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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 32312 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 38743 | . . . . 5 ⊢ (𝜑 → (𝑄 ≠ 𝑅 ↔ (𝑄 ∩ 𝑅) = { 0 })) |
7 | eqid 2726 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
8 | lsatcv0eq.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
9 | lsatcv0eq.c | . . . . . 6 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
10 | lveclmod 21084 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
11 | 3, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod) |
12 | 7, 2, 11, 4 | lsatlssel 38695 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑊)) |
13 | 7, 8, 1, 2, 9, 3, 12, 5 | lcvp 38738 | . . . . 5 ⊢ (𝜑 → ((𝑄 ∩ 𝑅) = { 0 } ↔ 𝑄𝐶(𝑄 ⊕ 𝑅))) |
14 | 1, 2, 9, 3, 4 | lsatcv0 38729 | . . . . . 6 ⊢ (𝜑 → { 0 }𝐶𝑄) |
15 | 14 | biantrurd 531 | . . . . 5 ⊢ (𝜑 → (𝑄𝐶(𝑄 ⊕ 𝑅) ↔ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅)))) |
16 | 6, 13, 15 | 3bitrd 304 | . . . 4 ⊢ (𝜑 → (𝑄 ≠ 𝑅 ↔ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅)))) |
17 | 3 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → 𝑊 ∈ LVec) |
18 | 1, 7 | lsssn0 20925 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → { 0 } ∈ (LSubSp‘𝑊)) |
19 | 11, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → { 0 } ∈ (LSubSp‘𝑊)) |
20 | 19 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → { 0 } ∈ (LSubSp‘𝑊)) |
21 | 12 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → 𝑄 ∈ (LSubSp‘𝑊)) |
22 | 7, 2, 11, 5 | lsatlssel 38695 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (LSubSp‘𝑊)) |
23 | 7, 8 | lsmcl 21061 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ (LSubSp‘𝑊) ∧ 𝑅 ∈ (LSubSp‘𝑊)) → (𝑄 ⊕ 𝑅) ∈ (LSubSp‘𝑊)) |
24 | 11, 12, 22, 23 | syl3anc 1368 | . . . . . . 7 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ (LSubSp‘𝑊)) |
25 | 24 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → (𝑄 ⊕ 𝑅) ∈ (LSubSp‘𝑊)) |
26 | simprl 769 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → { 0 }𝐶𝑄) | |
27 | simprr 771 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → 𝑄𝐶(𝑄 ⊕ 𝑅)) | |
28 | 7, 9, 17, 20, 21, 25, 26, 27 | lcvntr 38724 | . . . . 5 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → ¬ { 0 }𝐶(𝑄 ⊕ 𝑅)) |
29 | 28 | ex 411 | . . . 4 ⊢ (𝜑 → (({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅)) → ¬ { 0 }𝐶(𝑄 ⊕ 𝑅))) |
30 | 16, 29 | sylbid 239 | . . 3 ⊢ (𝜑 → (𝑄 ≠ 𝑅 → ¬ { 0 }𝐶(𝑄 ⊕ 𝑅))) |
31 | 30 | necon4ad 2949 | . 2 ⊢ (𝜑 → ({ 0 }𝐶(𝑄 ⊕ 𝑅) → 𝑄 = 𝑅)) |
32 | 7 | lsssssubg 20935 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
33 | 11, 32 | syl 17 | . . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
34 | 33, 12 | sseldd 3980 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
35 | 8 | lsmidm 19661 | . . . . 5 ⊢ (𝑄 ∈ (SubGrp‘𝑊) → (𝑄 ⊕ 𝑄) = 𝑄) |
36 | 34, 35 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑄 ⊕ 𝑄) = 𝑄) |
37 | 14, 36 | breqtrrd 5181 | . . 3 ⊢ (𝜑 → { 0 }𝐶(𝑄 ⊕ 𝑄)) |
38 | oveq2 7432 | . . . 4 ⊢ (𝑄 = 𝑅 → (𝑄 ⊕ 𝑄) = (𝑄 ⊕ 𝑅)) | |
39 | 38 | breq2d 5165 | . . 3 ⊢ (𝑄 = 𝑅 → ({ 0 }𝐶(𝑄 ⊕ 𝑄) ↔ { 0 }𝐶(𝑄 ⊕ 𝑅))) |
40 | 37, 39 | syl5ibcom 244 | . 2 ⊢ (𝜑 → (𝑄 = 𝑅 → { 0 }𝐶(𝑄 ⊕ 𝑅))) |
41 | 31, 40 | impbid 211 | 1 ⊢ (𝜑 → ({ 0 }𝐶(𝑄 ⊕ 𝑅) ↔ 𝑄 = 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∩ cin 3946 ⊆ wss 3947 {csn 4633 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 0gc0g 17454 SubGrpcsubg 19114 LSSumclsm 19632 LModclmod 20836 LSubSpclss 20908 LVecclvec 21080 LSAtomsclsa 38672 ⋖L clcv 38716 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-0g 17456 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-subg 19117 df-cntz 19311 df-oppg 19340 df-lsm 19634 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-drng 20709 df-lmod 20838 df-lss 20909 df-lsp 20949 df-lvec 21081 df-lsatoms 38674 df-lcv 38717 |
This theorem is referenced by: lsatcv1 38746 |
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