lclkrlem2v - Mathbox for Norm Megill

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Theorem lclkrlem2v 39542

Description: Lemma for lclkr 39547. When the hypotheses of lclkrlem2u 39541 and lclkrlem2u 39541 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 39482, which requires the orthomodular law dihoml4 39391 (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 39124	. . 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 37144	. . 3 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
13	1, 2, 3, 7, 12	lkrssv 37110	. 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 20240	. . . 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 39440	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	1, 15	lssss 20198	. . . . . . 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 39383	. . . . 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 39482	. . 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 39123	. . . . 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 39534	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺)))
38	18	snssd 4742	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
39	19	snssd 4742	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
40	4, 5, 1, 14	dochdmj1 39404	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
41	6, 38, 39, 40	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
42		df-pr 4564	. . . . . . . . 9 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
43	42	fveq2i 6777	. . . . . . . 8 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
44	43	fveq2i 6777	. . . . . . 7 ⊢ ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌})))
45	38, 39	unssd 4120	. . . . . . . 8 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ 𝑉)
46	4, 5, 14, 1, 17, 6, 45	dochocsp 39393	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))) = ( ⊥ ‘({𝑋} ∪ {𝑌})))
47	44, 46	eqtrid 2790	. . . . . 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 4147	. . . . . 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 37178	. . . . 5 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊆ (𝐿‘(𝐸 + 𝐺)))
53	51, 52	eqsstrd 3959	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊆ (𝐿‘(𝐸 + 𝐺)))
54	15	lsssssubg 20220	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
55	7, 54	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
56	55, 20	sseldd 3922	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈))
57	4, 5, 1, 15, 14	dochlss 39368	. . . . . . 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 3922	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (SubGrp‘𝑈))
60	2, 3, 15	lkrlss 37109	. . . . . . 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 3922	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ (SubGrp‘𝑈))
63	16	lsmlub 19270	. . . . 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 1370	. . . 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 709	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ⊕ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊆ (𝐿‘(𝐸 + 𝐺)))
66	27, 65	eqsstrrd 3960	. 2 ⊢ (𝜑 → 𝑉 ⊆ (𝐿‘(𝐸 + 𝐺)))
67	13, 66	eqssd 3938	1 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∪ cun 3885 ∩ cin 3886 ⊆ wss 3887 {csn 4561 {cpr 4563 ran crn 5590 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +_gcplusg 16962 ._rcmulr 16963 Scalarcsca 16965 ·_𝑠 cvsca 16966 0_gc0g 17150 -_gcsg 18579 SubGrpcsubg 18749 LSSumclsm 19239 inv_rcinvr 19913 LModclmod 20123 LSubSpclss 20193 LSpanclspn 20233 LFnlclfn 37071 LKerclk 37099 LDualcld 37137 HLchlt 37364 LHypclh 37998 DVecHcdvh 39092 DIsoHcdih 39242 ocHcoch 39361

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-mre 17295 df-mrc 17296 df-acs 17298 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-oppg 18950 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lsatoms 36990 df-lcv 37033 df-lfl 37072 df-lkr 37100 df-ldual 37138 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tgrp 38757 df-tendo 38769 df-edring 38771 df-dveca 39017 df-disoa 39043 df-dvech 39093 df-dib 39153 df-dic 39187 df-dih 39243 df-doch 39362 df-djh 39409

This theorem is referenced by: lclkrlem2w 39543

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