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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2n | Structured version Visualization version GIF version |
Description: Lemma for lclkr 37341. (Contributed by NM, 12-Jan-2015.) |
Ref | Expression |
---|---|
lclkrlem2m.v | ⊢ 𝑉 = (Base‘𝑈) |
lclkrlem2m.t | ⊢ · = ( ·𝑠 ‘𝑈) |
lclkrlem2m.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lclkrlem2m.q | ⊢ × = (.r‘𝑆) |
lclkrlem2m.z | ⊢ 0 = (0g‘𝑆) |
lclkrlem2m.i | ⊢ 𝐼 = (invr‘𝑆) |
lclkrlem2m.m | ⊢ − = (-g‘𝑈) |
lclkrlem2m.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lclkrlem2m.d | ⊢ 𝐷 = (LDual‘𝑈) |
lclkrlem2m.p | ⊢ + = (+g‘𝐷) |
lclkrlem2m.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lclkrlem2m.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lclkrlem2m.e | ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
lclkrlem2m.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lclkrlem2n.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lclkrlem2n.l | ⊢ 𝐿 = (LKer‘𝑈) |
lclkrlem2n.w | ⊢ (𝜑 → 𝑈 ∈ LVec) |
lclkrlem2n.j | ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) = 0 ) |
lclkrlem2n.k | ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) = 0 ) |
Ref | Expression |
---|---|
lclkrlem2n | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2771 | . 2 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
2 | lclkrlem2n.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
3 | lclkrlem2n.w | . . 3 ⊢ (𝜑 → 𝑈 ∈ LVec) | |
4 | lveclmod 19319 | . . 3 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑈 ∈ LMod) |
6 | lclkrlem2m.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑈) | |
7 | lclkrlem2m.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑈) | |
8 | lclkrlem2m.p | . . . 4 ⊢ + = (+g‘𝐷) | |
9 | lclkrlem2m.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
10 | lclkrlem2m.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
11 | 6, 7, 8, 5, 9, 10 | ldualvaddcl 34937 | . . 3 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹) |
12 | lclkrlem2n.l | . . . 4 ⊢ 𝐿 = (LKer‘𝑈) | |
13 | 6, 12, 1 | lkrlss 34902 | . . 3 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹) → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈)) |
14 | 5, 11, 13 | syl2anc 573 | . 2 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈)) |
15 | lclkrlem2n.j | . . 3 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) = 0 ) | |
16 | lclkrlem2m.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
17 | lclkrlem2m.s | . . . 4 ⊢ 𝑆 = (Scalar‘𝑈) | |
18 | lclkrlem2m.z | . . . 4 ⊢ 0 = (0g‘𝑆) | |
19 | lclkrlem2m.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
20 | 16, 17, 18, 6, 12, 3, 11, 19 | ellkr2 34898 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐿‘(𝐸 + 𝐺)) ↔ ((𝐸 + 𝐺)‘𝑋) = 0 )) |
21 | 15, 20 | mpbird 247 | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝐿‘(𝐸 + 𝐺))) |
22 | lclkrlem2n.k | . . 3 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) = 0 ) | |
23 | lclkrlem2m.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
24 | 16, 17, 18, 6, 12, 3, 11, 23 | ellkr2 34898 | . . 3 ⊢ (𝜑 → (𝑌 ∈ (𝐿‘(𝐸 + 𝐺)) ↔ ((𝐸 + 𝐺)‘𝑌) = 0 )) |
25 | 22, 24 | mpbird 247 | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝐿‘(𝐸 + 𝐺))) |
26 | 1, 2, 5, 14, 21, 25 | lspprss 19205 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 {cpr 4319 ‘cfv 6030 (class class class)co 6796 Basecbs 16064 +gcplusg 16149 .rcmulr 16150 Scalarcsca 16152 ·𝑠 cvsca 16153 0gc0g 16308 -gcsg 17632 invrcinvr 18879 LModclmod 19073 LSubSpclss 19142 LSpanclspn 19184 LVecclvec 19315 LFnlclfn 34864 LKerclk 34892 LDualcld 34930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-of 7048 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7900 df-map 8015 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-n0 11500 df-z 11585 df-uz 11894 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-plusg 16162 df-sca 16165 df-vsca 16166 df-0g 16310 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-minusg 17634 df-sbg 17635 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-lmod 19075 df-lss 19143 df-lsp 19185 df-lvec 19316 df-lfl 34865 df-lkr 34893 df-ldual 34931 |
This theorem is referenced by: lclkrlem2v 37336 |
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