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Mirrors > Home > MPE Home > Th. List > Mathboxes > lshpkrlem3 | Structured version Visualization version GIF version |
Description: Lemma for lshpkrex 36826. 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 36817 | . . . 4 ⊢ (𝜑 → ∃!𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍))) |
15 | riotasbc 7178 | . . . 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 2738 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 = (𝑧 + (𝑙 · 𝑍)) ↔ 𝑋 = (𝑧 + (𝑙 · 𝑍)))) | |
18 | 17 | rexbidv 3209 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑧 ∈ 𝑈 𝑥 = (𝑧 + (𝑙 · 𝑍)) ↔ ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)))) |
19 | 18 | riotabidv 7161 | . . . . 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 7209 | . . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝑘 · 𝑍) = (𝑙 · 𝑍)) | |
22 | 21 | oveq2d 7218 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑙 → (𝑦 + (𝑘 · 𝑍)) = (𝑦 + (𝑙 · 𝑍))) |
23 | 22 | eqeq2d 2745 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑙 → (𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ 𝑥 = (𝑦 + (𝑙 · 𝑍)))) |
24 | 23 | rexbidv 3209 | . . . . . . . . 9 ⊢ (𝑘 = 𝑙 → (∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑙 · 𝑍)))) |
25 | oveq1 7209 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦 + (𝑙 · 𝑍)) = (𝑧 + (𝑙 · 𝑍))) | |
26 | 25 | eqeq2d 2745 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑥 = (𝑦 + (𝑙 · 𝑍)) ↔ 𝑥 = (𝑧 + (𝑙 · 𝑍)))) |
27 | 26 | cbvrexvw 3352 | . . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑙 · 𝑍)) ↔ ∃𝑧 ∈ 𝑈 𝑥 = (𝑧 + (𝑙 · 𝑍))) |
28 | 24, 27 | bitrdi 290 | . . . . . . . 8 ⊢ (𝑘 = 𝑙 → (∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ ∃𝑧 ∈ 𝑈 𝑥 = (𝑧 + (𝑙 · 𝑍)))) |
29 | 28 | cbvriotavw 7169 | . . . . . . 7 ⊢ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))) = (℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑥 = (𝑧 + (𝑙 · 𝑍))) |
30 | 29 | mpteq2i 5136 | . . . . . 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 2762 | . . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑥 = (𝑧 + (𝑙 · 𝑍)))) |
32 | riotaex 7163 | . . . . 5 ⊢ (℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍))) ∈ V | |
33 | 19, 31, 32 | fvmpt 6807 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝐺‘𝑋) = (℩𝑙 ∈ 𝐾 ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)))) |
34 | dfsbcq 3689 | . . . 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 260 | . 2 ⊢ (𝜑 → [(𝐺‘𝑋) / 𝑙]∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍))) |
37 | fvex 6719 | . . 3 ⊢ (𝐺‘𝑋) ∈ V | |
38 | oveq1 7209 | . . . . . 6 ⊢ (𝑙 = (𝐺‘𝑋) → (𝑙 · 𝑍) = ((𝐺‘𝑋) · 𝑍)) | |
39 | 38 | oveq2d 7218 | . . . . 5 ⊢ (𝑙 = (𝐺‘𝑋) → (𝑧 + (𝑙 · 𝑍)) = (𝑧 + ((𝐺‘𝑋) · 𝑍))) |
40 | 39 | eqeq2d 2745 | . . . 4 ⊢ (𝑙 = (𝐺‘𝑋) → (𝑋 = (𝑧 + (𝑙 · 𝑍)) ↔ 𝑋 = (𝑧 + ((𝐺‘𝑋) · 𝑍)))) |
41 | 40 | rexbidv 3209 | . . 3 ⊢ (𝑙 = (𝐺‘𝑋) → (∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)) ↔ ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + ((𝐺‘𝑋) · 𝑍)))) |
42 | 37, 41 | sbcie 3730 | . 2 ⊢ ([(𝐺‘𝑋) / 𝑙]∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + (𝑙 · 𝑍)) ↔ ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + ((𝐺‘𝑋) · 𝑍))) |
43 | 36, 42 | sylib 221 | 1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + ((𝐺‘𝑋) · 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 ∃wrex 3055 ∃!wreu 3056 [wsbc 3687 {csn 4531 ↦ cmpt 5124 ‘cfv 6369 ℩crio 7158 (class class class)co 7202 Basecbs 16684 +gcplusg 16767 Scalarcsca 16770 ·𝑠 cvsca 16771 0gc0g 16916 LSSumclsm 18995 LSpanclspn 19980 LVecclvec 20111 LSHypclsh 36683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-tpos 7957 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-0g 16918 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-grp 18340 df-minusg 18341 df-sbg 18342 df-subg 18512 df-cntz 18683 df-lsm 18997 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-oppr 19613 df-dvdsr 19631 df-unit 19632 df-invr 19662 df-drng 19741 df-lmod 19873 df-lss 19941 df-lsp 19981 df-lvec 20112 df-lshyp 36685 |
This theorem is referenced by: lshpkrlem6 36823 |
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