lclkrlem2l - Mathbox for Norm Megill

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Theorem lclkrlem2l 41512

Description: Lemma for lclkr 41527. 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 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		lclkrlem2f.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
18	17	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → 𝐵 ∈ (𝑉 ∖ { 0 }))
19		lclkrlem2f.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
20	19	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → 𝐸 ∈ 𝐹)
21		lclkrlem2f.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
22	21	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → 𝐺 ∈ 𝐹)
23		lclkrlem2f.le	. . . 4 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
24	23	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
25		lclkrlem2f.lg	. . . 4 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
26	25	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
27		lclkrlem2f.kb	. . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
28	27	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
29		lclkrlem2f.nx	. . . 4 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
30	29	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
31		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → 𝑋 = 0 )
32		lclkrlem2l.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
33	32	adantr 480	. . 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 41511	. 2 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
35	15	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
36	17	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝐵 ∈ (𝑉 ∖ { 0 }))
37	19	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝐸 ∈ 𝐹)
38	21	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝐺 ∈ 𝐹)
39	23	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
40	25	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
41	27	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
42	29	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
43		lclkrlem2l.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
44	43	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑋 ∈ 𝑉)
45		simpr 484	. . 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 41510	. 2 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
47	15	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
48	17	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝐵 ∈ (𝑉 ∖ { 0 }))
49	19	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝐸 ∈ 𝐹)
50	21	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝐺 ∈ 𝐹)
51	23	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
52	25	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
53	27	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
54	29	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
55	43	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝑋 ∈ 𝑉)
56		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝑋 ≠ 0 )
57		eldifsn 4750	. . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ))
58	55, 56, 57	sylanbrc 583	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝑋 ∈ (𝑉 ∖ { 0 }))
59	32	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝑌 ∈ 𝑉)
60		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → 𝑌 ≠ 0 )
61		eldifsn 4750	. . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ))
62	59, 60, 61	sylanbrc 583	. . 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 41509	. 2 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
64	34, 46, 63	pm2.61da2ne 3013	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3911 {csn 4589 ‘cfv 6511 (class class class)co 7387 Basecbs 17179 +_gcplusg 17220 Scalarcsca 17223 0_gc0g 17402 LSSumclsm 19564 LSpanclspn 20877 LSHypclsh 38968 LFnlclfn 39050 LKerclk 39078 LDualcld 39116 HLchlt 39343 LHypclh 39978 DVecHcdvh 41072 ocHcoch 41341

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-riotaBAD 38946

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-undef 8252 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-mre 17547 df-mrc 17548 df-acs 17550 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18391 df-clat 18458 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cntz 19249 df-oppg 19278 df-lsm 19566 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-drng 20640 df-lmod 20768 df-lss 20838 df-lsp 20878 df-lvec 21010 df-lsatoms 38969 df-lshyp 38970 df-lcv 39012 df-lfl 39051 df-lkr 39079 df-ldual 39117 df-oposet 39169 df-ol 39171 df-oml 39172 df-covers 39259 df-ats 39260 df-atl 39291 df-cvlat 39315 df-hlat 39344 df-llines 39492 df-lplanes 39493 df-lvols 39494 df-lines 39495 df-psubsp 39497 df-pmap 39498 df-padd 39790 df-lhyp 39982 df-laut 39983 df-ldil 40098 df-ltrn 40099 df-trl 40153 df-tgrp 40737 df-tendo 40749 df-edring 40751 df-dveca 40997 df-disoa 41023 df-dvech 41073 df-dib 41133 df-dic 41167 df-dih 41223 df-doch 41342 df-djh 41389

This theorem is referenced by: lclkrlem2q 41517

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