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Mirrors > Home > MPE Home > Th. List > pthdepisspth | Structured version Visualization version GIF version |
Description: A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 12-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
pthdepisspth | ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → 𝐹(SPaths‘𝐺)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ispth 27507 | . . . 4 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅)) | |
2 | simplll 773 | . . . . . 6 ⊢ ((((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → 𝐹(Trails‘𝐺)𝑃) | |
3 | trliswlk 27482 | . . . . . . . . 9 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
4 | wlkcl 27400 | . . . . . . . . 9 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
5 | 3, 4 | syl 17 | . . . . . . . 8 ⊢ (𝐹(Trails‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) |
6 | 5 | ad3antrrr 728 | . . . . . . 7 ⊢ ((((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → (♯‘𝐹) ∈ ℕ0) |
7 | eqid 2824 | . . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
8 | 7 | wlkp 27401 | . . . . . . . . . 10 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) |
9 | 3, 8 | syl 17 | . . . . . . . . 9 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) |
10 | 9 | ad3antrrr 728 | . . . . . . . 8 ⊢ ((((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) |
11 | simpllr 774 | . . . . . . . 8 ⊢ ((((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) | |
12 | simpr 487 | . . . . . . . 8 ⊢ ((((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) | |
13 | 10, 11, 12 | 3jca 1124 | . . . . . . 7 ⊢ ((((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) |
14 | simplr 767 | . . . . . . 7 ⊢ ((((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) | |
15 | injresinj 13161 | . . . . . . 7 ⊢ ((♯‘𝐹) ∈ ℕ0 → ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → Fun ◡𝑃))) | |
16 | 6, 13, 14, 15 | syl3c 66 | . . . . . 6 ⊢ ((((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → Fun ◡𝑃) |
17 | 2, 16 | jca 514 | . . . . 5 ⊢ ((((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) |
18 | 17 | ex3 1342 | . . . 4 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → ((𝑃‘0) ≠ (𝑃‘(♯‘𝐹)) → (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃))) |
19 | 1, 18 | sylbi 219 | . . 3 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝑃‘0) ≠ (𝑃‘(♯‘𝐹)) → (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃))) |
20 | 19 | imp 409 | . 2 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) |
21 | isspth 27508 | . 2 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) | |
22 | 20, 21 | sylibr 236 | 1 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → 𝐹(SPaths‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∩ cin 3938 ∅c0 4294 {cpr 4572 class class class wbr 5069 ◡ccnv 5557 ↾ cres 5560 “ cima 5561 Fun wfun 6352 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 0cc0 10540 1c1 10541 ℕ0cn0 11900 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Vtxcvtx 26784 Walkscwlks 27381 Trailsctrls 27475 Pathscpths 27496 SPathscspths 27497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
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 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-wlks 27384 df-trls 27477 df-pths 27500 df-spths 27501 |
This theorem is referenced by: pthisspthorcycl 32379 |
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