lclkrlem2l - Mathbox for Norm Megill

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Theorem lclkrlem2l 42325

Description: Lemma for lclkr 42340. 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 42324	. 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 42323	. 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 4755	. . . 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 4755	. . . 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 42322	. 2 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
64	34, 46, 63	pm2.61da2ne 3048	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 2146 ≠ wne 2960 ∖ cdif 3903 {csn 4591 ‘cfv 6540 (class class class)co 7416 Basecbs 17287 +_gcplusg 17328 Scalarcsca 17331 0_gc0g 17510 LSSumclsm 19728 LSpanclspn 21122 LSHypclsh 39782 LFnlclfn 39864 LKerclk 39892 LDualcld 39930 HLchlt 40157 LHypclh 40791 DVecHcdvh 41885 ocHcoch 42154

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-riotaBAD 39760

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 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 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-n0 12516 df-z 12603 df-uz 12875 df-fz 13548 df-struct 17225 df-sets 17242 df-slot 17260 df-ndx 17272 df-base 17288 df-ress 17309 df-plusg 17341 df-mulr 17342 df-sca 17344 df-vsca 17345 df-0g 17512 df-mre 17656 df-mrc 17657 df-acs 17659 df-proset 18368 df-poset 18387 df-plt 18402 df-lub 18418 df-glb 18419 df-join 18420 df-meet 18421 df-p0 18497 df-p1 18498 df-lat 18506 df-clat 18573 df-mgm 18716 df-sgrp 18799 df-mnd 18815 df-submnd 18866 df-grp 19027 df-minusg 19028 df-sbg 19029 df-subg 19213 df-cntz 19411 df-oppg 19440 df-lsm 19730 df-cmn 19876 df-abl 19877 df-mgp 20241 df-rng 20255 df-ur 20288 df-ring 20341 df-oppr 20445 df-dvdsr 20465 df-unit 20466 df-invr 20496 df-dvr 20509 df-drng 20859 df-lmod 21013 df-lss 21083 df-lsp 21123 df-lvec 21254 df-lsatoms 39783 df-lshyp 39784 df-lcv 39826 df-lfl 39865 df-lkr 39893 df-ldual 39931 df-oposet 39983 df-ol 39985 df-oml 39986 df-covers 40073 df-ats 40074 df-atl 40105 df-cvlat 40129 df-hlat 40158 df-llines 40305 df-lplanes 40306 df-lvols 40307 df-lines 40308 df-psubsp 40310 df-pmap 40311 df-padd 40603 df-lhyp 40795 df-laut 40796 df-ldil 40911 df-ltrn 40912 df-trl 40966 df-tgrp 41550 df-tendo 41562 df-edring 41564 df-dveca 41810 df-disoa 41836 df-dvech 41886 df-dib 41946 df-dic 41980 df-dih 42036 df-doch 42155 df-djh 42202

This theorem is used by: lclkrlem2q 42330

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