lclkrlem2h - Mathbox for Norm Megill

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Theorem lclkrlem2h 42390

Description: Lemma for lclkr 42409. 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 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		lclkrlem2f.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
18	17	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐵 ∈ (𝑉 ∖ { 0 }))
19		lclkrlem2f.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
20	19	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐸 ∈ 𝐹)
21		lclkrlem2f.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
22	21	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐺 ∈ 𝐹)
23		lclkrlem2f.le	. . . 4 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
24	23	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
25		lclkrlem2f.lg	. . . 4 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
26	25	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
27		lclkrlem2f.kb	. . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
28	27	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
29		lclkrlem2f.nx	. . . 4 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
30	29	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
31		lclkrlem2h.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
32	31	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝑋 ∈ (𝑉 ∖ { 0 }))
33		lclkrlem2h.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
34	33	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝑌 ∈ (𝑉 ∖ { 0 }))
35		lclkrlem2h.ne	. . . 4 ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))
36	35	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐸) ≠ (𝐿‘𝐺))
37		simpr 490	. . 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 42389	. 2 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
39	1, 3, 2, 4, 15	dochoc1 42237	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
40	39	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
41	1, 3, 15	dvhlvec 41985	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
42	1, 3, 15	dvhlmod 41986	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
43	10, 13, 14, 42, 19, 21	ldualvaddcl 40006	. . . . . . 7 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
44	4, 11, 10, 12, 41, 43	lkrshpor 39983	. . . . . 6 ⊢ (𝜑 → ((𝐿‘(𝐸 + 𝐺)) ∈ 𝐽 ∨ (𝐿‘(𝐸 + 𝐺)) = 𝑉))
45	44	orcanai 1018	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘(𝐸 + 𝐺)) = 𝑉)
46	45	fveq2d 6883	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘𝑉))
47	46	fveq2d 6883	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘𝑉)))
48	40, 47, 45	3eqtr4d 2805	. 2 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
49	38, 48	pm2.61dan 825	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∖ cdif 3896 {csn 4584 ‘cfv 6533 (class class class)co 7414 Basecbs 17304 +_gcplusg 17345 Scalarcsca 17348 0_gc0g 17527 LSSumclsm 19764 LSpanclspn 21158 LSHypclsh 39851 LFnlclfn 39933 LKerclk 39961 LDualcld 39999 HLchlt 40226 LHypclh 40860 DVecHcdvh 41954 ocHcoch 42223

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-riotaBAD 39829

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-n0 12532 df-z 12619 df-uz 12891 df-fz 13565 df-struct 17242 df-sets 17259 df-slot 17277 df-ndx 17289 df-base 17305 df-ress 17326 df-plusg 17358 df-mulr 17359 df-sca 17361 df-vsca 17362 df-0g 17529 df-mre 17673 df-mrc 17674 df-acs 17676 df-proset 18385 df-poset 18404 df-plt 18419 df-lub 18435 df-glb 18436 df-join 18437 df-meet 18438 df-p0 18514 df-p1 18515 df-lat 18523 df-clat 18590 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-submnd 18895 df-grp 19063 df-minusg 19064 df-sbg 19065 df-subg 19249 df-cntz 19447 df-oppg 19476 df-lsm 19766 df-cmn 19912 df-abl 19913 df-mgp 20277 df-rng 20291 df-ur 20324 df-ring 20377 df-oppr 20481 df-dvdsr 20501 df-unit 20502 df-invr 20532 df-dvr 20545 df-drng 20895 df-lmod 21049 df-lss 21119 df-lsp 21159 df-lvec 21290 df-lsatoms 39852 df-lshyp 39853 df-lcv 39895 df-lfl 39934 df-lkr 39962 df-ldual 40000 df-oposet 40052 df-ol 40054 df-oml 40055 df-covers 40142 df-ats 40143 df-atl 40174 df-cvlat 40198 df-hlat 40227 df-llines 40374 df-lplanes 40375 df-lvols 40376 df-lines 40377 df-psubsp 40379 df-pmap 40380 df-padd 40672 df-lhyp 40864 df-laut 40865 df-ldil 40980 df-ltrn 40981 df-trl 41035 df-tgrp 41619 df-tendo 41631 df-edring 41633 df-dveca 41879 df-disoa 41905 df-dvech 41955 df-dib 42015 df-dic 42049 df-dih 42105 df-doch 42224 df-djh 42271

This theorem is used by: lclkrlem2i 42391

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