lclkrlem2b - Mathbox for Norm Megill

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Theorem lclkrlem2b 39449

Description: Lemma for lclkr 39474. (Contributed by NM, 17-Jan-2015.)

**Hypotheses**
Ref	Expression
lclkrlem2a.h	⊢ 𝐻 = (LHyp‘𝐾)
lclkrlem2a.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lclkrlem2a.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lclkrlem2a.v	⊢ 𝑉 = (Base‘𝑈)
lclkrlem2a.z	⊢ 0 = (0_g‘𝑈)
lclkrlem2a.p	⊢ ⊕ = (LSSum‘𝑈)
lclkrlem2a.n	⊢ 𝑁 = (LSpan‘𝑈)
lclkrlem2a.a	⊢ 𝐴 = (LSAtoms‘𝑈)
lclkrlem2a.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lclkrlem2a.b	⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
lclkrlem2a.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lclkrlem2a.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lclkrlem2a.e	⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))
lclkrlem2b.da	⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))

**Assertion**
Ref	Expression
lclkrlem2b	⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)

**Proof of Theorem lclkrlem2b**
Step	Hyp	Ref	Expression
1		lclkrlem2a.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lclkrlem2a.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lclkrlem2a.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lclkrlem2a.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
5		lclkrlem2a.z	. . 3 ⊢ 0 = (0_g‘𝑈)
6		lclkrlem2a.p	. . 3 ⊢ ⊕ = (LSSum‘𝑈)
7		lclkrlem2a.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
8		lclkrlem2a.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑈)
9		lclkrlem2a.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		lclkrlem2a.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → 𝐵 ∈ (𝑉 ∖ { 0 }))
13		lclkrlem2a.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
14	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
15		lclkrlem2a.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
16	15	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
17		lclkrlem2a.e	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))
18	17	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))
19		simpr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → ¬ 𝑋 ∈ ( ⊥ ‘{𝐵}))
20	1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 19	lclkrlem2a 39448	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
21	1, 3, 9	dvhlmod 39051	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod)
22		lmodabl 20085	. . . . . . 7 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel)
23	21, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Abel)
24		eqid 2738	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
25	24	lsssssubg 20135	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
26	21, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
27	13	eldifad 3895	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
28	4, 24, 7	lspsncl 20154	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
29	21, 27, 28	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
30	26, 29	sseldd 3918	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑈))
31	15	eldifad 3895	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
32	4, 24, 7	lspsncl 20154	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
33	21, 31, 32	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
34	26, 33	sseldd 3918	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑈))
35	6	lsmcom 19374	. . . . . 6 ⊢ ((𝑈 ∈ Abel ∧ (𝑁‘{𝑋}) ∈ (SubGrp‘𝑈) ∧ (𝑁‘{𝑌}) ∈ (SubGrp‘𝑈)) → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = ((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋})))
36	23, 30, 34, 35	syl3anc 1369	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = ((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋})))
37	36	ineq1d 4142	. . . 4 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋})) ∩ ( ⊥ ‘{𝐵})))
38	37	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋})) ∩ ( ⊥ ‘{𝐵})))
39	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
40	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → 𝐵 ∈ (𝑉 ∖ { 0 }))
41	15	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
42	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
43	17	necomd 2998	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ≠ ( ⊥ ‘{𝑋}))
44	43	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → ( ⊥ ‘{𝑌}) ≠ ( ⊥ ‘{𝑋}))
45		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))
46	1, 2, 3, 4, 5, 6, 7, 8, 39, 40, 41, 42, 44, 45	lclkrlem2a 39448	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
47	38, 46	eqeltrd 2839	. 2 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
48		lclkrlem2b.da	. 2 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
49	20, 47, 48	mpjaodan 955	1 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∖ cdif 3880 ∩ cin 3882 ⊆ wss 3883 {csn 4558 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 0_gc0g 17067 SubGrpcsubg 18664 LSSumclsm 19154 Abelcabl 19302 LModclmod 20038 LSubSpclss 20108 LSpanclspn 20148 LSAtomsclsa 36915 HLchlt 37291 LHypclh 37925 DVecHcdvh 39019 ocHcoch 39288

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-undef 8060 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-mre 17212 df-mrc 17213 df-acs 17215 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cntz 18838 df-oppg 18865 df-lsm 19156 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lvec 20280 df-lsatoms 36917 df-lshyp 36918 df-lcv 36960 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-tgrp 38684 df-tendo 38696 df-edring 38698 df-dveca 38944 df-disoa 38970 df-dvech 39020 df-dib 39080 df-dic 39114 df-dih 39170 df-doch 39289 df-djh 39336

This theorem is referenced by: lclkrlem2c 39450

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