lclkrlem2k - Mathbox for Norm Megill

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Theorem lclkrlem2k 41556

Description: Lemma for lclkr 41572. Kernel closure when 𝑋 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})))
lclkrlem2k.x	⊢ (𝜑 → 𝑋 = 0 )
lclkrlem2k.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)

**Assertion**
Ref	Expression
lclkrlem2k	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

**Proof of Theorem lclkrlem2k**
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 = (0_g‘𝑆)
7		lclkrlem2f.z	. . 3 ⊢ 0 = (0_g‘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 41149	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
22	10, 13, 14, 21, 18, 17	ldualvaddcom 39179	. . . . 5 ⊢ (𝜑 → (𝐸 + 𝐺) = (𝐺 + 𝐸))
23	22	fveq1d 6819	. . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = ((𝐺 + 𝐸)‘𝐵))
24		lclkrlem2f.kb	. . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
25	23, 24	eqtr3d 2768	. . 3 ⊢ (𝜑 → ((𝐺 + 𝐸)‘𝐵) = 𝑄)
26		lclkrlem2f.nx	. . . 4 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
27	26	orcomd 871	. . 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 41555	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐺 + 𝐸)))) = (𝐿‘(𝐺 + 𝐸)))
31	22	fveq2d 6821	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = (𝐿‘(𝐺 + 𝐸)))
32	31	fveq2d 6821	. . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘(𝐿‘(𝐺 + 𝐸))))
33	32	fveq2d 6821	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐺 + 𝐸)))))
34	30, 33, 31	3eqtr4d 2776	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2111 ∖ cdif 3894 {csn 4571 ‘cfv 6476 (class class class)co 7341 Basecbs 17115 +_gcplusg 17156 Scalarcsca 17159 0_gc0g 17338 LSSumclsm 19541 LSpanclspn 20899 LSHypclsh 39014 LFnlclfn 39096 LKerclk 39124 LDualcld 39162 HLchlt 39389 LHypclh 40023 DVecHcdvh 41117 ocHcoch 41386

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-riotaBAD 38992

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-undef 8198 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-n0 12377 df-z 12464 df-uz 12728 df-fz 13403 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-sca 17172 df-vsca 17173 df-0g 17340 df-proset 18195 df-poset 18214 df-plt 18229 df-lub 18245 df-glb 18246 df-join 18247 df-meet 18248 df-p0 18324 df-p1 18325 df-lat 18333 df-clat 18400 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-grp 18844 df-minusg 18845 df-sbg 18846 df-subg 19031 df-cntz 19224 df-lsm 19543 df-cmn 19689 df-abl 19690 df-mgp 20054 df-rng 20066 df-ur 20095 df-ring 20148 df-oppr 20250 df-dvdsr 20270 df-unit 20271 df-invr 20301 df-dvr 20314 df-drng 20641 df-lmod 20790 df-lss 20860 df-lsp 20900 df-lvec 21032 df-lfl 39097 df-lkr 39125 df-ldual 39163 df-oposet 39215 df-ol 39217 df-oml 39218 df-covers 39305 df-ats 39306 df-atl 39337 df-cvlat 39361 df-hlat 39390 df-llines 39537 df-lplanes 39538 df-lvols 39539 df-lines 39540 df-psubsp 39542 df-pmap 39543 df-padd 39835 df-lhyp 40027 df-laut 40028 df-ldil 40143 df-ltrn 40144 df-trl 40198 df-tendo 40794 df-edring 40796 df-disoa 41068 df-dvech 41118 df-dib 41178 df-dic 41212 df-dih 41268 df-doch 41387

This theorem is referenced by: lclkrlem2l 41557

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