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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2j | Structured version Visualization version GIF version |
Description: Lemma for lclkr 38549. 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 4734 | . . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
4 | lclkrlem2f.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | eqid 2818 | . . . . 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 38369 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
10 | 1, 3, 9 | syl2anc 584 | . . 3 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
11 | 4, 5, 8 | dochoc 38383 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋})) |
12 | 1, 10, 11 | syl2anc 584 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋})) |
13 | lclkrlem2f.lg | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) | |
14 | lclkrlem2j.y | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 = 0 ) | |
15 | 14 | sneqd 4569 | . . . . . . . . . . . 12 ⊢ (𝜑 → {𝑌} = { 0 }) |
16 | 15 | fveq2d 6667 | . . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) = ( ⊥ ‘{ 0 })) |
17 | eqid 2818 | . . . . . . . . . . . . 13 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
18 | lclkrlem2f.z | . . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝑈) | |
19 | 4, 6, 8, 17, 18 | doch0 38374 | . . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = (Base‘𝑈)) |
20 | 1, 19 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘{ 0 }) = (Base‘𝑈)) |
21 | 13, 16, 20 | 3eqtrd 2857 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘𝐺) = (Base‘𝑈)) |
22 | 4, 6, 1 | dvhlmod 38126 | . . . . . . . . . . 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 36110 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐿‘𝐺) = (Base‘𝑈) ↔ 𝐺 = ((Base‘𝑈) × {𝑄}))) |
29 | 22, 23, 28 | syl2anc 584 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐿‘𝐺) = (Base‘𝑈) ↔ 𝐺 = ((Base‘𝑈) × {𝑄}))) |
30 | 21, 29 | mpbid 233 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((Base‘𝑈) × {𝑄})) |
31 | lclkrlem2f.d | . . . . . . . . . 10 ⊢ 𝐷 = (LDual‘𝑈) | |
32 | eqid 2818 | . . . . . . . . . 10 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
33 | 17, 24, 25, 31, 32, 22 | ldual0v 36166 | . . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐷) = ((Base‘𝑈) × {𝑄})) |
34 | 30, 33 | eqtr4d 2856 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 = (0g‘𝐷)) |
35 | 34 | oveq2d 7161 | . . . . . . 7 ⊢ (𝜑 → (𝐸 + 𝐺) = (𝐸 + (0g‘𝐷))) |
36 | 31, 22 | lduallmod 36169 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ LMod) |
37 | eqid 2818 | . . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
38 | lclkrlem2f.e | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
39 | 26, 31, 37, 22, 38 | ldualelvbase 36143 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (Base‘𝐷)) |
40 | lclkrlem2f.p | . . . . . . . . 9 ⊢ + = (+g‘𝐷) | |
41 | 37, 40, 32 | lmod0vrid 19594 | . . . . . . . 8 ⊢ ((𝐷 ∈ LMod ∧ 𝐸 ∈ (Base‘𝐷)) → (𝐸 + (0g‘𝐷)) = 𝐸) |
42 | 36, 39, 41 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → (𝐸 + (0g‘𝐷)) = 𝐸) |
43 | 35, 42 | eqtrd 2853 | . . . . . 6 ⊢ (𝜑 → (𝐸 + 𝐺) = 𝐸) |
44 | 43 | fveq2d 6667 | . . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = (𝐿‘𝐸)) |
45 | lclkrlem2f.le | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) | |
46 | 44, 45 | eqtr2d 2854 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) = (𝐿‘(𝐸 + 𝐺))) |
47 | 46 | fveq2d 6667 | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝑋})) = ( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) |
48 | 47 | fveq2d 6667 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺))))) |
49 | 12, 48, 46 | 3eqtr3d 2861 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ⊆ wss 3933 {csn 4557 × cxp 5546 ran crn 5549 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Scalarcsca 16556 0gc0g 16701 LSSumclsm 18688 LModclmod 19563 LSpanclspn 19672 LSHypclsh 35991 LFnlclfn 36073 LKerclk 36101 LDualcld 36139 HLchlt 36366 LHypclh 37000 DVecHcdvh 38094 DIsoHcdih 38244 ocHcoch 38363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-undef 7928 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 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 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-lfl 36074 df-lkr 36102 df-ldual 36140 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-tendo 37771 df-edring 37773 df-disoa 38045 df-dvech 38095 df-dib 38155 df-dic 38189 df-dih 38245 df-doch 38364 |
This theorem is referenced by: lclkrlem2k 38533 lclkrlem2l 38534 |
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