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Mirrors > Home > MPE Home > Th. List > 1pthond | Structured version Visualization version GIF version |
Description: In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.) |
Ref | Expression |
---|---|
1wlkd.p | ⊢ 𝑃 = 〈“𝑋𝑌”〉 |
1wlkd.f | ⊢ 𝐹 = 〈“𝐽”〉 |
1wlkd.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
1wlkd.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
1wlkd.l | ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) |
1wlkd.j | ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) |
1wlkd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
1wlkd.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
1pthond | ⊢ (𝜑 → 𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1wlkd.p | . . . . 5 ⊢ 𝑃 = 〈“𝑋𝑌”〉 | |
2 | 1wlkd.f | . . . . 5 ⊢ 𝐹 = 〈“𝐽”〉 | |
3 | 1wlkd.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | 1wlkd.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
5 | 1wlkd.l | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) | |
6 | 1wlkd.j | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) | |
7 | 1wlkd.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | 1wlkd.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | 1wlkd 28971 | . . . 4 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
10 | 1 | fveq1i 6840 | . . . . . 6 ⊢ (𝑃‘0) = (〈“𝑋𝑌”〉‘0) |
11 | s2fv0 14768 | . . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (〈“𝑋𝑌”〉‘0) = 𝑋) | |
12 | 10, 11 | eqtrid 2788 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑃‘0) = 𝑋) |
13 | 3, 12 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑃‘0) = 𝑋) |
14 | 2 | fveq2i 6842 | . . . . . . 7 ⊢ (♯‘𝐹) = (♯‘〈“𝐽”〉) |
15 | s1len 14486 | . . . . . . 7 ⊢ (♯‘〈“𝐽”〉) = 1 | |
16 | 14, 15 | eqtri 2764 | . . . . . 6 ⊢ (♯‘𝐹) = 1 |
17 | 1, 16 | fveq12i 6845 | . . . . 5 ⊢ (𝑃‘(♯‘𝐹)) = (〈“𝑋𝑌”〉‘1) |
18 | s2fv1 14769 | . . . . . 6 ⊢ (𝑌 ∈ 𝑉 → (〈“𝑋𝑌”〉‘1) = 𝑌) | |
19 | 4, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → (〈“𝑋𝑌”〉‘1) = 𝑌) |
20 | 17, 19 | eqtrid 2788 | . . . 4 ⊢ (𝜑 → (𝑃‘(♯‘𝐹)) = 𝑌) |
21 | wlkv 28446 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
22 | 3simpc 1150 | . . . . . . 7 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
23 | 9, 21, 22 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝑃 ∈ V)) |
24 | 3, 4, 23 | jca31 515 | . . . . 5 ⊢ (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
25 | 7 | iswlkon 28491 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(♯‘𝐹)) = 𝑌))) |
26 | 24, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(♯‘𝐹)) = 𝑌))) |
27 | 9, 13, 20, 26 | mpbir3and 1342 | . . 3 ⊢ (𝜑 → 𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃) |
28 | 1, 2, 3, 4, 5, 6, 7, 8 | 1trld 28972 | . . 3 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
29 | 7 | istrlson 28541 | . . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑋(TrailsOn‘𝐺)𝑌)𝑃 ↔ (𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃))) |
30 | 24, 29 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹(𝑋(TrailsOn‘𝐺)𝑌)𝑃 ↔ (𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃))) |
31 | 27, 28, 30 | mpbir2and 711 | . 2 ⊢ (𝜑 → 𝐹(𝑋(TrailsOn‘𝐺)𝑌)𝑃) |
32 | 1, 2, 3, 4, 5, 6, 7, 8 | 1pthd 28973 | . 2 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
33 | 3 | adantl 482 | . . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → 𝑋 ∈ 𝑉) |
34 | 4 | adantl 482 | . . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → 𝑌 ∈ 𝑉) |
35 | simpl 483 | . . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → (𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
36 | 33, 34, 35 | jca31 515 | . . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
37 | 36 | ex 413 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))) |
38 | 21, 22, 37 | 3syl 18 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))) |
39 | 9, 38 | mpcom 38 | . . 3 ⊢ (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
40 | 7 | ispthson 28576 | . . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃 ↔ (𝐹(𝑋(TrailsOn‘𝐺)𝑌)𝑃 ∧ 𝐹(Paths‘𝐺)𝑃))) |
41 | 39, 40 | syl 17 | . 2 ⊢ (𝜑 → (𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃 ↔ (𝐹(𝑋(TrailsOn‘𝐺)𝑌)𝑃 ∧ 𝐹(Paths‘𝐺)𝑃))) |
42 | 31, 32, 41 | mpbir2and 711 | 1 ⊢ (𝜑 → 𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 Vcvv 3443 ⊆ wss 3908 {csn 4584 {cpr 4586 class class class wbr 5103 ‘cfv 6493 (class class class)co 7353 0cc0 11047 1c1 11048 ♯chash 14222 〈“cs1 14475 〈“cs2 14722 Vtxcvtx 27833 iEdgciedg 27834 Walkscwlks 28430 WalksOncwlkson 28431 Trailsctrls 28524 TrailsOnctrlson 28525 Pathscpths 28546 PathsOncpthson 28548 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-map 8763 df-pm 8764 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-fzo 13560 df-hash 14223 df-word 14395 df-concat 14451 df-s1 14476 df-s2 14729 df-wlks 28433 df-wlkson 28434 df-trls 28526 df-trlson 28527 df-pths 28550 df-pthson 28552 |
This theorem is referenced by: upgr1pthond 28980 lppthon 28981 1pthon2v 28983 |
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