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Mirrors > Home > MPE Home > Th. List > usgr2wspthon | Structured version Visualization version GIF version |
Description: A simple path of length 2 between two vertices corresponds to two adjacent edges in a simple graph. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-May-2021.) |
Ref | Expression |
---|---|
usgr2wspthon0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
usgr2wspthon0.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
usgr2wspthon | ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 ((𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrupgr 26292 | . . . 4 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph) | |
2 | 1 | adantr 466 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐺 ∈ UPGraph) |
3 | simpl 468 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
4 | 3 | adantl 467 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉) |
5 | simpr 471 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ 𝑉) | |
6 | 5 | adantl 467 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉) |
7 | usgr2wspthon0.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | 7 | elwspths2on 27101 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑇 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))) |
9 | 2, 4, 6, 8 | syl3anc 1476 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))) |
10 | simpl 468 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐺 ∈ USGraph) | |
11 | 10 | adantr 466 | . . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝐺 ∈ USGraph) |
12 | simplrl 762 | . . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
13 | simpr 471 | . . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉) | |
14 | simplrr 763 | . . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝐶 ∈ 𝑉) | |
15 | usgr2wspthon0.e | . . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺) | |
16 | 7, 15 | usgr2wspthons3 27106 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))) |
17 | 11, 12, 13, 14, 16 | syl13anc 1478 | . . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))) |
18 | 17 | anbi2d 614 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) ↔ (𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))) |
19 | anass 454 | . . . . 5 ⊢ (((𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)) ↔ (𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ (𝐴 ≠ 𝐶 ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))) | |
20 | 3anass 1080 | . . . . . . 7 ⊢ ((𝐴 ≠ 𝐶 ∧ {𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸) ↔ (𝐴 ≠ 𝐶 ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))) | |
21 | 20 | bicomi 214 | . . . . . 6 ⊢ ((𝐴 ≠ 𝐶 ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)) ↔ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)) |
22 | 21 | anbi2i 609 | . . . . 5 ⊢ ((𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ (𝐴 ≠ 𝐶 ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))) ↔ (𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))) |
23 | 19, 22 | bitri 264 | . . . 4 ⊢ (((𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)) ↔ (𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))) |
24 | 18, 23 | syl6bbr 278 | . . 3 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) ↔ ((𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))) |
25 | 24 | rexbidva 3197 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (∃𝑏 ∈ 𝑉 (𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) ↔ ∃𝑏 ∈ 𝑉 ((𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))) |
26 | 9, 25 | bitrd 268 | 1 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 ((𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∃wrex 3062 {cpr 4318 ‘cfv 6029 (class class class)co 6791 2c2 11270 〈“cs3 13789 Vtxcvtx 26088 Edgcedg 26153 UPGraphcupgr 26189 USGraphcusgr 26259 WSPathsNOn cwwspthsnon 26950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-ac2 9485 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-ifp 1050 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-2o 7712 df-oadd 7715 df-er 7894 df-map 8009 df-pm 8010 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-card 8963 df-ac 9137 df-cda 9190 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-nn 11221 df-2 11279 df-3 11280 df-n0 11493 df-xnn0 11564 df-z 11578 df-uz 11887 df-fz 12527 df-fzo 12667 df-hash 13315 df-word 13488 df-concat 13490 df-s1 13491 df-s2 13795 df-s3 13796 df-edg 26154 df-uhgr 26167 df-upgr 26191 df-umgr 26192 df-uspgr 26260 df-usgr 26261 df-wlks 26723 df-wlkson 26724 df-trls 26817 df-trlson 26818 df-pths 26840 df-spths 26841 df-pthson 26842 df-spthson 26843 df-wwlks 26951 df-wwlksn 26952 df-wwlksnon 26953 df-wspthsnon 26955 |
This theorem is referenced by: fusgr2wsp2nb 27509 |
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