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Mirrors > Home > MPE Home > Th. List > 2pthd | Structured version Visualization version GIF version |
Description: A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.) |
Ref | Expression |
---|---|
2wlkd.p | ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 |
2wlkd.f | ⊢ 𝐹 = 〈“𝐽𝐾”〉 |
2wlkd.s | ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
2wlkd.n | ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) |
2wlkd.e | ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) |
2wlkd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
2wlkd.i | ⊢ 𝐼 = (iEdg‘𝐺) |
2trld.n | ⊢ (𝜑 → 𝐽 ≠ 𝐾) |
Ref | Expression |
---|---|
2pthd | ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2wlkd.p | . . . 4 ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 | |
2 | s3cli 14604 | . . . 4 ⊢ 〈“𝐴𝐵𝐶”〉 ∈ Word V | |
3 | 1, 2 | eqeltri 2835 | . . 3 ⊢ 𝑃 ∈ Word V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝑃 ∈ Word V) |
5 | 2wlkd.f | . . . . 5 ⊢ 𝐹 = 〈“𝐽𝐾”〉 | |
6 | 5 | fveq2i 6769 | . . . 4 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾”〉) |
7 | s2len 14612 | . . . 4 ⊢ (♯‘〈“𝐽𝐾”〉) = 2 | |
8 | 6, 7 | eqtri 2766 | . . 3 ⊢ (♯‘𝐹) = 2 |
9 | 3m1e2 12111 | . . 3 ⊢ (3 − 1) = 2 | |
10 | 1 | fveq2i 6769 | . . . . 5 ⊢ (♯‘𝑃) = (♯‘〈“𝐴𝐵𝐶”〉) |
11 | s3len 14617 | . . . . 5 ⊢ (♯‘〈“𝐴𝐵𝐶”〉) = 3 | |
12 | 10, 11 | eqtr2i 2767 | . . . 4 ⊢ 3 = (♯‘𝑃) |
13 | 12 | oveq1i 7277 | . . 3 ⊢ (3 − 1) = ((♯‘𝑃) − 1) |
14 | 8, 9, 13 | 3eqtr2i 2772 | . 2 ⊢ (♯‘𝐹) = ((♯‘𝑃) − 1) |
15 | 2wlkd.s | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
16 | 2wlkd.n | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) | |
17 | 1, 5, 15, 16 | 2pthdlem1 28303 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗))) |
18 | eqid 2738 | . 2 ⊢ (♯‘𝐹) = (♯‘𝐹) | |
19 | 2wlkd.e | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) | |
20 | 2wlkd.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
21 | 2wlkd.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
22 | 2trld.n | . . 3 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
23 | 1, 5, 15, 16, 19, 20, 21, 22 | 2trld 28311 | . 2 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
24 | 4, 14, 17, 18, 23 | pthd 28145 | 1 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 Vcvv 3429 ⊆ wss 3886 {cpr 4563 class class class wbr 5073 ‘cfv 6426 (class class class)co 7267 1c1 10882 − cmin 11215 2c2 12038 3c3 12039 ♯chash 14054 Word cword 14227 〈“cs2 14564 〈“cs3 14565 Vtxcvtx 27376 iEdgciedg 27377 Pathscpths 28088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-n0 12244 df-z 12330 df-uz 12593 df-fz 13250 df-fzo 13393 df-hash 14055 df-word 14228 df-concat 14284 df-s1 14311 df-s2 14571 df-s3 14572 df-wlks 27976 df-trls 28069 df-pths 28092 |
This theorem is referenced by: 2cycld 33108 |
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