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Mirrors > Home > MPE Home > Th. List > usgr2trlspth | Structured version Visualization version GIF version |
Description: In a simple graph, any trail of length 2 is a simple path. (Contributed by AV, 5-Jun-2021.) |
Ref | Expression |
---|---|
usgr2trlspth | ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 ↔ 𝐹(SPaths‘𝐺)𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgr2trlncl 29449 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2))) | |
2 | 1 | imp 406 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) ∧ 𝐹(Trails‘𝐺)𝑃) → (𝑃‘0) ≠ (𝑃‘2)) |
3 | trliswlk 29386 | . . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
4 | wlkonwlk 29351 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃) | |
5 | simpll 764 | . . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) ∧ (𝑃‘0) ≠ (𝑃‘2)) → 𝐺 ∈ USGraph) | |
6 | simplr 766 | . . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) ∧ (𝑃‘0) ≠ (𝑃‘2)) → (♯‘𝐹) = 2) | |
7 | fveq2 6891 | . . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = (𝑃‘2)) | |
8 | 7 | eqcomd 2737 | . . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) = 2 → (𝑃‘2) = (𝑃‘(♯‘𝐹))) |
9 | 8 | neeq2d 3000 | . . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 2 → ((𝑃‘0) ≠ (𝑃‘2) ↔ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) |
10 | 9 | biimpd 228 | . . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 2 → ((𝑃‘0) ≠ (𝑃‘2) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) |
11 | 10 | adantl 481 | . . . . . . . . . . 11 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ((𝑃‘0) ≠ (𝑃‘2) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) |
12 | 11 | imp 406 | . . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) ∧ (𝑃‘0) ≠ (𝑃‘2)) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) |
13 | usgr2wlkspth 29448 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → (𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃 ↔ 𝐹((𝑃‘0)(SPathsOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃)) | |
14 | 5, 6, 12, 13 | syl3anc 1370 | . . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) ∧ (𝑃‘0) ≠ (𝑃‘2)) → (𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃 ↔ 𝐹((𝑃‘0)(SPathsOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃)) |
15 | spthonisspth 29439 | . . . . . . . . 9 ⊢ (𝐹((𝑃‘0)(SPathsOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃 → 𝐹(SPaths‘𝐺)𝑃) | |
16 | 14, 15 | syl6bi 253 | . . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) ∧ (𝑃‘0) ≠ (𝑃‘2)) → (𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃 → 𝐹(SPaths‘𝐺)𝑃)) |
17 | 16 | expcom 413 | . . . . . . 7 ⊢ ((𝑃‘0) ≠ (𝑃‘2) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃 → 𝐹(SPaths‘𝐺)𝑃))) |
18 | 17 | com13 88 | . . . . . 6 ⊢ (𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ((𝑃‘0) ≠ (𝑃‘2) → 𝐹(SPaths‘𝐺)𝑃))) |
19 | 3, 4, 18 | 3syl 18 | . . . . 5 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ((𝑃‘0) ≠ (𝑃‘2) → 𝐹(SPaths‘𝐺)𝑃))) |
20 | 19 | impcom 407 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) ∧ 𝐹(Trails‘𝐺)𝑃) → ((𝑃‘0) ≠ (𝑃‘2) → 𝐹(SPaths‘𝐺)𝑃)) |
21 | 2, 20 | mpd 15 | . . 3 ⊢ (((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) ∧ 𝐹(Trails‘𝐺)𝑃) → 𝐹(SPaths‘𝐺)𝑃) |
22 | 21 | ex 412 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → 𝐹(SPaths‘𝐺)𝑃)) |
23 | spthispth 29415 | . . 3 ⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃) | |
24 | pthistrl 29414 | . . 3 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | |
25 | 23, 24 | syl 17 | . 2 ⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) |
26 | 22, 25 | impbid1 224 | 1 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 ↔ 𝐹(SPaths‘𝐺)𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 0cc0 11116 2c2 12274 ♯chash 14297 USGraphcusgr 28841 Walkscwlks 29285 WalksOncwlkson 29286 Trailsctrls 29379 Pathscpths 29401 SPathscspths 29402 SPathsOncspthson 29404 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-oadd 8476 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-dju 9902 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-xnn0 12552 df-z 12566 df-uz 12830 df-fz 13492 df-fzo 13635 df-hash 14298 df-word 14472 df-concat 14528 df-s1 14553 df-s2 14806 df-s3 14807 df-edg 28740 df-uhgr 28750 df-upgr 28774 df-umgr 28775 df-uspgr 28842 df-usgr 28843 df-wlks 29288 df-wlkson 29289 df-trls 29381 df-trlson 29382 df-pths 29405 df-spths 29406 df-pthson 29407 df-spthson 29408 |
This theorem is referenced by: usgr2pthspth 29451 |
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