lclkrlem2t - Mathbox for Norm Megill

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Theorem lclkrlem2t 41513

Description: Lemma for lclkr 41520. 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 480	. . 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 480	. . 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 480	. . 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 480	. . 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 480	. . 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 480	. . 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 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈)) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
31		eqid 2729	. . 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 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈)) → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )
34		simpr 484	. . 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 41512	. 2 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
36	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → 𝑋 ∈ 𝑉)
37	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → 𝑌 ∈ 𝑉)
38	15	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → 𝐸 ∈ 𝐹)
39	17	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → 𝐺 ∈ 𝐹)
40	25	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
41	27	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
42	29	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
43	32	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0_g‘𝑈)) → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )
44		simpr 484	. . 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 41510	. 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 3012	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {csn 4585 ‘cfv 6499 (class class class)co 7369 Basecbs 17155 +_gcplusg 17196 ._rcmulr 17197 Scalarcsca 17199 ·_𝑠 cvsca 17200 0_gc0g 17378 -_gcsg 18849 LSSumclsm 19548 inv_rcinvr 20307 LSpanclspn 20909 LFnlclfn 39043 LKerclk 39071 LDualcld 39109 HLchlt 39336 LHypclh 39971 DVecHcdvh 41065 ocHcoch 41334

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-riotaBAD 38939

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-tpos 8182 df-undef 8229 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-0g 17380 df-mre 17523 df-mrc 17524 df-acs 17526 df-proset 18235 df-poset 18254 df-plt 18269 df-lub 18285 df-glb 18286 df-join 18287 df-meet 18288 df-p0 18364 df-p1 18365 df-lat 18373 df-clat 18440 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-grp 18850 df-minusg 18851 df-sbg 18852 df-subg 19037 df-cntz 19231 df-oppg 19260 df-lsm 19550 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-dvr 20321 df-drng 20651 df-lmod 20800 df-lss 20870 df-lsp 20910 df-lvec 21042 df-lsatoms 38962 df-lshyp 38963 df-lcv 39005 df-lfl 39044 df-lkr 39072 df-ldual 39110 df-oposet 39162 df-ol 39164 df-oml 39165 df-covers 39252 df-ats 39253 df-atl 39284 df-cvlat 39308 df-hlat 39337 df-llines 39485 df-lplanes 39486 df-lvols 39487 df-lines 39488 df-psubsp 39490 df-pmap 39491 df-padd 39783 df-lhyp 39975 df-laut 39976 df-ldil 40091 df-ltrn 40092 df-trl 40146 df-tgrp 40730 df-tendo 40742 df-edring 40744 df-dveca 40990 df-disoa 41016 df-dvech 41066 df-dib 41126 df-dic 41160 df-dih 41216 df-doch 41335 df-djh 41382

This theorem is referenced by: lclkrlem2u 41514 lclkrlem2x 41517

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