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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcv0eq | Structured version Visualization version GIF version |
Description: If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 29843 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsatcv0eq.o | ⊢ 0 = (0g‘𝑊) |
lsatcv0eq.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsatcv0eq.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatcv0eq.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
lsatcv0eq.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatcv0eq.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
lsatcv0eq.r | ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
Ref | Expression |
---|---|
lsatcv0eq | ⊢ (𝜑 → ({ 0 }𝐶(𝑄 ⊕ 𝑅) ↔ 𝑄 = 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatcv0eq.o | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
2 | lsatcv0eq.a | . . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
3 | lsatcv0eq.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lsatcv0eq.q | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
5 | lsatcv0eq.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
6 | 1, 2, 3, 4, 5 | lsatnem0 35733 | . . . . 5 ⊢ (𝜑 → (𝑄 ≠ 𝑅 ↔ (𝑄 ∩ 𝑅) = { 0 })) |
7 | eqid 2797 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
8 | lsatcv0eq.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
9 | lsatcv0eq.c | . . . . . 6 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
10 | lveclmod 19572 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
11 | 3, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod) |
12 | 7, 2, 11, 4 | lsatlssel 35685 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑊)) |
13 | 7, 8, 1, 2, 9, 3, 12, 5 | lcvp 35728 | . . . . 5 ⊢ (𝜑 → ((𝑄 ∩ 𝑅) = { 0 } ↔ 𝑄𝐶(𝑄 ⊕ 𝑅))) |
14 | 1, 2, 9, 3, 4 | lsatcv0 35719 | . . . . . 6 ⊢ (𝜑 → { 0 }𝐶𝑄) |
15 | 14 | biantrurd 533 | . . . . 5 ⊢ (𝜑 → (𝑄𝐶(𝑄 ⊕ 𝑅) ↔ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅)))) |
16 | 6, 13, 15 | 3bitrd 306 | . . . 4 ⊢ (𝜑 → (𝑄 ≠ 𝑅 ↔ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅)))) |
17 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → 𝑊 ∈ LVec) |
18 | 1, 7 | lsssn0 19413 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → { 0 } ∈ (LSubSp‘𝑊)) |
19 | 11, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → { 0 } ∈ (LSubSp‘𝑊)) |
20 | 19 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → { 0 } ∈ (LSubSp‘𝑊)) |
21 | 12 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → 𝑄 ∈ (LSubSp‘𝑊)) |
22 | 7, 2, 11, 5 | lsatlssel 35685 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (LSubSp‘𝑊)) |
23 | 7, 8 | lsmcl 19549 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ (LSubSp‘𝑊) ∧ 𝑅 ∈ (LSubSp‘𝑊)) → (𝑄 ⊕ 𝑅) ∈ (LSubSp‘𝑊)) |
24 | 11, 12, 22, 23 | syl3anc 1364 | . . . . . . 7 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ (LSubSp‘𝑊)) |
25 | 24 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → (𝑄 ⊕ 𝑅) ∈ (LSubSp‘𝑊)) |
26 | simprl 767 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → { 0 }𝐶𝑄) | |
27 | simprr 769 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → 𝑄𝐶(𝑄 ⊕ 𝑅)) | |
28 | 7, 9, 17, 20, 21, 25, 26, 27 | lcvntr 35714 | . . . . 5 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → ¬ { 0 }𝐶(𝑄 ⊕ 𝑅)) |
29 | 28 | ex 413 | . . . 4 ⊢ (𝜑 → (({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅)) → ¬ { 0 }𝐶(𝑄 ⊕ 𝑅))) |
30 | 16, 29 | sylbid 241 | . . 3 ⊢ (𝜑 → (𝑄 ≠ 𝑅 → ¬ { 0 }𝐶(𝑄 ⊕ 𝑅))) |
31 | 30 | necon4ad 3005 | . 2 ⊢ (𝜑 → ({ 0 }𝐶(𝑄 ⊕ 𝑅) → 𝑄 = 𝑅)) |
32 | 7 | lsssssubg 19424 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
33 | 11, 32 | syl 17 | . . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
34 | 33, 12 | sseldd 3896 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
35 | 8 | lsmidm 18521 | . . . . 5 ⊢ (𝑄 ∈ (SubGrp‘𝑊) → (𝑄 ⊕ 𝑄) = 𝑄) |
36 | 34, 35 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑄 ⊕ 𝑄) = 𝑄) |
37 | 14, 36 | breqtrrd 4996 | . . 3 ⊢ (𝜑 → { 0 }𝐶(𝑄 ⊕ 𝑄)) |
38 | oveq2 7031 | . . . 4 ⊢ (𝑄 = 𝑅 → (𝑄 ⊕ 𝑄) = (𝑄 ⊕ 𝑅)) | |
39 | 38 | breq2d 4980 | . . 3 ⊢ (𝑄 = 𝑅 → ({ 0 }𝐶(𝑄 ⊕ 𝑄) ↔ { 0 }𝐶(𝑄 ⊕ 𝑅))) |
40 | 37, 39 | syl5ibcom 246 | . 2 ⊢ (𝜑 → (𝑄 = 𝑅 → { 0 }𝐶(𝑄 ⊕ 𝑅))) |
41 | 31, 40 | impbid 213 | 1 ⊢ (𝜑 → ({ 0 }𝐶(𝑄 ⊕ 𝑅) ↔ 𝑄 = 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 ∩ cin 3864 ⊆ wss 3865 {csn 4478 class class class wbr 4968 ‘cfv 6232 (class class class)co 7023 0gc0g 16546 SubGrpcsubg 18031 LSSumclsm 18493 LModclmod 19328 LSubSpclss 19397 LVecclvec 19568 LSAtomsclsa 35662 ⋖L clcv 35706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-tpos 7750 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-0g 16548 df-mre 16690 df-mrc 16691 df-acs 16693 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-grp 17868 df-minusg 17869 df-sbg 17870 df-subg 18034 df-cntz 18192 df-oppg 18219 df-lsm 18495 df-cmn 18639 df-abl 18640 df-mgp 18934 df-ur 18946 df-ring 18993 df-oppr 19067 df-dvdsr 19085 df-unit 19086 df-invr 19116 df-drng 19198 df-lmod 19330 df-lss 19398 df-lsp 19438 df-lvec 19569 df-lsatoms 35664 df-lcv 35707 |
This theorem is referenced by: lsatcv1 35736 |
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