lclkrlem2l - Mathbox for Norm Megill

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Theorem lclkrlem2l 42391

Description: Lemma for lclkr 42406. Eliminate the 𝑋 ≠ 0, 𝑌 ≠ 0 hypotheses. (Contributed by NM, 18-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})))
lclkrlem2l.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
lclkrlem2l.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)

**Assertion**
Ref	Expression
lclkrlem2l	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

**Proof of Theorem lclkrlem2l**
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 ∧ 𝑋 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		lclkrlem2f.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
18	17	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → 𝐵 ∈ (𝑉 ∖ { 0 }))
19		lclkrlem2f.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
20	19	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → 𝐸 ∈ 𝐹)
21		lclkrlem2f.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
22	21	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → 𝐺 ∈ 𝐹)
23		lclkrlem2f.le	. . . 4 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
24	23	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
25		lclkrlem2f.lg	. . . 4 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
26	25	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
27		lclkrlem2f.kb	. . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
28	27	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
29		lclkrlem2f.nx	. . . 4 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
30	29	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
31		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → 𝑋 = 0 )
32		lclkrlem2l.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
33	32	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → 𝑌 ∈ 𝑉)
34	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 20, 22, 24, 26, 28, 30, 31, 33	lclkrlem2k 42390	. 2 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
35	15	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
36	17	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝐵 ∈ (𝑉 ∖ { 0 }))
37	19	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝐸 ∈ 𝐹)
38	21	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝐺 ∈ 𝐹)
39	23	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
40	25	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
41	27	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
42	29	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
43		lclkrlem2l.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
44	43	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑋 ∈ 𝑉)
45		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑌 = 0 )
46	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45	lclkrlem2j 42389	. 2 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
47	15	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
48	17	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝐵 ∈ (𝑉 ∖ { 0 }))
49	19	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝐸 ∈ 𝐹)
50	21	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝐺 ∈ 𝐹)
51	23	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
52	25	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
53	27	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
54	29	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
55	43	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝑋 ∈ 𝑉)
56		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝑋 ≠ 0 )
57		eldifsn 4748	. . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ))
58	55, 56, 57	sylanbrc 595	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝑋 ∈ (𝑉 ∖ { 0 }))
59	32	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝑌 ∈ 𝑉)
60		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝑌 ≠ 0 )
61		eldifsn 4748	. . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ))
62	59, 60, 61	sylanbrc 595	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝑌 ∈ (𝑉 ∖ { 0 }))
63	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 47, 48, 49, 50, 51, 52, 53, 54, 58, 62	lclkrlem2i 42388	. 2 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
64	34, 46, 63	pm2.61da2ne 3043	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 7413 Basecbs 17301 +_gcplusg 17342 Scalarcsca 17345 0_gc0g 17524 LSSumclsm 19761 LSpanclspn 21155 LSHypclsh 39848 LFnlclfn 39930 LKerclk 39958 LDualcld 39996 HLchlt 40223 LHypclh 40857 DVecHcdvh 41951 ocHcoch 42220

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 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-riotaBAD 39826

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 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-tpos 8224 df-undef 8271 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12529 df-z 12616 df-uz 12888 df-fz 13562 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-sca 17358 df-vsca 17359 df-0g 17526 df-mre 17670 df-mrc 17671 df-acs 17673 df-proset 18382 df-poset 18401 df-plt 18416 df-lub 18432 df-glb 18433 df-join 18434 df-meet 18435 df-p0 18511 df-p1 18512 df-lat 18520 df-clat 18587 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-grp 19060 df-minusg 19061 df-sbg 19062 df-subg 19246 df-cntz 19444 df-oppg 19473 df-lsm 19763 df-cmn 19909 df-abl 19910 df-mgp 20274 df-rng 20288 df-ur 20321 df-ring 20374 df-oppr 20478 df-dvdsr 20498 df-unit 20499 df-invr 20529 df-dvr 20542 df-drng 20892 df-lmod 21046 df-lss 21116 df-lsp 21156 df-lvec 21287 df-lsatoms 39849 df-lshyp 39850 df-lcv 39892 df-lfl 39931 df-lkr 39959 df-ldual 39997 df-oposet 40049 df-ol 40051 df-oml 40052 df-covers 40139 df-ats 40140 df-atl 40171 df-cvlat 40195 df-hlat 40224 df-llines 40371 df-lplanes 40372 df-lvols 40373 df-lines 40374 df-psubsp 40376 df-pmap 40377 df-padd 40669 df-lhyp 40861 df-laut 40862 df-ldil 40977 df-ltrn 40978 df-trl 41032 df-tgrp 41616 df-tendo 41628 df-edring 41630 df-dveca 41876 df-disoa 41902 df-dvech 41952 df-dib 42012 df-dic 42046 df-dih 42102 df-doch 42221 df-djh 42268

This theorem is used by: lclkrlem2q 42396

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