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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2j | Structured version Visualization version GIF version |
Description: Lemma for lclkr 38829. Kernel closure when 𝑌 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}))) |
lclkrlem2j.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lclkrlem2j.y | ⊢ (𝜑 → 𝑌 = 0 ) |
Ref | Expression |
---|---|
lclkrlem2j | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lclkrlem2f.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | lclkrlem2j.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | 2 | snssd 4702 | . . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
4 | lclkrlem2f.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | eqid 2798 | . . . . 5 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
6 | lclkrlem2f.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | lclkrlem2f.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
8 | lclkrlem2f.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
9 | 4, 5, 6, 7, 8 | dochcl 38649 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
10 | 1, 3, 9 | syl2anc 587 | . . 3 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
11 | 4, 5, 8 | dochoc 38663 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋})) |
12 | 1, 10, 11 | syl2anc 587 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋})) |
13 | lclkrlem2f.lg | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) | |
14 | lclkrlem2j.y | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 = 0 ) | |
15 | 14 | sneqd 4537 | . . . . . . . . . . . 12 ⊢ (𝜑 → {𝑌} = { 0 }) |
16 | 15 | fveq2d 6649 | . . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) = ( ⊥ ‘{ 0 })) |
17 | eqid 2798 | . . . . . . . . . . . . 13 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
18 | lclkrlem2f.z | . . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝑈) | |
19 | 4, 6, 8, 17, 18 | doch0 38654 | . . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = (Base‘𝑈)) |
20 | 1, 19 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘{ 0 }) = (Base‘𝑈)) |
21 | 13, 16, 20 | 3eqtrd 2837 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘𝐺) = (Base‘𝑈)) |
22 | 4, 6, 1 | dvhlmod 38406 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod) |
23 | lclkrlem2f.g | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
24 | lclkrlem2f.s | . . . . . . . . . . . 12 ⊢ 𝑆 = (Scalar‘𝑈) | |
25 | lclkrlem2f.q | . . . . . . . . . . . 12 ⊢ 𝑄 = (0g‘𝑆) | |
26 | lclkrlem2f.f | . . . . . . . . . . . 12 ⊢ 𝐹 = (LFnl‘𝑈) | |
27 | lclkrlem2f.l | . . . . . . . . . . . 12 ⊢ 𝐿 = (LKer‘𝑈) | |
28 | 24, 25, 17, 26, 27 | lkr0f 36390 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐿‘𝐺) = (Base‘𝑈) ↔ 𝐺 = ((Base‘𝑈) × {𝑄}))) |
29 | 22, 23, 28 | syl2anc 587 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐿‘𝐺) = (Base‘𝑈) ↔ 𝐺 = ((Base‘𝑈) × {𝑄}))) |
30 | 21, 29 | mpbid 235 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((Base‘𝑈) × {𝑄})) |
31 | lclkrlem2f.d | . . . . . . . . . 10 ⊢ 𝐷 = (LDual‘𝑈) | |
32 | eqid 2798 | . . . . . . . . . 10 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
33 | 17, 24, 25, 31, 32, 22 | ldual0v 36446 | . . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐷) = ((Base‘𝑈) × {𝑄})) |
34 | 30, 33 | eqtr4d 2836 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 = (0g‘𝐷)) |
35 | 34 | oveq2d 7151 | . . . . . . 7 ⊢ (𝜑 → (𝐸 + 𝐺) = (𝐸 + (0g‘𝐷))) |
36 | 31, 22 | lduallmod 36449 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ LMod) |
37 | eqid 2798 | . . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
38 | lclkrlem2f.e | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
39 | 26, 31, 37, 22, 38 | ldualelvbase 36423 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (Base‘𝐷)) |
40 | lclkrlem2f.p | . . . . . . . . 9 ⊢ + = (+g‘𝐷) | |
41 | 37, 40, 32 | lmod0vrid 19658 | . . . . . . . 8 ⊢ ((𝐷 ∈ LMod ∧ 𝐸 ∈ (Base‘𝐷)) → (𝐸 + (0g‘𝐷)) = 𝐸) |
42 | 36, 39, 41 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → (𝐸 + (0g‘𝐷)) = 𝐸) |
43 | 35, 42 | eqtrd 2833 | . . . . . 6 ⊢ (𝜑 → (𝐸 + 𝐺) = 𝐸) |
44 | 43 | fveq2d 6649 | . . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = (𝐿‘𝐸)) |
45 | lclkrlem2f.le | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) | |
46 | 44, 45 | eqtr2d 2834 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) = (𝐿‘(𝐸 + 𝐺))) |
47 | 46 | fveq2d 6649 | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝑋})) = ( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) |
48 | 47 | fveq2d 6649 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺))))) |
49 | 12, 48, 46 | 3eqtr3d 2841 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 {csn 4525 × cxp 5517 ran crn 5520 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Scalarcsca 16560 0gc0g 16705 LSSumclsm 18751 LModclmod 19627 LSpanclspn 19736 LSHypclsh 36271 LFnlclfn 36353 LKerclk 36381 LDualcld 36419 HLchlt 36646 LHypclh 37280 DVecHcdvh 38374 DIsoHcdih 38524 ocHcoch 38643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-undef 7922 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lfl 36354 df-lkr 36382 df-ldual 36420 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tendo 38051 df-edring 38053 df-disoa 38325 df-dvech 38375 df-dib 38435 df-dic 38469 df-dih 38525 df-doch 38644 |
This theorem is referenced by: lclkrlem2k 38813 lclkrlem2l 38814 |
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