 1339 | 1 ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 Vcvv 3461 ⊆ wss 3944 {cpr 4632 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 0cc0 11140 2c2 12300 ♯chash 14325 Word cword 14500 〈“cs2 14828 〈“cs3 14829 Vtxcvtx 28881 iEdgciedg 28882 Walkscwlks 29482 WalksOncwlkson 29483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-hash 14326 df-word 14501 df-concat 14557 df-s1 14582 df-s2 14835 df-s3 14836 df-wlks 29485 df-wlkson 29486 |
This theorem is referenced by: 2trlond 29822 umgr2adedgwlkon 29829 |
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