lclkrlem2b - Mathbox for Norm Megill

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Theorem lclkrlem2b 38126

Description: Lemma for lclkr 38151. (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 473	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		lclkrlem2a.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
12	11	adantr 473	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → 𝐵 ∈ (𝑉 ∖ { 0 }))
13		lclkrlem2a.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
14	13	adantr 473	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
15		lclkrlem2a.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
16	15	adantr 473	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
17		lclkrlem2a.e	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))
18	17	adantr 473	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))
19		simpr 477	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → ¬ 𝑋 ∈ ( ⊥ ‘{𝐵}))
20	1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 19	lclkrlem2a 38125	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
21	1, 3, 9	dvhlmod 37728	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod)
22		lmodabl 19415	. . . . . . 7 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel)
23	21, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Abel)
24		eqid 2771	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
25	24	lsssssubg 19464	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
26	21, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
27	13	eldifad 3834	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
28	4, 24, 7	lspsncl 19483	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
29	21, 27, 28	syl2anc 576	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
30	26, 29	sseldd 3852	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑈))
31	15	eldifad 3834	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
32	4, 24, 7	lspsncl 19483	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
33	21, 31, 32	syl2anc 576	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
34	26, 33	sseldd 3852	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑈))
35	6	lsmcom 18746	. . . . . 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 1352	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = ((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋})))
37	36	ineq1d 4069	. . . 4 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋})) ∩ ( ⊥ ‘{𝐵})))
38	37	adantr 473	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋})) ∩ ( ⊥ ‘{𝐵})))
39	9	adantr 473	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
40	11	adantr 473	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → 𝐵 ∈ (𝑉 ∖ { 0 }))
41	15	adantr 473	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
42	13	adantr 473	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
43	17	necomd 3015	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ≠ ( ⊥ ‘{𝑋}))
44	43	adantr 473	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → ( ⊥ ‘{𝑌}) ≠ ( ⊥ ‘{𝑋}))
45		simpr 477	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))
46	1, 2, 3, 4, 5, 6, 7, 8, 39, 40, 41, 42, 44, 45	lclkrlem2a 38125	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
47	38, 46	eqeltrd 2859	. 2 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
48		lclkrlem2b.da	. 2 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
49	20, 47, 48	mpjaodan 942	1 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∨ wo 834 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 ∖ cdif 3819 ∩ cin 3821 ⊆ wss 3822 {csn 4435 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 0_gc0g 16567 SubGrpcsubg 18069 LSSumclsm 18532 Abelcabl 18679 LModclmod 19368 LSubSpclss 19437 LSpanclspn 19477 LSAtomsclsa 35592 HLchlt 35968 LHypclh 36602 DVecHcdvh 37696 ocHcoch 37965

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-riotaBAD 35571

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-tpos 7693 df-undef 7740 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-n0 11706 df-z 11792 df-uz 12057 df-fz 12707 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-sca 16435 df-vsca 16436 df-0g 16569 df-mre 16727 df-mrc 16728 df-acs 16730 df-proset 17408 df-poset 17426 df-plt 17438 df-lub 17454 df-glb 17455 df-join 17456 df-meet 17457 df-p0 17519 df-p1 17520 df-lat 17526 df-clat 17588 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-subg 18072 df-cntz 18230 df-oppg 18257 df-lsm 18534 df-cmn 18680 df-abl 18681 df-mgp 18975 df-ur 18987 df-ring 19034 df-oppr 19108 df-dvdsr 19126 df-unit 19127 df-invr 19157 df-dvr 19168 df-drng 19239 df-lmod 19370 df-lss 19438 df-lsp 19478 df-lvec 19609 df-lsatoms 35594 df-lshyp 35595 df-lcv 35637 df-oposet 35794 df-ol 35796 df-oml 35797 df-covers 35884 df-ats 35885 df-atl 35916 df-cvlat 35940 df-hlat 35969 df-llines 36116 df-lplanes 36117 df-lvols 36118 df-lines 36119 df-psubsp 36121 df-pmap 36122 df-padd 36414 df-lhyp 36606 df-laut 36607 df-ldil 36722 df-ltrn 36723 df-trl 36777 df-tgrp 37361 df-tendo 37373 df-edring 37375 df-dveca 37621 df-disoa 37647 df-dvech 37697 df-dib 37757 df-dic 37791 df-dih 37847 df-doch 37966 df-djh 38013

This theorem is referenced by: lclkrlem2c 38127

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