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Mirrors > Home > MPE Home > Th. List > wlkp1lem4 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 27462. (Contributed by AV, 6-Mar-2021.) |
Ref | Expression |
---|---|
wlkp1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
wlkp1.i | ⊢ 𝐼 = (iEdg‘𝐺) |
wlkp1.f | ⊢ (𝜑 → Fun 𝐼) |
wlkp1.a | ⊢ (𝜑 → 𝐼 ∈ Fin) |
wlkp1.b | ⊢ (𝜑 → 𝐵 ∈ V) |
wlkp1.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
wlkp1.d | ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) |
wlkp1.w | ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
wlkp1.n | ⊢ 𝑁 = (♯‘𝐹) |
wlkp1.e | ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) |
wlkp1.x | ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) |
wlkp1.u | ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) |
wlkp1.h | ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) |
wlkp1.q | ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) |
wlkp1.s | ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
Ref | Expression |
---|---|
wlkp1lem4 | ⊢ (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkp1.w | . . 3 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | |
2 | eqid 2821 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 2 | wlkf 27395 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom (iEdg‘𝐺)) |
4 | eqid 2821 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
5 | 4 | wlkp 27397 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) |
6 | 3, 5 | jca 514 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))) |
7 | 1, 6 | syl 17 | . 2 ⊢ (𝜑 → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))) |
8 | wlkp1.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
9 | wlkp1.s | . . . . . 6 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
10 | 8, 9 | eleqtrrd 2916 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Vtx‘𝑆)) |
11 | 10 | elfvexd 6703 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) |
12 | 11 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))) → 𝑆 ∈ V) |
13 | wlkp1.h | . . . 4 ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) | |
14 | simprl 769 | . . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))) → 𝐹 ∈ Word dom (iEdg‘𝐺)) | |
15 | snex 5331 | . . . . 5 ⊢ {〈𝑁, 𝐵〉} ∈ V | |
16 | unexg 7471 | . . . . 5 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ {〈𝑁, 𝐵〉} ∈ V) → (𝐹 ∪ {〈𝑁, 𝐵〉}) ∈ V) | |
17 | 14, 15, 16 | sylancl 588 | . . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))) → (𝐹 ∪ {〈𝑁, 𝐵〉}) ∈ V) |
18 | 13, 17 | eqeltrid 2917 | . . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))) → 𝐻 ∈ V) |
19 | wlkp1.q | . . . 4 ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) | |
20 | ovex 7188 | . . . . . . 7 ⊢ (0...(♯‘𝐹)) ∈ V | |
21 | fvex 6682 | . . . . . . 7 ⊢ (Vtx‘𝐺) ∈ V | |
22 | 20, 21 | fpm 8438 | . . . . . 6 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → 𝑃 ∈ ((Vtx‘𝐺) ↑pm (0...(♯‘𝐹)))) |
23 | 22 | ad2antll 727 | . . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))) → 𝑃 ∈ ((Vtx‘𝐺) ↑pm (0...(♯‘𝐹)))) |
24 | snex 5331 | . . . . 5 ⊢ {〈(𝑁 + 1), 𝐶〉} ∈ V | |
25 | unexg 7471 | . . . . 5 ⊢ ((𝑃 ∈ ((Vtx‘𝐺) ↑pm (0...(♯‘𝐹))) ∧ {〈(𝑁 + 1), 𝐶〉} ∈ V) → (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) ∈ V) | |
26 | 23, 24, 25 | sylancl 588 | . . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))) → (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) ∈ V) |
27 | 19, 26 | eqeltrid 2917 | . . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))) → 𝑄 ∈ V) |
28 | 12, 18, 27 | 3jca 1124 | . 2 ⊢ ((𝜑 ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))) → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V)) |
29 | 7, 28 | mpdan 685 | 1 ⊢ (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∪ cun 3933 ⊆ wss 3935 {csn 4566 {cpr 4568 〈cop 4572 class class class wbr 5065 dom cdm 5554 Fun wfun 6348 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ↑pm cpm 8406 Fincfn 8508 0cc0 10536 1c1 10537 + caddc 10539 ...cfz 12891 ♯chash 13689 Word cword 13860 Vtxcvtx 26780 iEdgciedg 26781 Edgcedg 26831 Walkscwlks 27377 |
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 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
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 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-wlks 27380 |
This theorem is referenced by: wlkp1 27462 |
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