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Mirrors > Home > MPE Home > Th. List > wksonproplem | Structured version Visualization version GIF version |
Description: Lemma for theorems for properties of walks between two vertices, e.g., trlsonprop 27483. (Contributed by AV, 16-Jan-2021.) |
Ref | Expression |
---|---|
wksonproplem.v | ⊢ 𝑉 = (Vtx‘𝐺) |
wksonproplem.b | ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃))) |
wksonproplem.d | ⊢ 𝑊 = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑎(𝑂‘𝑔)𝑏)𝑝 ∧ 𝑓(𝑄‘𝑔)𝑝)})) |
wksonproplem.w | ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑓(𝑄‘𝐺)𝑝) → 𝑓(Walks‘𝐺)𝑝) |
Ref | Expression |
---|---|
wksonproplem | ⊢ (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wksonproplem.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | fvexi 6678 | . . . . 5 ⊢ 𝑉 ∈ V |
3 | wksonproplem.d | . . . . . 6 ⊢ 𝑊 = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑎(𝑂‘𝑔)𝑏)𝑝 ∧ 𝑓(𝑄‘𝑔)𝑝)})) | |
4 | simp1 1132 | . . . . . . 7 ⊢ ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V) | |
5 | simp2 1133 | . . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
6 | 5, 1 | eleqtrdi 2923 | . . . . . . 7 ⊢ ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺)) |
7 | simp3 1134 | . . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
8 | 7, 1 | eleqtrdi 2923 | . . . . . . 7 ⊢ ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ (Vtx‘𝐺)) |
9 | wksv 27395 | . . . . . . . 8 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V | |
10 | 9 | a1i 11 | . . . . . . 7 ⊢ ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V) |
11 | wksonproplem.w | . . . . . . 7 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑓(𝑄‘𝐺)𝑝) → 𝑓(Walks‘𝐺)𝑝) | |
12 | 4, 6, 8, 10, 11, 3 | mptmpoopabovd 7773 | . . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑊‘𝐺)𝐵) = {〈𝑓, 𝑝〉 ∣ (𝑓(𝐴(𝑂‘𝐺)𝐵)𝑝 ∧ 𝑓(𝑄‘𝐺)𝑝)}) |
13 | fveq2 6664 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) | |
14 | 13, 1 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉) |
15 | fveq2 6664 | . . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑂‘𝑔) = (𝑂‘𝐺)) | |
16 | 15 | oveqd 7167 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑎(𝑂‘𝑔)𝑏) = (𝑎(𝑂‘𝐺)𝑏)) |
17 | 16 | breqd 5069 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑓(𝑎(𝑂‘𝑔)𝑏)𝑝 ↔ 𝑓(𝑎(𝑂‘𝐺)𝑏)𝑝)) |
18 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑄‘𝑔) = (𝑄‘𝐺)) | |
19 | 18 | breqd 5069 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑓(𝑄‘𝑔)𝑝 ↔ 𝑓(𝑄‘𝐺)𝑝)) |
20 | 17, 19 | anbi12d 632 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑓(𝑎(𝑂‘𝑔)𝑏)𝑝 ∧ 𝑓(𝑄‘𝑔)𝑝) ↔ (𝑓(𝑎(𝑂‘𝐺)𝑏)𝑝 ∧ 𝑓(𝑄‘𝐺)𝑝))) |
21 | 3, 12, 14, 14, 20 | bropfvvvv 7781 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝑉 ∈ V) → (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 → (𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))) |
22 | 2, 2, 21 | mp2an 690 | . . . 4 ⊢ (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 → (𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
23 | 3anass 1091 | . . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ↔ (𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) | |
24 | 23 | anbi1i 625 | . . . . 5 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ↔ ((𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
25 | df-3an 1085 | . . . . 5 ⊢ ((𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ↔ ((𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) | |
26 | 24, 25 | bitr4i 280 | . . . 4 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ↔ (𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
27 | 22, 26 | sylibr 236 | . . 3 ⊢ (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
28 | wksonproplem.b | . . . . 5 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃))) | |
29 | 28 | biimpd 231 | . . . 4 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 → (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃))) |
30 | 29 | imdistani 571 | . . 3 ⊢ ((((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ 𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃) → (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃))) |
31 | 27, 30 | mpancom 686 | . 2 ⊢ (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 → (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃))) |
32 | df-3an 1085 | . 2 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃)) ↔ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃))) | |
33 | 31, 32 | sylibr 236 | 1 ⊢ (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 class class class wbr 5058 {copab 5120 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 Vtxcvtx 26775 Walkscwlks 27372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
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 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-wlks 27375 |
This theorem is referenced by: trlsonprop 27483 pthsonprop 27519 spthonprop 27520 |
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