lclkrlem2t - Mathbox for Norm Megill

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Theorem lclkrlem2t 42088

Description: Lemma for lclkr 42095. We eliminate all hypotheses with 𝐵 here. (Contributed by NM, 18-Jan-2015.)

**Hypotheses**
Ref	Expression
lclkrlem2m.v	⊢ 𝑉 = (Base‘𝑈)
lclkrlem2m.t	⊢ · = ( ·_𝑠 ‘𝑈)
lclkrlem2m.s	⊢ 𝑆 = (Scalar‘𝑈)
lclkrlem2m.q	⊢ × = (._r‘𝑆)
lclkrlem2m.z	⊢ 0 = (0_g‘𝑆)
lclkrlem2m.i	⊢ 𝐼 = (inv_r‘𝑆)
lclkrlem2m.m	⊢ − = (-_g‘𝑈)
lclkrlem2m.f	⊢ 𝐹 = (LFnl‘𝑈)
lclkrlem2m.d	⊢ 𝐷 = (LDual‘𝑈)
lclkrlem2m.p	⊢ + = (+_g‘𝐷)
lclkrlem2m.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
lclkrlem2m.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
lclkrlem2m.e	⊢ (𝜑 → 𝐸 ∈ 𝐹)
lclkrlem2m.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
lclkrlem2n.n	⊢ 𝑁 = (LSpan‘𝑈)
lclkrlem2n.l	⊢ 𝐿 = (LKer‘𝑈)
lclkrlem2o.h	⊢ 𝐻 = (LHyp‘𝐾)
lclkrlem2o.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lclkrlem2o.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lclkrlem2o.a	⊢ ⊕ = (LSSum‘𝑈)
lclkrlem2o.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lclkrlem2q.le	⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
lclkrlem2q.lg	⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
lclkrlem2t.n	⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )

**Assertion**
Ref	Expression
lclkrlem2t	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

**Proof of Theorem lclkrlem2t**
Step	Hyp	Ref	Expression
1		lclkrlem2m.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
2		lclkrlem2m.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝑈)
3		lclkrlem2m.s	. . 3 ⊢ 𝑆 = (Scalar‘𝑈)
4		lclkrlem2m.q	. . 3 ⊢ × = (._r‘𝑆)
5		lclkrlem2m.z	. . 3 ⊢ 0 = (0_g‘𝑆)
6		lclkrlem2m.i	. . 3 ⊢ 𝐼 = (inv_r‘𝑆)
7		lclkrlem2m.m	. . 3 ⊢ − = (-_g‘𝑈)
8		lclkrlem2m.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
9		lclkrlem2m.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
10		lclkrlem2m.p	. . 3 ⊢ + = (+_g‘𝐷)
11		lclkrlem2m.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12	11	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈)) → 𝑋 ∈ 𝑉)
13		lclkrlem2m.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
14	13	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈)) → 𝑌 ∈ 𝑉)
15		lclkrlem2m.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
16	15	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈)) → 𝐸 ∈ 𝐹)
17		lclkrlem2m.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
18	17	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈)) → 𝐺 ∈ 𝐹)
19		lclkrlem2n.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
20		lclkrlem2n.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
21		lclkrlem2o.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
22		lclkrlem2o.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
23		lclkrlem2o.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
24		lclkrlem2o.a	. . 3 ⊢ ⊕ = (LSSum‘𝑈)
25		lclkrlem2o.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
26	25	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
27		lclkrlem2q.le	. . . 4 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
28	27	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈)) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
29		lclkrlem2q.lg	. . . 4 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
30	29	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈)) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
31		eqid 2752	. . 3 ⊢ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))
32		lclkrlem2t.n	. . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )
33	32	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈)) → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )
34		simpr 487	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈)) → (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈))
35	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 19, 20, 21, 22, 23, 24, 26, 28, 30, 31, 33, 34	lclkrlem2s 42087	. 2 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
36	11	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → 𝑋 ∈ 𝑉)
37	13	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → 𝑌 ∈ 𝑉)
38	15	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → 𝐸 ∈ 𝐹)
39	17	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → 𝐺 ∈ 𝐹)
40	25	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
41	27	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
42	29	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
43	32	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )
44		simpr 487	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈))
45	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 36, 37, 38, 39, 19, 20, 21, 22, 23, 24, 40, 41, 42, 31, 43, 44	lclkrlem2q 42085	. 2 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
46	35, 45	pm2.61dane 3034	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1550 ∈ wcel 2132 ≠ wne 2947 {csn 4572 ‘cfv 6506 (class class class)co 7381 Basecbs 17217 +_gcplusg 17258 ._rcmulr 17259 Scalarcsca 17261 ·_𝑠 cvsca 17262 0_gc0g 17440 -_gcsg 18949 LSSumclsm 19646 inv_rcinvr 20404 LSpanclspn 21007 LFnlclfn 39619 LKerclk 39647 LDualcld 39685 HLchlt 39912 LHypclh 40546 DVecHcdvh 41640 ocHcoch 41909

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-riotaBAD 39515

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4856 df-int 4896 df-iun 4941 df-iin 4942 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-of 7645 df-om 7832 df-1st 7955 df-2nd 7956 df-tpos 8190 df-undef 8237 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-2o 8422 df-er 8662 df-map 8794 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-nn 12197 df-2 12266 df-3 12267 df-4 12268 df-5 12269 df-6 12270 df-n0 12468 df-z 12555 df-uz 12826 df-fz 13499 df-struct 17155 df-sets 17172 df-slot 17190 df-ndx 17202 df-base 17218 df-ress 17239 df-plusg 17271 df-mulr 17272 df-sca 17274 df-vsca 17275 df-0g 17442 df-mre 17586 df-mrc 17587 df-acs 17589 df-proset 18298 df-poset 18317 df-plt 18332 df-lub 18348 df-glb 18349 df-join 18350 df-meet 18351 df-p0 18427 df-p1 18428 df-lat 18436 df-clat 18503 df-mgm 18646 df-sgrp 18725 df-mnd 18741 df-submnd 18790 df-grp 18950 df-minusg 18951 df-sbg 18952 df-subg 19137 df-cntz 19329 df-oppg 19358 df-lsm 19648 df-cmn 19794 df-abl 19795 df-mgp 20159 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20354 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-drng 20749 df-lmod 20898 df-lss 20968 df-lsp 21008 df-lvec 21139 df-lsatoms 39538 df-lshyp 39539 df-lcv 39581 df-lfl 39620 df-lkr 39648 df-ldual 39686 df-oposet 39738 df-ol 39740 df-oml 39741 df-covers 39828 df-ats 39829 df-atl 39860 df-cvlat 39884 df-hlat 39913 df-llines 40060 df-lplanes 40061 df-lvols 40062 df-lines 40063 df-psubsp 40065 df-pmap 40066 df-padd 40358 df-lhyp 40550 df-laut 40551 df-ldil 40666 df-ltrn 40667 df-trl 40721 df-tgrp 41305 df-tendo 41317 df-edring 41319 df-dveca 41565 df-disoa 41591 df-dvech 41641 df-dib 41701 df-dic 41735 df-dih 41791 df-doch 41910 df-djh 41957

This theorem is referenced by: lclkrlem2u 42089 lclkrlem2x 42092

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