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Mirrors > Home > MPE Home > Th. List > 2spthd | Structured version Visualization version GIF version |
Description: A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-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) |
2spthd.n | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Ref | Expression |
---|---|
2spthd | ⊢ (𝜑 → 𝐹(SPaths‘𝐺)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2wlkd.p | . . 3 ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 | |
2 | 2wlkd.f | . . 3 ⊢ 𝐹 = 〈“𝐽𝐾”〉 | |
3 | 2wlkd.s | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
4 | 2wlkd.n | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) | |
5 | 2wlkd.e | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) | |
6 | 2wlkd.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
7 | 2wlkd.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
8 | 2trld.n | . . 3 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | 2trld 27720 | . 2 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
10 | 2spthd.n | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
11 | 3anan32 1093 | . . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ↔ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶)) | |
12 | 4, 10, 11 | sylanbrc 585 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) |
13 | funcnvs3 14279 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → Fun ◡〈“𝐴𝐵𝐶”〉) | |
14 | 3, 12, 13 | syl2anc 586 | . . 3 ⊢ (𝜑 → Fun ◡〈“𝐴𝐵𝐶”〉) |
15 | 1 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑃 = 〈“𝐴𝐵𝐶”〉) |
16 | 15 | cnveqd 5749 | . . . 4 ⊢ (𝜑 → ◡𝑃 = ◡〈“𝐴𝐵𝐶”〉) |
17 | 16 | funeqd 6380 | . . 3 ⊢ (𝜑 → (Fun ◡𝑃 ↔ Fun ◡〈“𝐴𝐵𝐶”〉)) |
18 | 14, 17 | mpbird 259 | . 2 ⊢ (𝜑 → Fun ◡𝑃) |
19 | isspth 27508 | . 2 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) | |
20 | 9, 18, 19 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐹(SPaths‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ⊆ wss 3939 {cpr 4572 class class class wbr 5069 ◡ccnv 5557 Fun wfun 6352 ‘cfv 6358 〈“cs2 14206 〈“cs3 14207 Vtxcvtx 26784 iEdgciedg 26785 Trailsctrls 27475 SPathscspths 27497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13926 df-s1 13953 df-s2 14213 df-s3 14214 df-wlks 27384 df-trls 27477 df-spths 27501 |
This theorem is referenced by: 2pthond 27724 umgr2adedgspth 27730 |
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