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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 26975 | . . . 4 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph) | |
2 | 1 | adantr 484 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐺 ∈ UPGraph) |
3 | simpl 486 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
4 | 3 | adantl 485 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉) |
5 | simpr 488 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ 𝑉) | |
6 | 5 | adantl 485 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉) |
7 | usgr2wspthon0.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | 7 | elwspths2on 27746 | . . 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 1368 | . 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 486 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐺 ∈ USGraph) | |
11 | 10 | adantr 484 | . . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝐺 ∈ USGraph) |
12 | simplrl 776 | . . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
13 | simpr 488 | . . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉) | |
14 | simplrr 777 | . . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝐶 ∈ 𝑉) | |
15 | usgr2wspthon0.e | . . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺) | |
16 | 7, 15 | usgr2wspthons3 27750 | . . . . . 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 1369 | . . . . 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 631 | . . . 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 472 | . . . . 5 ⊢ (((𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)) ↔ (𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ (𝐴 ≠ 𝐶 ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))) | |
20 | 3anass 1092 | . . . . . . 7 ⊢ ((𝐴 ≠ 𝐶 ∧ {𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸) ↔ (𝐴 ≠ 𝐶 ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))) | |
21 | 20 | bicomi 227 | . . . . . 6 ⊢ ((𝐴 ≠ 𝐶 ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)) ↔ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)) |
22 | 21 | anbi2i 625 | . . . . 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 278 | . . . 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 292 | . . 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 3255 | . 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 282 | 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 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 {cpr 4527 ‘cfv 6324 (class class class)co 7135 2c2 11680 〈“cs3 14195 Vtxcvtx 26789 Edgcedg 26840 UPGraphcupgr 26873 USGraphcusgr 26942 WSPathsNOn cwwspthsnon 27615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-ac2 9874 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ifp 1059 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-ac 9527 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-edg 26841 df-uhgr 26851 df-upgr 26875 df-umgr 26876 df-uspgr 26943 df-usgr 26944 df-wlks 27389 df-wlkson 27390 df-trls 27482 df-trlson 27483 df-pths 27505 df-spths 27506 df-pthson 27507 df-spthson 27508 df-wwlks 27616 df-wwlksn 27617 df-wwlksnon 27618 df-wspthsnon 27620 |
This theorem is referenced by: fusgr2wsp2nb 28119 |
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