lclkrlem2v - Mathbox for Norm Megill

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Theorem lclkrlem2v 41991

Description: Lemma for lclkr 41996. When the hypotheses of lclkrlem2u 41990 and lclkrlem2u 41990 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 41931, which requires the orthomodular law dihoml4 41840 (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 41573	. . 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 39593	. . 3 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
13	1, 2, 3, 7, 12	lkrssv 39559	. 2 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉)
14		lclkrlem2o.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
15		eqid 2737	. . . 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 20967	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
21		eqid 2737	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
22	4, 5, 1, 17, 21, 6, 18, 19	dihprrn 41889	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	1, 15	lssss 20925	. . . . . . 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 41832	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = (𝑁‘{𝑋, 𝑌})))
26	22, 25	mpbid 232	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = (𝑁‘{𝑋, 𝑌}))
27	4, 14, 5, 1, 15, 16, 6, 20, 26	dochexmid 41931	. . 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 41572	. . . . 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 41983	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺)))
38	18	snssd 4753	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
39	19	snssd 4753	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
40	4, 5, 1, 14	dochdmj1 41853	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
41	6, 38, 39, 40	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
42		df-pr 4571	. . . . . . . . 9 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
43	42	fveq2i 6838	. . . . . . . 8 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
44	43	fveq2i 6838	. . . . . . 7 ⊢ ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌})))
45	38, 39	unssd 4133	. . . . . . . 8 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ 𝑉)
46	4, 5, 14, 1, 17, 6, 45	dochocsp 41842	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))) = ( ⊥ ‘({𝑋} ∪ {𝑌})))
47	44, 46	eqtrid 2784	. . . . . 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 4162	. . . . . 6 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
51	41, 47, 50	3eqtr4d 2782	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ((𝐿‘𝐸) ∩ (𝐿‘𝐺)))
52	2, 3, 8, 9, 7, 10, 11	lkrin 39627	. . . . 5 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊆ (𝐿‘(𝐸 + 𝐺)))
53	51, 52	eqsstrd 3957	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊆ (𝐿‘(𝐸 + 𝐺)))
54	15	lsssssubg 20947	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
55	7, 54	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
56	55, 20	sseldd 3923	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈))
57	4, 5, 1, 15, 14	dochlss 41817	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉) → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
58	6, 24, 57	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
59	55, 58	sseldd 3923	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (SubGrp‘𝑈))
60	2, 3, 15	lkrlss 39558	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹) → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈))
61	7, 12, 60	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈))
62	55, 61	sseldd 3923	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ (SubGrp‘𝑈))
63	16	lsmlub 19633	. . . . 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 1374	. . . 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 713	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ⊕ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊆ (𝐿‘(𝐸 + 𝐺)))
66	27, 65	eqsstrrd 3958	. 2 ⊢ (𝜑 → 𝑉 ⊆ (𝐿‘(𝐸 + 𝐺)))
67	13, 66	eqssd 3940	1 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 {csn 4568 {cpr 4570 ran crn 5626 ‘cfv 6493 (class class class)co 7361 Basecbs 17173 +_gcplusg 17214 ._rcmulr 17215 Scalarcsca 17217 ·_𝑠 cvsca 17218 0_gc0g 17396 -_gcsg 18905 SubGrpcsubg 19090 LSSumclsm 19603 inv_rcinvr 20361 LModclmod 20849 LSubSpclss 20920 LSpanclspn 20960 LFnlclfn 39520 LKerclk 39548 LDualcld 39586 HLchlt 39813 LHypclh 40447 DVecHcdvh 41541 DIsoHcdih 41691 ocHcoch 41810

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-riotaBAD 39416

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-tpos 8170 df-undef 8217 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-n0 12432 df-z 12519 df-uz 12783 df-fz 13456 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-sca 17230 df-vsca 17231 df-0g 17398 df-mre 17542 df-mrc 17543 df-acs 17545 df-proset 18254 df-poset 18273 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18392 df-clat 18459 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19093 df-cntz 19286 df-oppg 19315 df-lsm 19605 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-dvr 20375 df-drng 20702 df-lmod 20851 df-lss 20921 df-lsp 20961 df-lvec 21093 df-lsatoms 39439 df-lcv 39482 df-lfl 39521 df-lkr 39549 df-ldual 39587 df-oposet 39639 df-ol 39641 df-oml 39642 df-covers 39729 df-ats 39730 df-atl 39761 df-cvlat 39785 df-hlat 39814 df-llines 39961 df-lplanes 39962 df-lvols 39963 df-lines 39964 df-psubsp 39966 df-pmap 39967 df-padd 40259 df-lhyp 40451 df-laut 40452 df-ldil 40567 df-ltrn 40568 df-trl 40622 df-tgrp 41206 df-tendo 41218 df-edring 41220 df-dveca 41466 df-disoa 41492 df-dvech 41542 df-dib 41602 df-dic 41636 df-dih 41692 df-doch 41811 df-djh 41858

This theorem is referenced by: lclkrlem2w 41992

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