lclkrlem2l - Mathbox for Norm Megill

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Theorem lclkrlem2l 42841

Description: Lemma for lclkr 42856. 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 42840	. 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 42839	. 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 4750	. . . 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 4750	. . . 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 42838	. 2 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
64	34, 46, 63	pm2.61da2ne 3047	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 2147 ≠ wne 2959 ∖ cdif 4048 {csn 4588 ‘cfv 6566 (class class class)co 7451 Basecbs 17431 +_gcplusg 17472 Scalarcsca 17475 0_gc0g 17660 LSSumclsm 19899 LSpanclspn 21297 LSHypclsh 40298 LFnlclfn 40380 LKerclk 40408 LDualcld 40446 HLchlt 40673 LHypclh 41307 DVecHcdvh 42401 ocHcoch 42670

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-riotaBAD 40276

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-tpos 8281 df-undef 8328 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-n0 12647 df-z 12734 df-uz 13006 df-fz 13680 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-sca 17489 df-vsca 17490 df-0g 17662 df-mre 17806 df-mrc 17807 df-acs 17809 df-proset 18518 df-poset 18537 df-plt 18552 df-lub 18568 df-glb 18569 df-join 18570 df-meet 18571 df-p0 18647 df-p1 18648 df-lat 18656 df-clat 18723 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-grp 19197 df-minusg 19198 df-sbg 19199 df-subg 19383 df-cntz 19582 df-oppg 19611 df-lsm 19901 df-cmn 20047 df-abl 20048 df-mgp 20412 df-rng 20426 df-ur 20459 df-ring 20512 df-oppr 20618 df-dvdsr 20638 df-unit 20639 df-invr 20669 df-dvr 20682 df-drng 21033 df-lmod 21188 df-lss 21258 df-lsp 21298 df-lvec 21429 df-lsatoms 40299 df-lshyp 40300 df-lcv 40342 df-lfl 40381 df-lkr 40409 df-ldual 40447 df-oposet 40499 df-ol 40501 df-oml 40502 df-covers 40589 df-ats 40590 df-atl 40621 df-cvlat 40645 df-hlat 40674 df-llines 40821 df-lplanes 40822 df-lvols 40823 df-lines 40824 df-psubsp 40826 df-pmap 40827 df-padd 41119 df-lhyp 41311 df-laut 41312 df-ldil 41427 df-ltrn 41428 df-trl 41482 df-tgrp 42066 df-tendo 42078 df-edring 42080 df-dveca 42326 df-disoa 42352 df-dvech 42402 df-dib 42462 df-dic 42496 df-dih 42552 df-doch 42671 df-djh 42718

This theorem is used by: lclkrlem2q 42846

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