Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 |
---|---|
11wlkd.p | ⊢ 𝑃 = 〈“𝑋𝑌”〉 |
11wlkd.f | ⊢ 𝐹 = 〈“𝐽”〉 |
11wlkd.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
11wlkd.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
11wlkd.l | ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) |
11wlkd.j | ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) |
11wlkd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
11wlkd.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
1pthond | ⊢ (𝜑 → 𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 11wlkd.p | . . . . 5 ⊢ 𝑃 = 〈“𝑋𝑌”〉 | |
2 | 11wlkd.f | . . . . 5 ⊢ 𝐹 = 〈“𝐽”〉 | |
3 | 11wlkd.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | 11wlkd.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
5 | 11wlkd.l | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) | |
6 | 11wlkd.j | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) | |
7 | 11wlkd.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | 11wlkd.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | 11wlkd 41308 | . . . 4 ⊢ (𝜑 → 𝐹(1Walks‘𝐺)𝑃) |
10 | 1 | fveq1i 6104 | . . . . . 6 ⊢ (𝑃‘0) = (〈“𝑋𝑌”〉‘0) |
11 | s2fv0 13482 | . . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (〈“𝑋𝑌”〉‘0) = 𝑋) | |
12 | 10, 11 | syl5eq 2656 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑃‘0) = 𝑋) |
13 | 3, 12 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑃‘0) = 𝑋) |
14 | 2 | fveq2i 6106 | . . . . . . 7 ⊢ (#‘𝐹) = (#‘〈“𝐽”〉) |
15 | s1len 13238 | . . . . . . 7 ⊢ (#‘〈“𝐽”〉) = 1 | |
16 | 14, 15 | eqtri 2632 | . . . . . 6 ⊢ (#‘𝐹) = 1 |
17 | 1, 16 | fveq12i 6108 | . . . . 5 ⊢ (𝑃‘(#‘𝐹)) = (〈“𝑋𝑌”〉‘1) |
18 | s2fv1 13483 | . . . . . 6 ⊢ (𝑌 ∈ 𝑉 → (〈“𝑋𝑌”〉‘1) = 𝑌) | |
19 | 4, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → (〈“𝑋𝑌”〉‘1) = 𝑌) |
20 | 17, 19 | syl5eq 2656 | . . . 4 ⊢ (𝜑 → (𝑃‘(#‘𝐹)) = 𝑌) |
21 | wlkv 40815 | . . . . . . 7 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
22 | 3simpc 1053 | . . . . . . 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 555 | . . . . 5 ⊢ (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
25 | 7 | iswlkOn 40865 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃 ↔ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(#‘𝐹)) = 𝑌))) |
26 | 24, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃 ↔ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(#‘𝐹)) = 𝑌))) |
27 | 9, 13, 20, 26 | mpbir3and 1238 | . . 3 ⊢ (𝜑 → 𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃) |
28 | 1, 2, 3, 4, 5, 6, 7, 8 | 1trld 41309 | . . 3 ⊢ (𝜑 → 𝐹(TrailS‘𝐺)𝑃) |
29 | 7 | istrlson 40914 | . . . 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 959 | . 2 ⊢ (𝜑 → 𝐹(𝑋(TrailsOn‘𝐺)𝑌)𝑃) |
32 | 1, 2, 3, 4, 5, 6, 7, 8 | 1pthd 41310 | . 2 ⊢ (𝜑 → 𝐹(PathS‘𝐺)𝑃) |
33 | 3 | adantl 481 | . . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → 𝑋 ∈ 𝑉) |
34 | 4 | adantl 481 | . . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → 𝑌 ∈ 𝑉) |
35 | simpl 472 | . . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → (𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
36 | 33, 34, 35 | jca31 555 | . . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
37 | 36 | ex 449 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))) |
38 | 21, 22, 37 | 3syl 18 | . . . 4 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))) |
39 | 9, 38 | mpcom 37 | . . 3 ⊢ (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
40 | 7 | ispthson 40948 | . . 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 959 | 1 ⊢ (𝜑 → 𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 {csn 4125 {cpr 4127 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 #chash 12979 〈“cs1 13149 〈“cs2 13437 Vtxcvtx 25673 iEdgciedg 25674 1Walksc1wlks 40796 WalksOncwlkson 40798 TrailSctrls 40899 TrailsOnctrlson 40900 PathScpths 40919 PathsOncpthson 40921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-1wlks 40800 df-wlkson 40802 df-trls 40901 df-trlson 40902 df-pths 40923 df-pthson 40925 |
This theorem is referenced by: upgr1pthond 41317 lppthon 41318 1pthon2v-av 41320 |
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