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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2v | Structured version Visualization version GIF version |
Description: Lemma for lclkr 38661. When the hypotheses of lclkrlem2u 38655 and lclkrlem2u 38655 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 38596, which requires the orthomodular law dihoml4 38505 (Lemma 3.3 of [Holland95] p. 214). (Contributed by NM, 16-Jan-2015.) |
Ref | Expression |
---|---|
lclkrlem2m.v | ⊢ 𝑉 = (Base‘𝑈) |
lclkrlem2m.t | ⊢ · = ( ·𝑠 ‘𝑈) |
lclkrlem2m.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lclkrlem2m.q | ⊢ × = (.r‘𝑆) |
lclkrlem2m.z | ⊢ 0 = (0g‘𝑆) |
lclkrlem2m.i | ⊢ 𝐼 = (invr‘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 ) |
Ref | Expression |
---|---|
lclkrlem2v | ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = 𝑉) |
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 38238 | . . 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 36258 | . . 3 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹) |
13 | 1, 2, 3, 7, 12 | lkrssv 36224 | . 2 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉) |
14 | lclkrlem2o.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
15 | eqid 2819 | . . . 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 19742 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
21 | eqid 2819 | . . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
22 | 4, 5, 1, 17, 21, 6, 18, 19 | dihprrn 38554 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
23 | 1, 15 | lssss 19700 | . . . . . . 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 38497 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = (𝑁‘{𝑋, 𝑌}))) |
26 | 22, 25 | mpbid 234 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = (𝑁‘{𝑋, 𝑌})) |
27 | 4, 14, 5, 1, 15, 16, 6, 20, 26 | dochexmid 38596 | . . 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 = (0g‘𝑆) | |
32 | lclkrlem2m.i | . . . . 5 ⊢ 𝐼 = (invr‘𝑆) | |
33 | lclkrlem2m.m | . . . . 5 ⊢ − = (-g‘𝑈) | |
34 | 4, 5, 6 | dvhlvec 38237 | . . . . 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 38648 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺))) |
38 | 18 | snssd 4734 | . . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
39 | 19 | snssd 4734 | . . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
40 | 4, 5, 1, 14 | dochdmj1 38518 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))) |
41 | 6, 38, 39, 40 | syl3anc 1366 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))) |
42 | df-pr 4562 | . . . . . . . . 9 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌}) | |
43 | 42 | fveq2i 6666 | . . . . . . . 8 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌})) |
44 | 43 | fveq2i 6666 | . . . . . . 7 ⊢ ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))) |
45 | 38, 39 | unssd 4160 | . . . . . . . 8 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ 𝑉) |
46 | 4, 5, 14, 1, 17, 6, 45 | dochocsp 38507 | . . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))) = ( ⊥ ‘({𝑋} ∪ {𝑌}))) |
47 | 44, 46 | syl5eq 2866 | . . . . . 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 4188 | . . . . . 6 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))) |
51 | 41, 47, 50 | 3eqtr4d 2864 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ((𝐿‘𝐸) ∩ (𝐿‘𝐺))) |
52 | 2, 3, 8, 9, 7, 10, 11 | lkrin 36292 | . . . . 5 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊆ (𝐿‘(𝐸 + 𝐺))) |
53 | 51, 52 | eqsstrd 4003 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊆ (𝐿‘(𝐸 + 𝐺))) |
54 | 15 | lsssssubg 19722 | . . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
55 | 7, 54 | syl 17 | . . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
56 | 55, 20 | sseldd 3966 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈)) |
57 | 4, 5, 1, 15, 14 | dochlss 38482 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉) → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈)) |
58 | 6, 24, 57 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈)) |
59 | 55, 58 | sseldd 3966 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (SubGrp‘𝑈)) |
60 | 2, 3, 15 | lkrlss 36223 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹) → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈)) |
61 | 7, 12, 60 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈)) |
62 | 55, 61 | sseldd 3966 | . . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ (SubGrp‘𝑈)) |
63 | 16 | lsmlub 18782 | . . . . 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 1366 | . . . 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 710 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ⊕ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊆ (𝐿‘(𝐸 + 𝐺))) |
66 | 27, 65 | eqsstrrd 4004 | . 2 ⊢ (𝜑 → 𝑉 ⊆ (𝐿‘(𝐸 + 𝐺))) |
67 | 13, 66 | eqssd 3982 | 1 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∪ cun 3932 ∩ cin 3933 ⊆ wss 3934 {csn 4559 {cpr 4561 ran crn 5549 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 -gcsg 18097 SubGrpcsubg 18265 LSSumclsm 18751 invrcinvr 19413 LModclmod 19626 LSubSpclss 19695 LSpanclspn 19735 LFnlclfn 36185 LKerclk 36213 LDualcld 36251 HLchlt 36478 LHypclh 37112 DVecHcdvh 38206 DIsoHcdih 38356 ocHcoch 38475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-riotaBAD 36081 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-tpos 7884 df-undef 7931 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-ring 19291 df-oppr 19365 df-dvdsr 19383 df-unit 19384 df-invr 19414 df-dvr 19425 df-drng 19496 df-lmod 19628 df-lss 19696 df-lsp 19736 df-lvec 19867 df-lsatoms 36104 df-lcv 36147 df-lfl 36186 df-lkr 36214 df-ldual 36252 df-oposet 36304 df-ol 36306 df-oml 36307 df-covers 36394 df-ats 36395 df-atl 36426 df-cvlat 36450 df-hlat 36479 df-llines 36626 df-lplanes 36627 df-lvols 36628 df-lines 36629 df-psubsp 36631 df-pmap 36632 df-padd 36924 df-lhyp 37116 df-laut 37117 df-ldil 37232 df-ltrn 37233 df-trl 37287 df-tgrp 37871 df-tendo 37883 df-edring 37885 df-dveca 38131 df-disoa 38157 df-dvech 38207 df-dib 38267 df-dic 38301 df-dih 38357 df-doch 38476 df-djh 38523 |
This theorem is referenced by: lclkrlem2w 38657 |
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