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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 30327 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 5043 | . . . 4 ⊢ (𝑈 = { 0 } → (𝑈𝐶(𝑄 ⊕ 𝑅) ↔ { 0 }𝐶(𝑄 ⊕ 𝑅))) | |
3 | 1, 2 | syl5ibcom 248 | . . 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 36716 | . . 3 ⊢ (𝜑 → ({ 0 }𝐶(𝑄 ⊕ 𝑅) ↔ 𝑄 = 𝑅)) |
12 | 3, 11 | sylibd 242 | . 2 ⊢ (𝜑 → (𝑈 = { 0 } → 𝑄 = 𝑅)) |
13 | 1 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑈𝐶(𝑄 ⊕ 𝑅)) |
14 | lsatcv1.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
15 | 8 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑊 ∈ LVec) |
16 | lsatcv1.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
17 | 16 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑈 ∈ 𝑆) |
18 | oveq1 7189 | . . . . . . 7 ⊢ (𝑄 = 𝑅 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑅)) | |
19 | lveclmod 20009 | . . . . . . . . . 10 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
20 | 8, 19 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ LMod) |
21 | 14, 6, 20, 10 | lsatlssel 36666 | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
22 | 14 | lsssubg 19860 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝑆) → 𝑅 ∈ (SubGrp‘𝑊)) |
23 | 20, 21, 22 | syl2anc 587 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
24 | 5 | lsmidm 18918 | . . . . . . . 8 ⊢ (𝑅 ∈ (SubGrp‘𝑊) → (𝑅 ⊕ 𝑅) = 𝑅) |
25 | 23, 24 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑅 ⊕ 𝑅) = 𝑅) |
26 | 18, 25 | sylan9eqr 2796 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → (𝑄 ⊕ 𝑅) = 𝑅) |
27 | 10 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑅 ∈ 𝐴) |
28 | 26, 27 | eqeltrd 2834 | . . . . 5 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → (𝑄 ⊕ 𝑅) ∈ 𝐴) |
29 | 4, 14, 6, 7, 15, 17, 28 | lsatcveq0 36701 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → (𝑈𝐶(𝑄 ⊕ 𝑅) ↔ 𝑈 = { 0 })) |
30 | 13, 29 | mpbid 235 | . . 3 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑈 = { 0 }) |
31 | 30 | ex 416 | . 2 ⊢ (𝜑 → (𝑄 = 𝑅 → 𝑈 = { 0 })) |
32 | 12, 31 | impbid 215 | 1 ⊢ (𝜑 → (𝑈 = { 0 } ↔ 𝑄 = 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {csn 4526 class class class wbr 5040 ‘cfv 6349 (class class class)co 7182 0gc0g 16828 SubGrpcsubg 18403 LSSumclsm 18889 LModclmod 19765 LSubSpclss 19834 LVecclvec 20005 LSAtomsclsa 36643 ⋖L clcv 36687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-tpos 7933 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-0g 16830 df-mre 16972 df-mrc 16973 df-acs 16975 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-submnd 18085 df-grp 18234 df-minusg 18235 df-sbg 18236 df-subg 18406 df-cntz 18577 df-oppg 18604 df-lsm 18891 df-cmn 19038 df-abl 19039 df-mgp 19371 df-ur 19383 df-ring 19430 df-oppr 19507 df-dvdsr 19525 df-unit 19526 df-invr 19556 df-drng 19635 df-lmod 19767 df-lss 19835 df-lsp 19875 df-lvec 20006 df-lsatoms 36645 df-lcv 36688 |
This theorem is referenced by: lsatcvat2 36720 |
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