lclkrlem2j - Mathbox for Norm Megill

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Theorem lclkrlem2j 41813

Description: Lemma for lclkr 41830. Kernel closure when 𝑌 is zero. (Contributed by NM, 18-Jan-2015.)

**Hypotheses**
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 = (0_g‘𝑆)
lclkrlem2f.z	⊢ 0 = (0_g‘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 )

**Assertion**
Ref	Expression
lclkrlem2j	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

**Proof of Theorem lclkrlem2j**
Step	Hyp	Ref	Expression
1		lclkrlem2f.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		lclkrlem2j.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
3	2	snssd 4766	. . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
4		lclkrlem2f.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2737	. . . . 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 41650	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
10	1, 3, 9	syl2anc 585	. . 3 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
11	4, 5, 8	dochoc 41664	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
12	1, 10, 11	syl2anc 585	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
13		lclkrlem2f.lg	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
14		lclkrlem2j.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 = 0 )
15	14	sneqd 4593	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑌} = { 0 })
16	15	fveq2d 6839	. . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) = ( ⊥ ‘{ 0 }))
17		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Base‘𝑈) = (Base‘𝑈)
18		lclkrlem2f.z	. . . . . . . . . . . . 13 ⊢ 0 = (0_g‘𝑈)
19	4, 6, 8, 17, 18	doch0 41655	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = (Base‘𝑈))
20	1, 19	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘{ 0 }) = (Base‘𝑈))
21	13, 16, 20	3eqtrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘𝐺) = (Base‘𝑈))
22	4, 6, 1	dvhlmod 41407	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
23		lclkrlem2f.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
24		lclkrlem2f.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (Scalar‘𝑈)
25		lclkrlem2f.q	. . . . . . . . . . . 12 ⊢ 𝑄 = (0_g‘𝑆)
26		lclkrlem2f.f	. . . . . . . . . . . 12 ⊢ 𝐹 = (LFnl‘𝑈)
27		lclkrlem2f.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (LKer‘𝑈)
28	24, 25, 17, 26, 27	lkr0f 39391	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐿‘𝐺) = (Base‘𝑈) ↔ 𝐺 = ((Base‘𝑈) × {𝑄})))
29	22, 23, 28	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿‘𝐺) = (Base‘𝑈) ↔ 𝐺 = ((Base‘𝑈) × {𝑄})))
30	21, 29	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((Base‘𝑈) × {𝑄}))
31		lclkrlem2f.d	. . . . . . . . . 10 ⊢ 𝐷 = (LDual‘𝑈)
32		eqid 2737	. . . . . . . . . 10 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
33	17, 24, 25, 31, 32, 22	ldual0v 39447	. . . . . . . . 9 ⊢ (𝜑 → (0_g‘𝐷) = ((Base‘𝑈) × {𝑄}))
34	30, 33	eqtr4d 2775	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (0_g‘𝐷))
35	34	oveq2d 7376	. . . . . . 7 ⊢ (𝜑 → (𝐸 + 𝐺) = (𝐸 + (0_g‘𝐷)))
36	31, 22	lduallmod 39450	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ LMod)
37		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
38		lclkrlem2f.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
39	26, 31, 37, 22, 38	ldualelvbase 39424	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (Base‘𝐷))
40		lclkrlem2f.p	. . . . . . . . 9 ⊢ + = (+_g‘𝐷)
41	37, 40, 32	lmod0vrid 20848	. . . . . . . 8 ⊢ ((𝐷 ∈ LMod ∧ 𝐸 ∈ (Base‘𝐷)) → (𝐸 + (0_g‘𝐷)) = 𝐸)
42	36, 39, 41	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝐸 + (0_g‘𝐷)) = 𝐸)
43	35, 42	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → (𝐸 + 𝐺) = 𝐸)
44	43	fveq2d 6839	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = (𝐿‘𝐸))
45		lclkrlem2f.le	. . . . 5 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
46	44, 45	eqtr2d 2773	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) = (𝐿‘(𝐸 + 𝐺)))
47	46	fveq2d 6839	. . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝑋})) = ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))))
48	47	fveq2d 6839	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))))
49	12, 48, 46	3eqtr3d 2780	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ∖ cdif 3899 ⊆ wss 3902 {csn 4581 × cxp 5623 ran crn 5626 ‘cfv 6493 (class class class)co 7360 Basecbs 17140 +_gcplusg 17181 Scalarcsca 17184 0_gc0g 17363 LSSumclsm 19567 LModclmod 20815 LSpanclspn 20926 LSHypclsh 39272 LFnlclfn 39354 LKerclk 39382 LDualcld 39420 HLchlt 39647 LHypclh 40281 DVecHcdvh 41375 DIsoHcdih 41525 ocHcoch 41644

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-riotaBAD 39250

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8170 df-undef 8217 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-n0 12406 df-z 12493 df-uz 12756 df-fz 13428 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-0g 17365 df-proset 18221 df-poset 18240 df-plt 18255 df-lub 18271 df-glb 18272 df-join 18273 df-meet 18274 df-p0 18350 df-p1 18351 df-lat 18359 df-clat 18426 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-subg 19057 df-cntz 19250 df-lsm 19569 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20277 df-dvdsr 20297 df-unit 20298 df-invr 20328 df-dvr 20341 df-drng 20668 df-lmod 20817 df-lss 20887 df-lsp 20927 df-lvec 21059 df-lfl 39355 df-lkr 39383 df-ldual 39421 df-oposet 39473 df-ol 39475 df-oml 39476 df-covers 39563 df-ats 39564 df-atl 39595 df-cvlat 39619 df-hlat 39648 df-llines 39795 df-lplanes 39796 df-lvols 39797 df-lines 39798 df-psubsp 39800 df-pmap 39801 df-padd 40093 df-lhyp 40285 df-laut 40286 df-ldil 40401 df-ltrn 40402 df-trl 40456 df-tendo 41052 df-edring 41054 df-disoa 41326 df-dvech 41376 df-dib 41436 df-dic 41470 df-dih 41526 df-doch 41645

This theorem is referenced by: lclkrlem2k 41814 lclkrlem2l 41815

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