lclkrlem2l - Mathbox for Norm Megill

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Theorem lclkrlem2l 41778

Description: Lemma for lclkr 41793. 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 41777	. 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 41776	. 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 4742	. . . 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 4742	. . . 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 41775	. 2 ⊢ ((𝜑 ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
64	34, 46, 63	pm2.61da2ne 3020	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 1541 ∈ wcel 2113 ≠ wne 2932 ∖ cdif 3898 {csn 4580 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 +_gcplusg 17177 Scalarcsca 17180 0_gc0g 17359 LSSumclsm 19563 LSpanclspn 20922 LSHypclsh 39235 LFnlclfn 39317 LKerclk 39345 LDualcld 39383 HLchlt 39610 LHypclh 40244 DVecHcdvh 41338 ocHcoch 41607

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-riotaBAD 39213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-undef 8215 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-0g 17361 df-mre 17505 df-mrc 17506 df-acs 17508 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-join 18269 df-meet 18270 df-p0 18346 df-p1 18347 df-lat 18355 df-clat 18422 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-cntz 19246 df-oppg 19275 df-lsm 19565 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-drng 20664 df-lmod 20813 df-lss 20883 df-lsp 20923 df-lvec 21055 df-lsatoms 39236 df-lshyp 39237 df-lcv 39279 df-lfl 39318 df-lkr 39346 df-ldual 39384 df-oposet 39436 df-ol 39438 df-oml 39439 df-covers 39526 df-ats 39527 df-atl 39558 df-cvlat 39582 df-hlat 39611 df-llines 39758 df-lplanes 39759 df-lvols 39760 df-lines 39761 df-psubsp 39763 df-pmap 39764 df-padd 40056 df-lhyp 40248 df-laut 40249 df-ldil 40364 df-ltrn 40365 df-trl 40419 df-tgrp 41003 df-tendo 41015 df-edring 41017 df-dveca 41263 df-disoa 41289 df-dvech 41339 df-dib 41399 df-dic 41433 df-dih 41489 df-doch 41608 df-djh 41655

This theorem is referenced by: lclkrlem2q 41783

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