lclkrlem2h - Mathbox for Norm Megill

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Theorem lclkrlem2h 41977

Description: Lemma for lclkr 41996. 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 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		lclkrlem2f.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
18	17	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐵 ∈ (𝑉 ∖ { 0 }))
19		lclkrlem2f.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
20	19	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐸 ∈ 𝐹)
21		lclkrlem2f.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
22	21	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐺 ∈ 𝐹)
23		lclkrlem2f.le	. . . 4 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
24	23	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
25		lclkrlem2f.lg	. . . 4 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
26	25	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
27		lclkrlem2f.kb	. . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
28	27	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
29		lclkrlem2f.nx	. . . 4 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
30	29	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
31		lclkrlem2h.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
32	31	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝑋 ∈ (𝑉 ∖ { 0 }))
33		lclkrlem2h.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
34	33	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝑌 ∈ (𝑉 ∖ { 0 }))
35		lclkrlem2h.ne	. . . 4 ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))
36	35	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐸) ≠ (𝐿‘𝐺))
37		simpr 484	. . 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 41976	. 2 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
39	1, 3, 2, 4, 15	dochoc1 41824	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
40	39	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
41	1, 3, 15	dvhlvec 41572	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
42	1, 3, 15	dvhlmod 41573	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
43	10, 13, 14, 42, 19, 21	ldualvaddcl 39593	. . . . . . 7 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
44	4, 11, 10, 12, 41, 43	lkrshpor 39570	. . . . . 6 ⊢ (𝜑 → ((𝐿‘(𝐸 + 𝐺)) ∈ 𝐽 ∨ (𝐿‘(𝐸 + 𝐺)) = 𝑉))
45	44	orcanai 1005	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘(𝐸 + 𝐺)) = 𝑉)
46	45	fveq2d 6839	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘𝑉))
47	46	fveq2d 6839	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘𝑉)))
48	40, 47, 45	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
49	38, 48	pm2.61dan 813	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 {csn 4568 ‘cfv 6493 (class class class)co 7361 Basecbs 17173 +_gcplusg 17214 Scalarcsca 17217 0_gc0g 17396 LSSumclsm 19603 LSpanclspn 20960 LSHypclsh 39438 LFnlclfn 39520 LKerclk 39548 LDualcld 39586 HLchlt 39813 LHypclh 40447 DVecHcdvh 41541 ocHcoch 41810

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 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-riotaBAD 39416

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 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-tpos 8170 df-undef 8217 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-n0 12432 df-z 12519 df-uz 12783 df-fz 13456 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-sca 17230 df-vsca 17231 df-0g 17398 df-mre 17542 df-mrc 17543 df-acs 17545 df-proset 18254 df-poset 18273 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18392 df-clat 18459 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19093 df-cntz 19286 df-oppg 19315 df-lsm 19605 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-dvr 20375 df-drng 20702 df-lmod 20851 df-lss 20921 df-lsp 20961 df-lvec 21093 df-lsatoms 39439 df-lshyp 39440 df-lcv 39482 df-lfl 39521 df-lkr 39549 df-ldual 39587 df-oposet 39639 df-ol 39641 df-oml 39642 df-covers 39729 df-ats 39730 df-atl 39761 df-cvlat 39785 df-hlat 39814 df-llines 39961 df-lplanes 39962 df-lvols 39963 df-lines 39964 df-psubsp 39966 df-pmap 39967 df-padd 40259 df-lhyp 40451 df-laut 40452 df-ldil 40567 df-ltrn 40568 df-trl 40622 df-tgrp 41206 df-tendo 41218 df-edring 41220 df-dveca 41466 df-disoa 41492 df-dvech 41542 df-dib 41602 df-dic 41636 df-dih 41692 df-doch 41811 df-djh 41858

This theorem is referenced by: lclkrlem2i 41978

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