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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcv1 | Structured version Visualization version GIF version |
Description: Two atoms covering the zero subspace are equal. (atcv1 30151 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsatcv1.o | ⊢ 0 = (0g‘𝑊) |
lsatcv1.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsatcv1.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsatcv1.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatcv1.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
lsatcv1.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatcv1.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lsatcv1.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
lsatcv1.r | ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
lsatcv1.l | ⊢ (𝜑 → 𝑈𝐶(𝑄 ⊕ 𝑅)) |
Ref | Expression |
---|---|
lsatcv1 | ⊢ (𝜑 → (𝑈 = { 0 } ↔ 𝑄 = 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatcv1.l | . . . 4 ⊢ (𝜑 → 𝑈𝐶(𝑄 ⊕ 𝑅)) | |
2 | breq1 5062 | . . . 4 ⊢ (𝑈 = { 0 } → (𝑈𝐶(𝑄 ⊕ 𝑅) ↔ { 0 }𝐶(𝑄 ⊕ 𝑅))) | |
3 | 1, 2 | syl5ibcom 247 | . . 3 ⊢ (𝜑 → (𝑈 = { 0 } → { 0 }𝐶(𝑄 ⊕ 𝑅))) |
4 | lsatcv1.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
5 | lsatcv1.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
6 | lsatcv1.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
7 | lsatcv1.c | . . . 4 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
8 | lsatcv1.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
9 | lsatcv1.q | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
10 | lsatcv1.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
11 | 4, 5, 6, 7, 8, 9, 10 | lsatcv0eq 36177 | . . 3 ⊢ (𝜑 → ({ 0 }𝐶(𝑄 ⊕ 𝑅) ↔ 𝑄 = 𝑅)) |
12 | 3, 11 | sylibd 241 | . 2 ⊢ (𝜑 → (𝑈 = { 0 } → 𝑄 = 𝑅)) |
13 | 1 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑈𝐶(𝑄 ⊕ 𝑅)) |
14 | lsatcv1.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
15 | 8 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑊 ∈ LVec) |
16 | lsatcv1.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
17 | 16 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑈 ∈ 𝑆) |
18 | oveq1 7157 | . . . . . . 7 ⊢ (𝑄 = 𝑅 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑅)) | |
19 | lveclmod 19872 | . . . . . . . . . 10 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
20 | 8, 19 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ LMod) |
21 | 14, 6, 20, 10 | lsatlssel 36127 | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
22 | 14 | lsssubg 19723 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝑆) → 𝑅 ∈ (SubGrp‘𝑊)) |
23 | 20, 21, 22 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
24 | 5 | lsmidm 18782 | . . . . . . . 8 ⊢ (𝑅 ∈ (SubGrp‘𝑊) → (𝑅 ⊕ 𝑅) = 𝑅) |
25 | 23, 24 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑅 ⊕ 𝑅) = 𝑅) |
26 | 18, 25 | sylan9eqr 2878 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → (𝑄 ⊕ 𝑅) = 𝑅) |
27 | 10 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑅 ∈ 𝐴) |
28 | 26, 27 | eqeltrd 2913 | . . . . 5 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → (𝑄 ⊕ 𝑅) ∈ 𝐴) |
29 | 4, 14, 6, 7, 15, 17, 28 | lsatcveq0 36162 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → (𝑈𝐶(𝑄 ⊕ 𝑅) ↔ 𝑈 = { 0 })) |
30 | 13, 29 | mpbid 234 | . . 3 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑈 = { 0 }) |
31 | 30 | ex 415 | . 2 ⊢ (𝜑 → (𝑄 = 𝑅 → 𝑈 = { 0 })) |
32 | 12, 31 | impbid 214 | 1 ⊢ (𝜑 → (𝑈 = { 0 } ↔ 𝑄 = 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {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: lsatcvat2 36181 |
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