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Mirrors > Home > MPE Home > Th. List > Mathboxes > lshpkrlem3 | Structured version Visualization version GIF version |
Description: Lemma for lshpkrex 35274. Defining property of 𝐺‘𝑋. (Contributed by NM, 15-Jul-2014.) |
Ref | Expression |
---|---|
lshpkrlem.v | ⊢ 𝑉 = (Base‘𝑊) |
lshpkrlem.a | ⊢ + = (+g‘𝑊) |
lshpkrlem.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lshpkrlem.p | ⊢ ⊕ = (LSSum‘𝑊) |
lshpkrlem.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
lshpkrlem.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lshpkrlem.u | ⊢ (𝜑 → 𝑈 ∈ 𝐻) |
lshpkrlem.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
lshpkrlem.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lshpkrlem.e | ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) |
lshpkrlem.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lshpkrlem.k | ⊢ 𝐾 = (Base‘𝐷) |
lshpkrlem.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lshpkrlem.o | ⊢ 0 = (0g‘𝐷) |
lshpkrlem.g | ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) |
Ref | Expression |
---|---|
lshpkrlem3 | ⊢ (𝜑 → ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + ((𝐺‘𝑋) · 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lshpkrlem.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lshpkrlem.a | . . . . 5 ⊢ + = (+g‘𝑊) | |
3 | lshpkrlem.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | lshpkrlem.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝑊) | |
5 | lshpkrlem.h | . . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊) | |
6 | lshpkrlem.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
7 | lshpkrlem.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐻) | |
8 | lshpkrlem.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
9 | lshpkrlem.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
10 | lshpkrlem.e | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) | |
11 | lshpkrlem.d | . . . . 5 ⊢ 𝐷 = (Scalar‘𝑊) | |
12 | lshpkrlem.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐷) | |
13 | lshpkrlem.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | lshpsmreu 35265 | . . . 4 ⊢ (𝜑 → ∃!𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍))) |
15 | riotasbc 6898 | . . . 4 ⊢ (∃!𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)) → [(℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍))) / 𝑙]∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍))) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → [(℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍))) / 𝑙]∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍))) |
17 | eqeq1 2782 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 = (𝑧 + (𝑙 · 𝑍)) ↔ 𝑋 = (𝑧 + (𝑙 · 𝑍)))) | |
18 | 17 | rexbidv 3237 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑧 ∈ 𝑈 𝑥 = (𝑧 + (𝑙 · 𝑍)) ↔ ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)))) |
19 | 18 | riotabidv 6885 | . . . . 5 ⊢ (𝑥 = 𝑋 → (℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑥 = (𝑧 + (𝑙 · 𝑍))) = (℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)))) |
20 | lshpkrlem.g | . . . . . 6 ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) | |
21 | oveq1 6929 | . . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝑘 · 𝑍) = (𝑙 · 𝑍)) | |
22 | 21 | oveq2d 6938 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑙 → (𝑦 + (𝑘 · 𝑍)) = (𝑦 + (𝑙 · 𝑍))) |
23 | 22 | eqeq2d 2788 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑙 → (𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ 𝑥 = (𝑦 + (𝑙 · 𝑍)))) |
24 | 23 | rexbidv 3237 | . . . . . . . . 9 ⊢ (𝑘 = 𝑙 → (∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑙 · 𝑍)))) |
25 | oveq1 6929 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦 + (𝑙 · 𝑍)) = (𝑧 + (𝑙 · 𝑍))) | |
26 | 25 | eqeq2d 2788 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑥 = (𝑦 + (𝑙 · 𝑍)) ↔ 𝑥 = (𝑧 + (𝑙 · 𝑍)))) |
27 | 26 | cbvrexv 3368 | . . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑙 · 𝑍)) ↔ ∃𝑧 ∈ 𝑈 𝑥 = (𝑧 + (𝑙 · 𝑍))) |
28 | 24, 27 | syl6bb 279 | . . . . . . . 8 ⊢ (𝑘 = 𝑙 → (∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ ∃𝑧 ∈ 𝑈 𝑥 = (𝑧 + (𝑙 · 𝑍)))) |
29 | 28 | cbvriotav 6894 | . . . . . . 7 ⊢ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))) = (℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑥 = (𝑧 + (𝑙 · 𝑍))) |
30 | 29 | mpteq2i 4976 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) = (𝑥 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑥 = (𝑧 + (𝑙 · 𝑍)))) |
31 | 20, 30 | eqtri 2802 | . . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑥 = (𝑧 + (𝑙 · 𝑍)))) |
32 | riotaex 6887 | . . . . 5 ⊢ (℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍))) ∈ V | |
33 | 19, 31, 32 | fvmpt 6542 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝐺‘𝑋) = (℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)))) |
34 | dfsbcq 3654 | . . . 4 ⊢ ((𝐺‘𝑋) = (℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍))) → ([(𝐺‘𝑋) / 𝑙]∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)) ↔ [(℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍))) / 𝑙]∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)))) | |
35 | 9, 33, 34 | 3syl 18 | . . 3 ⊢ (𝜑 → ([(𝐺‘𝑋) / 𝑙]∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)) ↔ [(℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍))) / 𝑙]∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)))) |
36 | 16, 35 | mpbird 249 | . 2 ⊢ (𝜑 → [(𝐺‘𝑋) / 𝑙]∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍))) |
37 | fvex 6459 | . . 3 ⊢ (𝐺‘𝑋) ∈ V | |
38 | oveq1 6929 | . . . . . 6 ⊢ (𝑙 = (𝐺‘𝑋) → (𝑙 · 𝑍) = ((𝐺‘𝑋) · 𝑍)) | |
39 | 38 | oveq2d 6938 | . . . . 5 ⊢ (𝑙 = (𝐺‘𝑋) → (𝑧 + (𝑙 · 𝑍)) = (𝑧 + ((𝐺‘𝑋) · 𝑍))) |
40 | 39 | eqeq2d 2788 | . . . 4 ⊢ (𝑙 = (𝐺‘𝑋) → (𝑋 = (𝑧 + (𝑙 · 𝑍)) ↔ 𝑋 = (𝑧 + ((𝐺‘𝑋) · 𝑍)))) |
41 | 40 | rexbidv 3237 | . . 3 ⊢ (𝑙 = (𝐺‘𝑋) → (∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)) ↔ ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + ((𝐺‘𝑋) · 𝑍)))) |
42 | 37, 41 | sbcie 3687 | . 2 ⊢ ([(𝐺‘𝑋) / 𝑙]∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)) ↔ ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + ((𝐺‘𝑋) · 𝑍))) |
43 | 36, 42 | sylib 210 | 1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + ((𝐺‘𝑋) · 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 ∃!wreu 3092 [wsbc 3652 {csn 4398 ↦ cmpt 4965 ‘cfv 6135 ℩crio 6882 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 Scalarcsca 16341 ·𝑠 cvsca 16342 0gc0g 16486 LSSumclsm 18433 LSpanclspn 19366 LVecclvec 19497 LSHypclsh 35131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-lshyp 35133 |
This theorem is referenced by: lshpkrlem6 35271 |
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