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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2k | Structured version Visualization version GIF version |
Description: Lemma for lclkr 38701. Kernel closure when 𝑋 is zero. (Contributed by NM, 18-Jan-2015.) |
Ref | Expression |
---|---|
lclkrlem2f.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lclkrlem2f.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lclkrlem2f.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lclkrlem2f.v | ⊢ 𝑉 = (Base‘𝑈) |
lclkrlem2f.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lclkrlem2f.q | ⊢ 𝑄 = (0g‘𝑆) |
lclkrlem2f.z | ⊢ 0 = (0g‘𝑈) |
lclkrlem2f.a | ⊢ ⊕ = (LSSum‘𝑈) |
lclkrlem2f.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lclkrlem2f.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lclkrlem2f.j | ⊢ 𝐽 = (LSHyp‘𝑈) |
lclkrlem2f.l | ⊢ 𝐿 = (LKer‘𝑈) |
lclkrlem2f.d | ⊢ 𝐷 = (LDual‘𝑈) |
lclkrlem2f.p | ⊢ + = (+g‘𝐷) |
lclkrlem2f.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lclkrlem2f.b | ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 })) |
lclkrlem2f.e | ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
lclkrlem2f.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lclkrlem2f.le | ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
lclkrlem2f.lg | ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
lclkrlem2f.kb | ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄) |
lclkrlem2f.nx | ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))) |
lclkrlem2k.x | ⊢ (𝜑 → 𝑋 = 0 ) |
lclkrlem2k.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lclkrlem2k | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lclkrlem2f.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lclkrlem2f.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | lclkrlem2f.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | lclkrlem2f.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
5 | lclkrlem2f.s | . . 3 ⊢ 𝑆 = (Scalar‘𝑈) | |
6 | lclkrlem2f.q | . . 3 ⊢ 𝑄 = (0g‘𝑆) | |
7 | lclkrlem2f.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
8 | lclkrlem2f.a | . . 3 ⊢ ⊕ = (LSSum‘𝑈) | |
9 | lclkrlem2f.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
10 | lclkrlem2f.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
11 | lclkrlem2f.j | . . 3 ⊢ 𝐽 = (LSHyp‘𝑈) | |
12 | lclkrlem2f.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
13 | lclkrlem2f.d | . . 3 ⊢ 𝐷 = (LDual‘𝑈) | |
14 | lclkrlem2f.p | . . 3 ⊢ + = (+g‘𝐷) | |
15 | lclkrlem2f.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
16 | lclkrlem2f.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 })) | |
17 | lclkrlem2f.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
18 | lclkrlem2f.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
19 | lclkrlem2f.lg | . . 3 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) | |
20 | lclkrlem2f.le | . . 3 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) | |
21 | 1, 3, 15 | dvhlmod 38278 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
22 | 10, 13, 14, 21, 18, 17 | ldualvaddcom 36308 | . . . . 5 ⊢ (𝜑 → (𝐸 + 𝐺) = (𝐺 + 𝐸)) |
23 | 22 | fveq1d 6658 | . . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = ((𝐺 + 𝐸)‘𝐵)) |
24 | lclkrlem2f.kb | . . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄) | |
25 | 23, 24 | eqtr3d 2858 | . . 3 ⊢ (𝜑 → ((𝐺 + 𝐸)‘𝐵) = 𝑄) |
26 | lclkrlem2f.nx | . . . 4 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))) | |
27 | 26 | orcomd 867 | . . 3 ⊢ (𝜑 → (¬ 𝑌 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵}))) |
28 | lclkrlem2k.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
29 | lclkrlem2k.x | . . 3 ⊢ (𝜑 → 𝑋 = 0 ) | |
30 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 25, 27, 28, 29 | lclkrlem2j 38684 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐺 + 𝐸)))) = (𝐿‘(𝐺 + 𝐸))) |
31 | 22 | fveq2d 6660 | . . . 4 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = (𝐿‘(𝐺 + 𝐸))) |
32 | 31 | fveq2d 6660 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘(𝐿‘(𝐺 + 𝐸)))) |
33 | 32 | fveq2d 6660 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐺 + 𝐸))))) |
34 | 30, 33, 31 | 3eqtr4d 2866 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∖ cdif 3921 {csn 4553 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 +gcplusg 16548 Scalarcsca 16551 0gc0g 16696 LSSumclsm 18742 LSpanclspn 19726 LSHypclsh 36143 LFnlclfn 36225 LKerclk 36253 LDualcld 36291 HLchlt 36518 LHypclh 37152 DVecHcdvh 38246 ocHcoch 38515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-riotaBAD 36121 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-om 7567 df-1st 7675 df-2nd 7676 df-tpos 7878 df-undef 7925 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-map 8394 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-n0 11885 df-z 11969 df-uz 12231 df-fz 12883 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-sca 16564 df-vsca 16565 df-0g 16698 df-proset 17521 df-poset 17539 df-plt 17551 df-lub 17567 df-glb 17568 df-join 17569 df-meet 17570 df-p0 17632 df-p1 17633 df-lat 17639 df-clat 17701 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-submnd 17940 df-grp 18089 df-minusg 18090 df-sbg 18091 df-subg 18259 df-cntz 18430 df-lsm 18744 df-cmn 18891 df-abl 18892 df-mgp 19223 df-ur 19235 df-ring 19282 df-oppr 19356 df-dvdsr 19374 df-unit 19375 df-invr 19405 df-dvr 19416 df-drng 19487 df-lmod 19619 df-lss 19687 df-lsp 19727 df-lvec 19858 df-lfl 36226 df-lkr 36254 df-ldual 36292 df-oposet 36344 df-ol 36346 df-oml 36347 df-covers 36434 df-ats 36435 df-atl 36466 df-cvlat 36490 df-hlat 36519 df-llines 36666 df-lplanes 36667 df-lvols 36668 df-lines 36669 df-psubsp 36671 df-pmap 36672 df-padd 36964 df-lhyp 37156 df-laut 37157 df-ldil 37272 df-ltrn 37273 df-trl 37327 df-tendo 37923 df-edring 37925 df-disoa 38197 df-dvech 38247 df-dib 38307 df-dic 38341 df-dih 38397 df-doch 38516 |
This theorem is referenced by: lclkrlem2l 38686 |
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