lclkrlem2h - Mathbox for Norm Megill

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Theorem lclkrlem2h 37584

Description: Lemma for lclkr 37603. Eliminate the (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽 hypothesis. (Contributed by NM, 16-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})))
lclkrlem2h.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lclkrlem2h.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lclkrlem2h.ne	⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))

**Assertion**
Ref	Expression
lclkrlem2h	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

**Proof of Theorem lclkrlem2h**
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	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16	15	adantr 474	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		lclkrlem2f.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
18	17	adantr 474	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐵 ∈ (𝑉 ∖ { 0 }))
19		lclkrlem2f.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
20	19	adantr 474	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐸 ∈ 𝐹)
21		lclkrlem2f.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
22	21	adantr 474	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐺 ∈ 𝐹)
23		lclkrlem2f.le	. . . 4 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
24	23	adantr 474	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
25		lclkrlem2f.lg	. . . 4 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
26	25	adantr 474	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
27		lclkrlem2f.kb	. . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
28	27	adantr 474	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
29		lclkrlem2f.nx	. . . 4 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
30	29	adantr 474	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
31		lclkrlem2h.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
32	31	adantr 474	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝑋 ∈ (𝑉 ∖ { 0 }))
33		lclkrlem2h.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
34	33	adantr 474	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝑌 ∈ (𝑉 ∖ { 0 }))
35		lclkrlem2h.ne	. . . 4 ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))
36	35	adantr 474	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐸) ≠ (𝐿‘𝐺))
37		simpr 479	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽)
38	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 37	lclkrlem2g 37583	. 2 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
39	1, 3, 2, 4, 15	dochoc1 37431	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
40	39	adantr 474	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
41	1, 3, 15	dvhlvec 37179	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
42	1, 3, 15	dvhlmod 37180	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
43	10, 13, 14, 42, 19, 21	ldualvaddcl 35200	. . . . . . 7 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
44	4, 11, 10, 12, 41, 43	lkrshpor 35177	. . . . . 6 ⊢ (𝜑 → ((𝐿‘(𝐸 + 𝐺)) ∈ 𝐽 ∨ (𝐿‘(𝐸 + 𝐺)) = 𝑉))
45	44	orcanai 1030	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘(𝐸 + 𝐺)) = 𝑉)
46	45	fveq2d 6441	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘𝑉))
47	46	fveq2d 6441	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘𝑉)))
48	40, 47, 45	3eqtr4d 2871	. 2 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
49	38, 48	pm2.61dan 847	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∨ wo 878 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∖ cdif 3795 {csn 4399 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 +_gcplusg 16312 Scalarcsca 16315 0_gc0g 16460 LSSumclsm 18407 LSpanclspn 19337 LSHypclsh 35045 LFnlclfn 35127 LKerclk 35155 LDualcld 35193 HLchlt 35420 LHypclh 36054 DVecHcdvh 37148 ocHcoch 37417

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-riotaBAD 35023

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-undef 7669 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-0g 16462 df-mre 16606 df-mrc 16607 df-acs 16609 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-clat 17468 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-cntz 18107 df-oppg 18133 df-lsm 18409 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-drng 19112 df-lmod 19228 df-lss 19296 df-lsp 19338 df-lvec 19469 df-lsatoms 35046 df-lshyp 35047 df-lcv 35089 df-lfl 35128 df-lkr 35156 df-ldual 35194 df-oposet 35246 df-ol 35248 df-oml 35249 df-covers 35336 df-ats 35337 df-atl 35368 df-cvlat 35392 df-hlat 35421 df-llines 35568 df-lplanes 35569 df-lvols 35570 df-lines 35571 df-psubsp 35573 df-pmap 35574 df-padd 35866 df-lhyp 36058 df-laut 36059 df-ldil 36174 df-ltrn 36175 df-trl 36229 df-tgrp 36813 df-tendo 36825 df-edring 36827 df-dveca 37073 df-disoa 37099 df-dvech 37149 df-dib 37209 df-dic 37243 df-dih 37299 df-doch 37418 df-djh 37465

This theorem is referenced by: lclkrlem2i 37585

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