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Mirrors > Home > MPE Home > Th. List > isspthonpth | Structured version Visualization version GIF version |
Description: A pair of functions is a simple path between two given vertices iff it is a simple path starting and ending at the two vertices. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-Jan-2021.) |
Ref | Expression |
---|---|
isspthonpth.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
isspthonpth | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(SPaths‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isspthonpth.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | isspthson 27523 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(SPaths‘𝐺)𝑃))) |
3 | 1 | istrlson 27487 | . . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃))) |
4 | 3 | adantr 483 | . . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) ∧ 𝐹(SPaths‘𝐺)𝑃) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃))) |
5 | spthispth 27506 | . . . . . . . . 9 ⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃) | |
6 | pthistrl 27505 | . . . . . . . . 9 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) |
8 | 7 | adantl 484 | . . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) ∧ 𝐹(SPaths‘𝐺)𝑃) → 𝐹(Trails‘𝐺)𝑃) |
9 | 8 | biantrud 534 | . . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) ∧ 𝐹(SPaths‘𝐺)𝑃) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃))) |
10 | spthiswlk 27508 | . . . . . . . 8 ⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
11 | 10 | adantl 484 | . . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) ∧ 𝐹(SPaths‘𝐺)𝑃) → 𝐹(Walks‘𝐺)𝑃) |
12 | 1 | iswlkon 27438 | . . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵))) |
13 | 3anass 1091 | . . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵))) | |
14 | 12, 13 | syl6bb 289 | . . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))) |
15 | 14 | adantr 483 | . . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) ∧ 𝐹(SPaths‘𝐺)𝑃) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))) |
16 | 11, 15 | mpbirand 705 | . . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) ∧ 𝐹(SPaths‘𝐺)𝑃) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵))) |
17 | 4, 9, 16 | 3bitr2d 309 | . . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) ∧ 𝐹(SPaths‘𝐺)𝑃) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵))) |
18 | 17 | ex 415 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → (𝐹(SPaths‘𝐺)𝑃 → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))) |
19 | 18 | pm5.32rd 580 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → ((𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(SPaths‘𝐺)𝑃) ↔ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ∧ 𝐹(SPaths‘𝐺)𝑃))) |
20 | 3anass 1091 | . . . 4 ⊢ ((𝐹(SPaths‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ↔ (𝐹(SPaths‘𝐺)𝑃 ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵))) | |
21 | ancom 463 | . . . 4 ⊢ ((𝐹(SPaths‘𝐺)𝑃 ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)) ↔ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ∧ 𝐹(SPaths‘𝐺)𝑃)) | |
22 | 20, 21 | bitr2i 278 | . . 3 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ∧ 𝐹(SPaths‘𝐺)𝑃) ↔ (𝐹(SPaths‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)) |
23 | 19, 22 | syl6bb 289 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → ((𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(SPaths‘𝐺)𝑃) ↔ (𝐹(SPaths‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵))) |
24 | 2, 23 | bitrd 281 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(SPaths‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 0cc0 10536 ♯chash 13689 Vtxcvtx 26780 Walkscwlks 27377 WalksOncwlkson 27378 Trailsctrls 27471 TrailsOnctrlson 27472 Pathscpths 27492 SPathscspths 27493 SPathsOncspthson 27495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
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 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-wlks 27380 df-wlkson 27381 df-trls 27473 df-trlson 27474 df-pths 27496 df-spths 27497 df-spthson 27499 |
This theorem is referenced by: uhgrwkspth 27535 usgr2wlkspth 27539 wspthsnwspthsnon 27694 elwspths2spth 27745 |
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