lclkrlem2v - Mathbox for Norm Megill

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Theorem lclkrlem2v 42725

Description: Lemma for lclkr 42730. When the hypotheses of lclkrlem2u 42724 and lclkrlem2u 42724 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 42665, which requires the orthomodular law dihoml4 42574 (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 42307	. . 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 40327	. . 3 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
13	1, 2, 3, 7, 12	lkrssv 40293	. 2 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉)
14		lclkrlem2o.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
15		eqid 2763	. . . 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 21265	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
21		eqid 2763	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
22	4, 5, 1, 17, 21, 6, 18, 19	dihprrn 42623	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	1, 15	lssss 21223	. . . . . . 7 ⊢ ((𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉)
24	20, 23	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉)
25	4, 21, 5, 1, 14, 6, 24	dochoccl 42566	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = (𝑁‘{𝑋, 𝑌})))
26	22, 25	mpbid 235	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = (𝑁‘{𝑋, 𝑌}))
27	4, 14, 5, 1, 15, 16, 6, 20, 26	dochexmid 42665	. . 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 42306	. . . . 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 42717	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺)))
38	18	snssd 4746	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
39	19	snssd 4746	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
40	4, 5, 1, 14	dochdmj1 42587	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
41	6, 38, 39, 40	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
42		df-pr 4590	. . . . . . . . 9 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
43	42	fveq2i 6896	. . . . . . . 8 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
44	43	fveq2i 6896	. . . . . . 7 ⊢ ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌})))
45	38, 39	unssd 4192	. . . . . . . 8 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ 𝑉)
46	4, 5, 14, 1, 17, 6, 45	dochocsp 42576	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))) = ( ⊥ ‘({𝑋} ∪ {𝑌})))
47	44, 46	eqtrid 2810	. . . . . 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 4120	. . . . . 6 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
51	41, 47, 50	3eqtr4d 2808	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ((𝐿‘𝐸) ∩ (𝐿‘𝐺)))
52	2, 3, 8, 9, 7, 10, 11	lkrin 40361	. . . . 5 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊆ (𝐿‘(𝐸 + 𝐺)))
53	51, 52	eqsstrd 3946	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊆ (𝐿‘(𝐸 + 𝐺)))
54	15	lsssssubg 21245	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
55	7, 54	syl 18	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
56	55, 20	sseldd 3913	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈))
57	4, 5, 1, 15, 14	dochlss 42551	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉) → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
58	6, 24, 57	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
59	55, 58	sseldd 3913	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (SubGrp‘𝑈))
60	2, 3, 15	lkrlss 40292	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹) → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈))
61	7, 12, 60	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈))
62	55, 61	sseldd 3913	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ (SubGrp‘𝑈))
63	16	lsmlub 19890	. . . . 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 1398	. . . 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 725	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ⊕ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊆ (𝐿‘(𝐸 + 𝐺)))
66	27, 65	eqsstrrd 3947	. 2 ⊢ (𝜑 → 𝑉 ⊆ (𝐿‘(𝐸 + 𝐺)))
67	13, 66	eqssd 3929	1 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = 𝑉)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2146 ⊆ wss 3899 ∩ cin 4091 ∪ cun 4151 {csn 4587 {cpr 4589 ran crn 5655 ‘cfv 6546 (class class class)co 7427 Basecbs 17399 +_gcplusg 17440 ._rcmulr 17441 Scalarcsca 17443 ·_𝑠 cvsca 17444 0_gc0g 17622 -_gcsg 19158 SubGrpcsubg 19342 LSSumclsm 19860 inv_rcinvr 20629 LModclmod 21147 LSubSpclss 21218 LSpanclspn 21258 LFnlclfn 40254 LKerclk 40282 LDualcld 40320 HLchlt 40547 LHypclh 41181 DVecHcdvh 42275 DIsoHcdih 42425 ocHcoch 42544

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-riotaBAD 40150

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-iin 4956 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-of 7700 df-om 7885 df-1st 8008 df-2nd 8009 df-tpos 8254 df-undef 8301 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-2o 8488 df-er 8728 df-map 8860 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-5 12418 df-6 12419 df-n0 12617 df-z 12704 df-uz 12976 df-fz 13650 df-struct 17337 df-sets 17354 df-slot 17372 df-ndx 17384 df-base 17400 df-ress 17421 df-plusg 17453 df-mulr 17454 df-sca 17456 df-vsca 17457 df-0g 17624 df-mre 17768 df-mrc 17769 df-acs 17771 df-proset 18480 df-poset 18499 df-plt 18514 df-lub 18530 df-glb 18531 df-join 18532 df-meet 18533 df-p0 18609 df-p1 18610 df-lat 18618 df-clat 18685 df-mgm 18828 df-sgrp 18920 df-mnd 18936 df-submnd 18991 df-grp 19159 df-minusg 19160 df-sbg 19161 df-subg 19345 df-cntz 19543 df-oppg 19572 df-lsm 19862 df-cmn 20008 df-abl 20009 df-mgp 20373 df-rng 20387 df-ur 20420 df-ring 20473 df-oppr 20579 df-dvdsr 20599 df-unit 20600 df-invr 20630 df-dvr 20643 df-drng 20994 df-lmod 21149 df-lss 21219 df-lsp 21259 df-lvec 21390 df-lsatoms 40173 df-lcv 40216 df-lfl 40255 df-lkr 40283 df-ldual 40321 df-oposet 40373 df-ol 40375 df-oml 40376 df-covers 40463 df-ats 40464 df-atl 40495 df-cvlat 40519 df-hlat 40548 df-llines 40695 df-lplanes 40696 df-lvols 40697 df-lines 40698 df-psubsp 40700 df-pmap 40701 df-padd 40993 df-lhyp 41185 df-laut 41186 df-ldil 41301 df-ltrn 41302 df-trl 41356 df-tgrp 41940 df-tendo 41952 df-edring 41954 df-dveca 42200 df-disoa 42226 df-dvech 42276 df-dib 42336 df-dic 42370 df-dih 42426 df-doch 42545 df-djh 42592

This theorem is used by: lclkrlem2w 42726

W3C validator