Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 |
---|---|
31wlkd.p | ⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 |
31wlkd.f | ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 |
31wlkd.s | ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) |
31wlkd.n | ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) |
31wlkd.e | ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) |
31wlkd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
31wlkd.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 | 31wlkd.p | . . . 4 ⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 | |
2 | s4cli 13477 | . . . 4 ⊢ 〈“𝐴𝐵𝐶𝐷”〉 ∈ Word V | |
3 | 1, 2 | eqeltri 2684 | . . 3 ⊢ 𝑃 ∈ Word V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝑃 ∈ Word V) |
5 | 31wlkd.f | . . . . 5 ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 | |
6 | 5 | fveq2i 6106 | . . . 4 ⊢ (#‘𝐹) = (#‘〈“𝐽𝐾𝐿”〉) |
7 | s3len 13489 | . . . 4 ⊢ (#‘〈“𝐽𝐾𝐿”〉) = 3 | |
8 | 6, 7 | eqtri 2632 | . . 3 ⊢ (#‘𝐹) = 3 |
9 | 4m1e3 11015 | . . 3 ⊢ (4 − 1) = 3 | |
10 | 1 | fveq2i 6106 | . . . . 5 ⊢ (#‘𝑃) = (#‘〈“𝐴𝐵𝐶𝐷”〉) |
11 | s4len 13494 | . . . . 5 ⊢ (#‘〈“𝐴𝐵𝐶𝐷”〉) = 4 | |
12 | 10, 11 | eqtr2i 2633 | . . . 4 ⊢ 4 = (#‘𝑃) |
13 | 12 | oveq1i 6559 | . . 3 ⊢ (4 − 1) = ((#‘𝑃) − 1) |
14 | 8, 9, 13 | 3eqtr2i 2638 | . 2 ⊢ (#‘𝐹) = ((#‘𝑃) − 1) |
15 | 31wlkd.s | . . 3 ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) | |
16 | 31wlkd.n | . . 3 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) | |
17 | 1, 5, 15, 16 | 3pthdlem1 41331 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(#‘𝑃))∀𝑗 ∈ (1..^(#‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗))) |
18 | eqid 2610 | . 2 ⊢ (#‘𝐹) = (#‘𝐹) | |
19 | 31wlkd.e | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) | |
20 | 31wlkd.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
21 | 31wlkd.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 41339 | . 2 ⊢ (𝜑 → 𝐹(TrailS‘𝐺)𝑃) |
24 | 4, 14, 17, 18, 23 | pthd 40975 | 1 ⊢ (𝜑 → 𝐹(PathS‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 {cpr 4127 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 1c1 9816 − cmin 10145 3c3 10948 4c4 10949 #chash 12979 Word cword 13146 〈“cs3 13438 〈“cs4 13439 Vtxcvtx 25673 iEdgciedg 25674 PathScpths 40919 |
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-3 10957 df-4 10958 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-s3 13445 df-s4 13446 df-1wlks 40800 df-trls 40901 df-pths 40923 |
This theorem is referenced by: 3pthond 41342 3cycld 41345 |
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