lclkrlem2j - Mathbox for Norm Megill

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Theorem lclkrlem2j 42680

Description: Lemma for lclkr 42697. 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 4743	. . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
4		lclkrlem2f.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2760	. . . . 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 42517	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
10	1, 3, 9	syl2anc 596	. . 3 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
11	4, 5, 8	dochoc 42531	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
12	1, 10, 11	syl2anc 596	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
13		lclkrlem2f.lg	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
14		lclkrlem2j.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 = 0 )
15	14	sneqd 4592	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑌} = { 0 })
16	15	fveq2d 6892	. . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) = ( ⊥ ‘{ 0 }))
17		eqid 2760	. . . . . . . . . . . . 13 ⊢ (Base‘𝑈) = (Base‘𝑈)
18		lclkrlem2f.z	. . . . . . . . . . . . 13 ⊢ 0 = (0_g‘𝑈)
19	4, 6, 8, 17, 18	doch0 42522	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = (Base‘𝑈))
20	1, 19	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘{ 0 }) = (Base‘𝑈))
21	13, 16, 20	3eqtrd 2799	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘𝐺) = (Base‘𝑈))
22	4, 6, 1	dvhlmod 42274	. . . . . . . . . . 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 40258	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐿‘𝐺) = (Base‘𝑈) ↔ 𝐺 = ((Base‘𝑈) × {𝑄})))
29	22, 23, 28	syl2anc 596	. . . . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
33	17, 24, 25, 31, 32, 22	ldual0v 40314	. . . . . . . . 9 ⊢ (𝜑 → (0_g‘𝐷) = ((Base‘𝑈) × {𝑄}))
34	30, 33	eqtr4d 2798	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (0_g‘𝐷))
35	34	oveq2d 7438	. . . . . . 7 ⊢ (𝜑 → (𝐸 + 𝐺) = (𝐸 + (0_g‘𝐷)))
36	31, 22	lduallmod 40317	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ LMod)
37		eqid 2760	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
38		lclkrlem2f.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
39	26, 31, 37, 22, 38	ldualelvbase 40291	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (Base‘𝐷))
40		lclkrlem2f.p	. . . . . . . . 9 ⊢ + = (+_g‘𝐷)
41	37, 40, 32	lmod0vrid 21175	. . . . . . . 8 ⊢ ((𝐷 ∈ LMod ∧ 𝐸 ∈ (Base‘𝐷)) → (𝐸 + (0_g‘𝐷)) = 𝐸)
42	36, 39, 41	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝐸 + (0_g‘𝐷)) = 𝐸)
43	35, 42	eqtrd 2795	. . . . . 6 ⊢ (𝜑 → (𝐸 + 𝐺) = 𝐸)
44	43	fveq2d 6892	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = (𝐿‘𝐸))
45		lclkrlem2f.le	. . . . 5 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
46	44, 45	eqtr2d 2796	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) = (𝐿‘(𝐸 + 𝐺)))
47	46	fveq2d 6892	. . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝑋})) = ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))))
48	47	fveq2d 6892	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))))
49	12, 48, 46	3eqtr3d 2803	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ⊆ wss 3896 ∖ cdif 4044 {csn 4584 × cxp 5648 ran crn 5651 ‘cfv 6541 (class class class)co 7422 Basecbs 17394 +_gcplusg 17435 Scalarcsca 17438 0_gc0g 17617 LSSumclsm 19855 LModclmod 21142 LSpanclspn 21253 LSHypclsh 40139 LFnlclfn 40221 LKerclk 40249 LDualcld 40287 HLchlt 40514 LHypclh 41148 DVecHcdvh 42242 DIsoHcdih 42392 ocHcoch 42511

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282 ax-riotaBAD 40117

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-of 7695 df-om 7880 df-1st 8003 df-2nd 8004 df-tpos 8248 df-undef 8295 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-er 8722 df-map 8854 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-nn 12341 df-2 12410 df-3 12411 df-4 12412 df-5 12413 df-6 12414 df-n0 12612 df-z 12699 df-uz 12971 df-fz 13645 df-struct 17332 df-sets 17349 df-slot 17367 df-ndx 17379 df-base 17395 df-ress 17416 df-plusg 17448 df-mulr 17449 df-sca 17451 df-vsca 17452 df-0g 17619 df-proset 18475 df-poset 18494 df-plt 18509 df-lub 18525 df-glb 18526 df-join 18527 df-meet 18528 df-p0 18604 df-p1 18605 df-lat 18613 df-clat 18680 df-mgm 18823 df-sgrp 18915 df-mnd 18931 df-submnd 18986 df-grp 19154 df-minusg 19155 df-sbg 19156 df-subg 19340 df-cntz 19538 df-lsm 19857 df-cmn 20003 df-abl 20004 df-mgp 20368 df-rng 20382 df-ur 20415 df-ring 20468 df-oppr 20574 df-dvdsr 20594 df-unit 20595 df-invr 20625 df-dvr 20638 df-drng 20989 df-lmod 21144 df-lss 21214 df-lsp 21254 df-lvec 21385 df-lfl 40222 df-lkr 40250 df-ldual 40288 df-oposet 40340 df-ol 40342 df-oml 40343 df-covers 40430 df-ats 40431 df-atl 40462 df-cvlat 40486 df-hlat 40515 df-llines 40662 df-lplanes 40663 df-lvols 40664 df-lines 40665 df-psubsp 40667 df-pmap 40668 df-padd 40960 df-lhyp 41152 df-laut 41153 df-ldil 41268 df-ltrn 41269 df-trl 41323 df-tendo 41919 df-edring 41921 df-disoa 42193 df-dvech 42243 df-dib 42303 df-dic 42337 df-dih 42393 df-doch 42512

This theorem is used by: lclkrlem2k 42681 lclkrlem2l 42682

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