lclkrlem2t - Mathbox for Norm Megill

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Theorem lclkrlem2t 41931

Description: Lemma for lclkr 41938. 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 2737	. . 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 41930	. 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 41928	. 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 3020	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 {csn 4582 ‘cfv 6502 (class class class)co 7370 Basecbs 17150 +_gcplusg 17191 ._rcmulr 17192 Scalarcsca 17194 ·_𝑠 cvsca 17195 0_gc0g 17373 -_gcsg 18882 LSSumclsm 19580 inv_rcinvr 20340 LSpanclspn 20939 LFnlclfn 39462 LKerclk 39490 LDualcld 39528 HLchlt 39755 LHypclh 40389 DVecHcdvh 41483 ocHcoch 41752

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 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-riotaBAD 39358

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 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-tpos 8180 df-undef 8227 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-n0 12416 df-z 12503 df-uz 12766 df-fz 13438 df-struct 17088 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-mulr 17205 df-sca 17207 df-vsca 17208 df-0g 17375 df-mre 17519 df-mrc 17520 df-acs 17522 df-proset 18231 df-poset 18250 df-plt 18265 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-p1 18361 df-lat 18369 df-clat 18436 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-submnd 18723 df-grp 18883 df-minusg 18884 df-sbg 18885 df-subg 19070 df-cntz 19263 df-oppg 19292 df-lsm 19582 df-cmn 19728 df-abl 19729 df-mgp 20093 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20290 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-dvr 20354 df-drng 20681 df-lmod 20830 df-lss 20900 df-lsp 20940 df-lvec 21072 df-lsatoms 39381 df-lshyp 39382 df-lcv 39424 df-lfl 39463 df-lkr 39491 df-ldual 39529 df-oposet 39581 df-ol 39583 df-oml 39584 df-covers 39671 df-ats 39672 df-atl 39703 df-cvlat 39727 df-hlat 39756 df-llines 39903 df-lplanes 39904 df-lvols 39905 df-lines 39906 df-psubsp 39908 df-pmap 39909 df-padd 40201 df-lhyp 40393 df-laut 40394 df-ldil 40509 df-ltrn 40510 df-trl 40564 df-tgrp 41148 df-tendo 41160 df-edring 41162 df-dveca 41408 df-disoa 41434 df-dvech 41484 df-dib 41544 df-dic 41578 df-dih 41634 df-doch 41753 df-djh 41800

This theorem is referenced by: lclkrlem2u 41932 lclkrlem2x 41935

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