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Mirrors > Home > MPE Home > Th. List > 3pthd | Structured version Visualization version GIF version |
Description: A path of length 3 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
Ref | Expression |
---|---|
3wlkd.p | ⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 |
3wlkd.f | ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 |
3wlkd.s | ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) |
3wlkd.n | ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) |
3wlkd.e | ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) |
3wlkd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
3wlkd.i | ⊢ 𝐼 = (iEdg‘𝐺) |
3trld.n | ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) |
Ref | Expression |
---|---|
3pthd | ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3wlkd.p | . . . 4 ⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 | |
2 | s4cli 13827 | . . . 4 ⊢ 〈“𝐴𝐵𝐶𝐷”〉 ∈ Word V | |
3 | 1, 2 | eqeltri 2835 | . . 3 ⊢ 𝑃 ∈ Word V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝑃 ∈ Word V) |
5 | 3wlkd.f | . . . . 5 ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 | |
6 | 5 | fveq2i 6355 | . . . 4 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾𝐿”〉) |
7 | s3len 13839 | . . . 4 ⊢ (♯‘〈“𝐽𝐾𝐿”〉) = 3 | |
8 | 6, 7 | eqtri 2782 | . . 3 ⊢ (♯‘𝐹) = 3 |
9 | 4m1e3 11330 | . . 3 ⊢ (4 − 1) = 3 | |
10 | 1 | fveq2i 6355 | . . . . 5 ⊢ (♯‘𝑃) = (♯‘〈“𝐴𝐵𝐶𝐷”〉) |
11 | s4len 13844 | . . . . 5 ⊢ (♯‘〈“𝐴𝐵𝐶𝐷”〉) = 4 | |
12 | 10, 11 | eqtr2i 2783 | . . . 4 ⊢ 4 = (♯‘𝑃) |
13 | 12 | oveq1i 6823 | . . 3 ⊢ (4 − 1) = ((♯‘𝑃) − 1) |
14 | 8, 9, 13 | 3eqtr2i 2788 | . 2 ⊢ (♯‘𝐹) = ((♯‘𝑃) − 1) |
15 | 3wlkd.s | . . 3 ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) | |
16 | 3wlkd.n | . . 3 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) | |
17 | 1, 5, 15, 16 | 3pthdlem1 27316 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗))) |
18 | eqid 2760 | . 2 ⊢ (♯‘𝐹) = (♯‘𝐹) | |
19 | 3wlkd.e | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) | |
20 | 3wlkd.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
21 | 3wlkd.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
22 | 3trld.n | . . 3 ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) | |
23 | 1, 5, 15, 16, 19, 20, 21, 22 | 3trld 27324 | . 2 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
24 | 4, 14, 17, 18, 23 | pthd 26875 | 1 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 Vcvv 3340 ⊆ wss 3715 {cpr 4323 class class class wbr 4804 ‘cfv 6049 (class class class)co 6813 1c1 10129 − cmin 10458 3c3 11263 4c4 11264 ♯chash 13311 Word cword 13477 〈“cs3 13787 〈“cs4 13788 Vtxcvtx 26073 iEdgciedg 26074 Pathscpths 26818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ifp 1051 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-map 8025 df-pm 8026 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-fzo 12660 df-hash 13312 df-word 13485 df-concat 13487 df-s1 13488 df-s2 13793 df-s3 13794 df-s4 13795 df-wlks 26705 df-trls 26799 df-pths 26822 |
This theorem is referenced by: 3pthond 27327 3cycld 27330 |
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