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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 30150 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 36175 | . . . . 5 ⊢ (𝜑 → (𝑄 ≠ 𝑅 ↔ (𝑄 ∩ 𝑅) = { 0 })) |
7 | eqid 2821 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
8 | lsatcv0eq.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
9 | lsatcv0eq.c | . . . . . 6 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
10 | lveclmod 19872 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
11 | 3, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod) |
12 | 7, 2, 11, 4 | lsatlssel 36127 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑊)) |
13 | 7, 8, 1, 2, 9, 3, 12, 5 | lcvp 36170 | . . . . 5 ⊢ (𝜑 → ((𝑄 ∩ 𝑅) = { 0 } ↔ 𝑄𝐶(𝑄 ⊕ 𝑅))) |
14 | 1, 2, 9, 3, 4 | lsatcv0 36161 | . . . . . 6 ⊢ (𝜑 → { 0 }𝐶𝑄) |
15 | 14 | biantrurd 535 | . . . . 5 ⊢ (𝜑 → (𝑄𝐶(𝑄 ⊕ 𝑅) ↔ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅)))) |
16 | 6, 13, 15 | 3bitrd 307 | . . . 4 ⊢ (𝜑 → (𝑄 ≠ 𝑅 ↔ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅)))) |
17 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → 𝑊 ∈ LVec) |
18 | 1, 7 | lsssn0 19713 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → { 0 } ∈ (LSubSp‘𝑊)) |
19 | 11, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → { 0 } ∈ (LSubSp‘𝑊)) |
20 | 19 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → { 0 } ∈ (LSubSp‘𝑊)) |
21 | 12 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → 𝑄 ∈ (LSubSp‘𝑊)) |
22 | 7, 2, 11, 5 | lsatlssel 36127 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (LSubSp‘𝑊)) |
23 | 7, 8 | lsmcl 19849 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ (LSubSp‘𝑊) ∧ 𝑅 ∈ (LSubSp‘𝑊)) → (𝑄 ⊕ 𝑅) ∈ (LSubSp‘𝑊)) |
24 | 11, 12, 22, 23 | syl3anc 1367 | . . . . . . 7 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ (LSubSp‘𝑊)) |
25 | 24 | adantr 483 | . . . . . 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 36156 | . . . . 5 ⊢ ((𝜑 ∧ ({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅))) → ¬ { 0 }𝐶(𝑄 ⊕ 𝑅)) |
29 | 28 | ex 415 | . . . 4 ⊢ (𝜑 → (({ 0 }𝐶𝑄 ∧ 𝑄𝐶(𝑄 ⊕ 𝑅)) → ¬ { 0 }𝐶(𝑄 ⊕ 𝑅))) |
30 | 16, 29 | sylbid 242 | . . 3 ⊢ (𝜑 → (𝑄 ≠ 𝑅 → ¬ { 0 }𝐶(𝑄 ⊕ 𝑅))) |
31 | 30 | necon4ad 3035 | . 2 ⊢ (𝜑 → ({ 0 }𝐶(𝑄 ⊕ 𝑅) → 𝑄 = 𝑅)) |
32 | 7 | lsssssubg 19724 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
33 | 11, 32 | syl 17 | . . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
34 | 33, 12 | sseldd 3968 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
35 | 8 | lsmidm 18782 | . . . . 5 ⊢ (𝑄 ∈ (SubGrp‘𝑊) → (𝑄 ⊕ 𝑄) = 𝑄) |
36 | 34, 35 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑄 ⊕ 𝑄) = 𝑄) |
37 | 14, 36 | breqtrrd 5087 | . . 3 ⊢ (𝜑 → { 0 }𝐶(𝑄 ⊕ 𝑄)) |
38 | oveq2 7158 | . . . 4 ⊢ (𝑄 = 𝑅 → (𝑄 ⊕ 𝑄) = (𝑄 ⊕ 𝑅)) | |
39 | 38 | breq2d 5071 | . . 3 ⊢ (𝑄 = 𝑅 → ({ 0 }𝐶(𝑄 ⊕ 𝑄) ↔ { 0 }𝐶(𝑄 ⊕ 𝑅))) |
40 | 37, 39 | syl5ibcom 247 | . 2 ⊢ (𝜑 → (𝑄 = 𝑅 → { 0 }𝐶(𝑄 ⊕ 𝑅))) |
41 | 31, 40 | impbid 214 | 1 ⊢ (𝜑 → ({ 0 }𝐶(𝑄 ⊕ 𝑅) ↔ 𝑄 = 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∩ cin 3935 ⊆ wss 3936 {csn 4561 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 0gc0g 16707 SubGrpcsubg 18267 LSSumclsm 18753 LModclmod 19628 LSubSpclss 19697 LVecclvec 19868 LSAtomsclsa 36104 ⋖L clcv 36148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-oppg 18468 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lsatoms 36106 df-lcv 36149 |
This theorem is referenced by: lsatcv1 36178 |
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