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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 14232 | . . . 4 ⊢ 〈“𝐴𝐵𝐶𝐷”〉 ∈ Word V | |
3 | 1, 2 | eqeltri 2906 | . . 3 ⊢ 𝑃 ∈ Word V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝑃 ∈ Word V) |
5 | 3wlkd.f | . . . . 5 ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 | |
6 | 5 | fveq2i 6666 | . . . 4 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾𝐿”〉) |
7 | s3len 14244 | . . . 4 ⊢ (♯‘〈“𝐽𝐾𝐿”〉) = 3 | |
8 | 6, 7 | eqtri 2841 | . . 3 ⊢ (♯‘𝐹) = 3 |
9 | 4m1e3 11754 | . . 3 ⊢ (4 − 1) = 3 | |
10 | 1 | fveq2i 6666 | . . . . 5 ⊢ (♯‘𝑃) = (♯‘〈“𝐴𝐵𝐶𝐷”〉) |
11 | s4len 14249 | . . . . 5 ⊢ (♯‘〈“𝐴𝐵𝐶𝐷”〉) = 4 | |
12 | 10, 11 | eqtr2i 2842 | . . . 4 ⊢ 4 = (♯‘𝑃) |
13 | 12 | oveq1i 7155 | . . 3 ⊢ (4 − 1) = ((♯‘𝑃) − 1) |
14 | 8, 9, 13 | 3eqtr2i 2847 | . 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 27870 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗))) |
18 | eqid 2818 | . 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 27878 | . 2 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
24 | 4, 14, 17, 18, 23 | pthd 27477 | 1 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ⊆ wss 3933 {cpr 4559 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 1c1 10526 − cmin 10858 3c3 11681 4c4 11682 ♯chash 13678 Word cword 13849 〈“cs3 14192 〈“cs4 14193 Vtxcvtx 26708 iEdgciedg 26709 Pathscpths 27420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-s2 14198 df-s3 14199 df-s4 14200 df-wlks 27308 df-trls 27401 df-pths 27424 |
This theorem is referenced by: 3pthond 27881 3cycld 27884 |
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