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Mirrors > Home > MPE Home > Th. List > usgr2wspthons3 | Structured version Visualization version GIF version |
Description: A simple path of length 2 between two vertices represented as length 3 string corresponds to two adjacent edges in a simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 16-Mar-2022.) |
Ref | Expression |
---|---|
usgr2wspthon0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
usgr2wspthon0.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
usgr2wspthons3 | ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn 11711 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
2 | ne0i 4300 | . . . . . . 7 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (𝐴(2 WSPathsNOn 𝐺)𝐶) ≠ ∅) | |
3 | wspthsnonn0vne 27696 | . . . . . . 7 ⊢ ((2 ∈ ℕ ∧ (𝐴(2 WSPathsNOn 𝐺)𝐶) ≠ ∅) → 𝐴 ≠ 𝐶) | |
4 | 1, 2, 3 | sylancr 589 | . . . . . 6 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → 𝐴 ≠ 𝐶) |
5 | simplr 767 | . . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐶) ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → 𝐴 ≠ 𝐶) | |
6 | wpthswwlks2on 27740 | . . . . . . . . . . 11 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐶) → (𝐴(2 WSPathsNOn 𝐺)𝐶) = (𝐴(2 WWalksNOn 𝐺)𝐶)) | |
7 | 6 | eleq2d 2898 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐶) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) |
8 | 7 | biimpa 479 | . . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐶) ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) |
9 | 5, 8 | jca 514 | . . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐶) ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → (𝐴 ≠ 𝐶 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) |
10 | 9 | exp31 422 | . . . . . . 7 ⊢ (𝐺 ∈ USGraph → (𝐴 ≠ 𝐶 → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (𝐴 ≠ 𝐶 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))) |
11 | 10 | com13 88 | . . . . . 6 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (𝐴 ≠ 𝐶 → (𝐺 ∈ USGraph → (𝐴 ≠ 𝐶 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))) |
12 | 4, 11 | mpd 15 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (𝐺 ∈ USGraph → (𝐴 ≠ 𝐶 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))) |
13 | 12 | com12 32 | . . . 4 ⊢ (𝐺 ∈ USGraph → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (𝐴 ≠ 𝐶 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))) |
14 | 7 | biimprd 250 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐶) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))) |
15 | 14 | expimpd 456 | . . . 4 ⊢ (𝐺 ∈ USGraph → ((𝐴 ≠ 𝐶 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))) |
16 | 13, 15 | impbid 214 | . . 3 ⊢ (𝐺 ∈ USGraph → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝐴 ≠ 𝐶 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))) |
17 | 16 | adantr 483 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝐴 ≠ 𝐶 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))) |
18 | usgrumgr 26964 | . . . . 5 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph) | |
19 | usgr2wspthon0.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
20 | usgr2wspthon0.e | . . . . . 6 ⊢ 𝐸 = (Edg‘𝐺) | |
21 | 19, 20 | umgrwwlks2on 27736 | . . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))) |
22 | 18, 21 | sylan 582 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))) |
23 | 22 | anbi2d 630 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 ≠ 𝐶 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝐴 ≠ 𝐶 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))) |
24 | 3anass 1091 | . . 3 ⊢ ((𝐴 ≠ 𝐶 ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (𝐴 ≠ 𝐶 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))) | |
25 | 23, 24 | syl6bbr 291 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 ≠ 𝐶 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))) |
26 | 17, 25 | bitrd 281 | 1 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 {cpr 4569 ‘cfv 6355 (class class class)co 7156 ℕcn 11638 2c2 11693 〈“cs3 14204 Vtxcvtx 26781 Edgcedg 26832 UMGraphcumgr 26866 USGraphcusgr 26934 WWalksNOn cwwlksnon 27605 WSPathsNOn cwwspthsnon 27607 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-ac2 9885 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-ac 9542 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 df-s3 14211 df-edg 26833 df-uhgr 26843 df-upgr 26867 df-umgr 26868 df-uspgr 26935 df-usgr 26936 df-wlks 27381 df-wlkson 27382 df-trls 27474 df-trlson 27475 df-pths 27497 df-spths 27498 df-pthson 27499 df-spthson 27500 df-wwlks 27608 df-wwlksn 27609 df-wwlksnon 27610 df-wspthsnon 27612 |
This theorem is referenced by: usgr2wspthon 27744 |
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