lclkrlem2v - Mathbox for Norm Megill

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Theorem lclkrlem2v 41727

Description: Lemma for lclkr 41732. When the hypotheses of lclkrlem2u 41726 and lclkrlem2u 41726 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 41667, which requires the orthomodular law dihoml4 41576 (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 41309	. . 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 39329	. . 3 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
13	1, 2, 3, 7, 12	lkrssv 39295	. 2 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉)
14		lclkrlem2o.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
15		eqid 2734	. . . 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 20927	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
21		eqid 2734	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
22	4, 5, 1, 17, 21, 6, 18, 19	dihprrn 41625	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	1, 15	lssss 20885	. . . . . . 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 41568	. . . . 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 41667	. . 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 41308	. . . . 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 41719	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺)))
38	18	snssd 4763	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
39	19	snssd 4763	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
40	4, 5, 1, 14	dochdmj1 41589	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
41	6, 38, 39, 40	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
42		df-pr 4581	. . . . . . . . 9 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
43	42	fveq2i 6835	. . . . . . . 8 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
44	43	fveq2i 6835	. . . . . . 7 ⊢ ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌})))
45	38, 39	unssd 4142	. . . . . . . 8 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ 𝑉)
46	4, 5, 14, 1, 17, 6, 45	dochocsp 41578	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))) = ( ⊥ ‘({𝑋} ∪ {𝑌})))
47	44, 46	eqtrid 2781	. . . . . 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 4171	. . . . . 6 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
51	41, 47, 50	3eqtr4d 2779	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ((𝐿‘𝐸) ∩ (𝐿‘𝐺)))
52	2, 3, 8, 9, 7, 10, 11	lkrin 39363	. . . . 5 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊆ (𝐿‘(𝐸 + 𝐺)))
53	51, 52	eqsstrd 3966	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊆ (𝐿‘(𝐸 + 𝐺)))
54	15	lsssssubg 20907	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
55	7, 54	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
56	55, 20	sseldd 3932	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈))
57	4, 5, 1, 15, 14	dochlss 41553	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉) → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
58	6, 24, 57	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
59	55, 58	sseldd 3932	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (SubGrp‘𝑈))
60	2, 3, 15	lkrlss 39294	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹) → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈))
61	7, 12, 60	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈))
62	55, 61	sseldd 3932	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ (SubGrp‘𝑈))
63	16	lsmlub 19591	. . . . 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 1373	. . . 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 712	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ⊕ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊆ (𝐿‘(𝐸 + 𝐺)))
66	27, 65	eqsstrrd 3967	. 2 ⊢ (𝜑 → 𝑉 ⊆ (𝐿‘(𝐸 + 𝐺)))
67	13, 66	eqssd 3949	1 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 {csn 4578 {cpr 4580 ran crn 5623 ‘cfv 6490 (class class class)co 7356 Basecbs 17134 +_gcplusg 17175 ._rcmulr 17176 Scalarcsca 17178 ·_𝑠 cvsca 17179 0_gc0g 17357 -_gcsg 18863 SubGrpcsubg 19048 LSSumclsm 19561 inv_rcinvr 20321 LModclmod 20809 LSubSpclss 20880 LSpanclspn 20920 LFnlclfn 39256 LKerclk 39284 LDualcld 39322 HLchlt 39549 LHypclh 40183 DVecHcdvh 41277 DIsoHcdih 41427 ocHcoch 41546

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 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-riotaBAD 39152

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-undef 8213 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-n0 12400 df-z 12487 df-uz 12750 df-fz 13422 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-sca 17191 df-vsca 17192 df-0g 17359 df-mre 17503 df-mrc 17504 df-acs 17506 df-proset 18215 df-poset 18234 df-plt 18249 df-lub 18265 df-glb 18266 df-join 18267 df-meet 18268 df-p0 18344 df-p1 18345 df-lat 18353 df-clat 18420 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19051 df-cntz 19244 df-oppg 19273 df-lsm 19563 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20271 df-dvdsr 20291 df-unit 20292 df-invr 20322 df-dvr 20335 df-drng 20662 df-lmod 20811 df-lss 20881 df-lsp 20921 df-lvec 21053 df-lsatoms 39175 df-lcv 39218 df-lfl 39257 df-lkr 39285 df-ldual 39323 df-oposet 39375 df-ol 39377 df-oml 39378 df-covers 39465 df-ats 39466 df-atl 39497 df-cvlat 39521 df-hlat 39550 df-llines 39697 df-lplanes 39698 df-lvols 39699 df-lines 39700 df-psubsp 39702 df-pmap 39703 df-padd 39995 df-lhyp 40187 df-laut 40188 df-ldil 40303 df-ltrn 40304 df-trl 40358 df-tgrp 40942 df-tendo 40954 df-edring 40956 df-dveca 41202 df-disoa 41228 df-dvech 41278 df-dib 41338 df-dic 41372 df-dih 41428 df-doch 41547 df-djh 41594

This theorem is referenced by: lclkrlem2w 41728

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