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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2n | Structured version Visualization version GIF version |
Description: Lemma for lclkr 41138. (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 2725 | . 2 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
2 | lclkrlem2n.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
3 | lclkrlem2n.w | . . 3 ⊢ (𝜑 → 𝑈 ∈ LVec) | |
4 | lveclmod 21008 | . . 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 38734 | . . 3 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹) |
12 | lclkrlem2n.l | . . . 4 ⊢ 𝐿 = (LKer‘𝑈) | |
13 | 6, 12, 1 | lkrlss 38699 | . . 3 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹) → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈)) |
14 | 5, 11, 13 | syl2anc 582 | . 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 38695 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐿‘(𝐸 + 𝐺)) ↔ ((𝐸 + 𝐺)‘𝑋) = 0 )) |
21 | 15, 20 | mpbird 256 | . 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 38695 | . . 3 ⊢ (𝜑 → (𝑌 ∈ (𝐿‘(𝐸 + 𝐺)) ↔ ((𝐸 + 𝐺)‘𝑌) = 0 )) |
25 | 22, 24 | mpbird 256 | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝐿‘(𝐸 + 𝐺))) |
26 | 1, 2, 5, 14, 21, 25 | lspprss 20893 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 {cpr 4632 ‘cfv 6549 (class class class)co 7419 Basecbs 17188 +gcplusg 17241 .rcmulr 17242 Scalarcsca 17244 ·𝑠 cvsca 17245 0gc0g 17429 -gcsg 18905 invrcinvr 20343 LModclmod 20760 LSubSpclss 20832 LSpanclspn 20872 LVecclvec 21004 LFnlclfn 38661 LKerclk 38689 LDualcld 38727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-plusg 17254 df-sca 17257 df-vsca 17258 df-0g 17431 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-grp 18906 df-minusg 18907 df-sbg 18908 df-cmn 19754 df-abl 19755 df-mgp 20092 df-rng 20110 df-ur 20139 df-ring 20192 df-lmod 20762 df-lss 20833 df-lsp 20873 df-lvec 21005 df-lfl 38662 df-lkr 38690 df-ldual 38728 |
This theorem is referenced by: lclkrlem2v 41133 |
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