lclkrlem2t - Mathbox for Norm Megill

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Theorem lclkrlem2t 41992

Description: Lemma for lclkr 41999. 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 41991	. 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 41989	. 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 4568 ‘cfv 6494 (class class class)co 7362 Basecbs 17174 +_gcplusg 17215 ._rcmulr 17216 Scalarcsca 17218 ·_𝑠 cvsca 17219 0_gc0g 17397 -_gcsg 18906 LSSumclsm 19604 inv_rcinvr 20362 LSpanclspn 20961 LFnlclfn 39523 LKerclk 39551 LDualcld 39589 HLchlt 39816 LHypclh 40450 DVecHcdvh 41544 ocHcoch 41813

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-riotaBAD 39419

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 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7626 df-om 7813 df-1st 7937 df-2nd 7938 df-tpos 8171 df-undef 8218 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-er 8638 df-map 8770 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-n0 12433 df-z 12520 df-uz 12784 df-fz 13457 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-sca 17231 df-vsca 17232 df-0g 17399 df-mre 17543 df-mrc 17544 df-acs 17546 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18393 df-clat 18460 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19094 df-cntz 19287 df-oppg 19316 df-lsm 19606 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-drng 20703 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lvec 21094 df-lsatoms 39442 df-lshyp 39443 df-lcv 39485 df-lfl 39524 df-lkr 39552 df-ldual 39590 df-oposet 39642 df-ol 39644 df-oml 39645 df-covers 39732 df-ats 39733 df-atl 39764 df-cvlat 39788 df-hlat 39817 df-llines 39964 df-lplanes 39965 df-lvols 39966 df-lines 39967 df-psubsp 39969 df-pmap 39970 df-padd 40262 df-lhyp 40454 df-laut 40455 df-ldil 40570 df-ltrn 40571 df-trl 40625 df-tgrp 41209 df-tendo 41221 df-edring 41223 df-dveca 41469 df-disoa 41495 df-dvech 41545 df-dib 41605 df-dic 41639 df-dih 41695 df-doch 41814 df-djh 41861

This theorem is referenced by: lclkrlem2u 41993 lclkrlem2x 41996

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