lclkrlem2v - Mathbox for Norm Megill

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Theorem lclkrlem2v 39469

Description: Lemma for lclkr 39474. When the hypotheses of lclkrlem2u 39468 and lclkrlem2u 39468 are negated, the functional sum must be zero, so the kernel is the vector space. We make use of the law of excluded middle, dochexmid 39409, which requires the orthomodular law dihoml4 39318 (Lemma 3.3 of [Holland95] p. 214). (Contributed by NM, 16-Jan-2015.)

**Hypotheses**
Ref	Expression
lclkrlem2m.v	⊢ 𝑉 = (Base‘𝑈)
lclkrlem2m.t	⊢ · = ( ·_𝑠 ‘𝑈)
lclkrlem2m.s	⊢ 𝑆 = (Scalar‘𝑈)
lclkrlem2m.q	⊢ × = (._r‘𝑆)
lclkrlem2m.z	⊢ 0 = (0_g‘𝑆)
lclkrlem2m.i	⊢ 𝐼 = (inv_r‘𝑆)
lclkrlem2m.m	⊢ − = (-_g‘𝑈)
lclkrlem2m.f	⊢ 𝐹 = (LFnl‘𝑈)
lclkrlem2m.d	⊢ 𝐷 = (LDual‘𝑈)
lclkrlem2m.p	⊢ + = (+_g‘𝐷)
lclkrlem2m.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
lclkrlem2m.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
lclkrlem2m.e	⊢ (𝜑 → 𝐸 ∈ 𝐹)
lclkrlem2m.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
lclkrlem2n.n	⊢ 𝑁 = (LSpan‘𝑈)
lclkrlem2n.l	⊢ 𝐿 = (LKer‘𝑈)
lclkrlem2o.h	⊢ 𝐻 = (LHyp‘𝐾)
lclkrlem2o.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lclkrlem2o.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lclkrlem2o.a	⊢ ⊕ = (LSSum‘𝑈)
lclkrlem2o.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lclkrlem2q.le	⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
lclkrlem2q.lg	⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
lclkrlem2v.j	⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) = 0 )
lclkrlem2v.k	⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) = 0 )

**Assertion**
Ref	Expression
lclkrlem2v	⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = 𝑉)

**Proof of Theorem lclkrlem2v**
Step	Hyp	Ref	Expression
1		lclkrlem2m.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
2		lclkrlem2m.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
3		lclkrlem2n.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
4		lclkrlem2o.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
5		lclkrlem2o.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		lclkrlem2o.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7	4, 5, 6	dvhlmod 39051	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
8		lclkrlem2m.d	. . . 4 ⊢ 𝐷 = (LDual‘𝑈)
9		lclkrlem2m.p	. . . 4 ⊢ + = (+_g‘𝐷)
10		lclkrlem2m.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
11		lclkrlem2m.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
12	2, 8, 9, 7, 10, 11	ldualvaddcl 37071	. . 3 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
13	1, 2, 3, 7, 12	lkrssv 37037	. 2 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉)
14		lclkrlem2o.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
15		eqid 2738	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
16		lclkrlem2o.a	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
17		lclkrlem2n.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
18		lclkrlem2m.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
19		lclkrlem2m.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
20	1, 15, 17, 7, 18, 19	lspprcl 20155	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
21		eqid 2738	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
22	4, 5, 1, 17, 21, 6, 18, 19	dihprrn 39367	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	1, 15	lssss 20113	. . . . . . 7 ⊢ ((𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉)
24	20, 23	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉)
25	4, 21, 5, 1, 14, 6, 24	dochoccl 39310	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = (𝑁‘{𝑋, 𝑌})))
26	22, 25	mpbid 231	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = (𝑁‘{𝑋, 𝑌}))
27	4, 14, 5, 1, 15, 16, 6, 20, 26	dochexmid 39409	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ⊕ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = 𝑉)
28		lclkrlem2m.t	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝑈)
29		lclkrlem2m.s	. . . . 5 ⊢ 𝑆 = (Scalar‘𝑈)
30		lclkrlem2m.q	. . . . 5 ⊢ × = (._r‘𝑆)
31		lclkrlem2m.z	. . . . 5 ⊢ 0 = (0_g‘𝑆)
32		lclkrlem2m.i	. . . . 5 ⊢ 𝐼 = (inv_r‘𝑆)
33		lclkrlem2m.m	. . . . 5 ⊢ − = (-_g‘𝑈)
34	4, 5, 6	dvhlvec 39050	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
35		lclkrlem2v.j	. . . . 5 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) = 0 )
36		lclkrlem2v.k	. . . . 5 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) = 0 )
37	1, 28, 29, 30, 31, 32, 33, 2, 8, 9, 18, 19, 10, 11, 17, 3, 34, 35, 36	lclkrlem2n 39461	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺)))
38	18	snssd 4739	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
39	19	snssd 4739	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
40	4, 5, 1, 14	dochdmj1 39331	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
41	6, 38, 39, 40	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
42		df-pr 4561	. . . . . . . . 9 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
43	42	fveq2i 6759	. . . . . . . 8 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
44	43	fveq2i 6759	. . . . . . 7 ⊢ ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌})))
45	38, 39	unssd 4116	. . . . . . . 8 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ 𝑉)
46	4, 5, 14, 1, 17, 6, 45	dochocsp 39320	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))) = ( ⊥ ‘({𝑋} ∪ {𝑌})))
47	44, 46	syl5eq 2791	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘({𝑋} ∪ {𝑌})))
48		lclkrlem2q.le	. . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
49		lclkrlem2q.lg	. . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
50	48, 49	ineq12d 4144	. . . . . 6 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
51	41, 47, 50	3eqtr4d 2788	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ((𝐿‘𝐸) ∩ (𝐿‘𝐺)))
52	2, 3, 8, 9, 7, 10, 11	lkrin 37105	. . . . 5 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊆ (𝐿‘(𝐸 + 𝐺)))
53	51, 52	eqsstrd 3955	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊆ (𝐿‘(𝐸 + 𝐺)))
54	15	lsssssubg 20135	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
55	7, 54	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
56	55, 20	sseldd 3918	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈))
57	4, 5, 1, 15, 14	dochlss 39295	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉) → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
58	6, 24, 57	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
59	55, 58	sseldd 3918	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (SubGrp‘𝑈))
60	2, 3, 15	lkrlss 37036	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹) → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈))
61	7, 12, 60	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈))
62	55, 61	sseldd 3918	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ (SubGrp‘𝑈))
63	16	lsmlub 19185	. . . . 5 ⊢ (((𝑁‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈) ∧ ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (SubGrp‘𝑈) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (SubGrp‘𝑈)) → (((𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺)) ∧ ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊆ (𝐿‘(𝐸 + 𝐺))) ↔ ((𝑁‘{𝑋, 𝑌}) ⊕ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊆ (𝐿‘(𝐸 + 𝐺))))
64	56, 59, 62, 63	syl3anc 1369	. . . 4 ⊢ (𝜑 → (((𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺)) ∧ ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊆ (𝐿‘(𝐸 + 𝐺))) ↔ ((𝑁‘{𝑋, 𝑌}) ⊕ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊆ (𝐿‘(𝐸 + 𝐺))))
65	37, 53, 64	mpbi2and 708	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ⊕ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊆ (𝐿‘(𝐸 + 𝐺)))
66	27, 65	eqsstrrd 3956	. 2 ⊢ (𝜑 → 𝑉 ⊆ (𝐿‘(𝐸 + 𝐺)))
67	13, 66	eqssd 3934	1 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 {csn 4558 {cpr 4560 ran crn 5581 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 +_gcplusg 16888 ._rcmulr 16889 Scalarcsca 16891 ·_𝑠 cvsca 16892 0_gc0g 17067 -_gcsg 18494 SubGrpcsubg 18664 LSSumclsm 19154 inv_rcinvr 19828 LModclmod 20038 LSubSpclss 20108 LSpanclspn 20148 LFnlclfn 36998 LKerclk 37026 LDualcld 37064 HLchlt 37291 LHypclh 37925 DVecHcdvh 39019 DIsoHcdih 39169 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-of 7511 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-lcv 36960 df-lfl 36999 df-lkr 37027 df-ldual 37065 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: lclkrlem2w 39470

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