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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 32208 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 5153 | . . . 4 ⊢ (𝑈 = { 0 } → (𝑈𝐶(𝑄 ⊕ 𝑅) ↔ { 0 }𝐶(𝑄 ⊕ 𝑅))) | |
3 | 1, 2 | syl5ibcom 244 | . . 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 38523 | . . 3 ⊢ (𝜑 → ({ 0 }𝐶(𝑄 ⊕ 𝑅) ↔ 𝑄 = 𝑅)) |
12 | 3, 11 | sylibd 238 | . 2 ⊢ (𝜑 → (𝑈 = { 0 } → 𝑄 = 𝑅)) |
13 | 1 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑈𝐶(𝑄 ⊕ 𝑅)) |
14 | lsatcv1.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
15 | 8 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑊 ∈ LVec) |
16 | lsatcv1.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
17 | 16 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑈 ∈ 𝑆) |
18 | oveq1 7431 | . . . . . . 7 ⊢ (𝑄 = 𝑅 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑅)) | |
19 | lveclmod 20996 | . . . . . . . . . 10 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
20 | 8, 19 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ LMod) |
21 | 14, 6, 20, 10 | lsatlssel 38473 | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
22 | 14 | lsssubg 20846 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝑆) → 𝑅 ∈ (SubGrp‘𝑊)) |
23 | 20, 21, 22 | syl2anc 582 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
24 | 5 | lsmidm 19623 | . . . . . . . 8 ⊢ (𝑅 ∈ (SubGrp‘𝑊) → (𝑅 ⊕ 𝑅) = 𝑅) |
25 | 23, 24 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑅 ⊕ 𝑅) = 𝑅) |
26 | 18, 25 | sylan9eqr 2789 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → (𝑄 ⊕ 𝑅) = 𝑅) |
27 | 10 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑅 ∈ 𝐴) |
28 | 26, 27 | eqeltrd 2828 | . . . . 5 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → (𝑄 ⊕ 𝑅) ∈ 𝐴) |
29 | 4, 14, 6, 7, 15, 17, 28 | lsatcveq0 38508 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → (𝑈𝐶(𝑄 ⊕ 𝑅) ↔ 𝑈 = { 0 })) |
30 | 13, 29 | mpbid 231 | . . 3 ⊢ ((𝜑 ∧ 𝑄 = 𝑅) → 𝑈 = { 0 }) |
31 | 30 | ex 411 | . 2 ⊢ (𝜑 → (𝑄 = 𝑅 → 𝑈 = { 0 })) |
32 | 12, 31 | impbid 211 | 1 ⊢ (𝜑 → (𝑈 = { 0 } ↔ 𝑄 = 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {csn 4630 class class class wbr 5150 ‘cfv 6551 (class class class)co 7424 0gc0g 17426 SubGrpcsubg 19080 LSSumclsm 19594 LModclmod 20748 LSubSpclss 20820 LVecclvec 20992 LSAtomsclsa 38450 ⋖L clcv 38494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-tpos 8236 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-0g 17428 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-grp 18898 df-minusg 18899 df-sbg 18900 df-subg 19083 df-cntz 19273 df-oppg 19302 df-lsm 19596 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180 df-oppr 20278 df-dvdsr 20301 df-unit 20302 df-invr 20332 df-drng 20631 df-lmod 20750 df-lss 20821 df-lsp 20861 df-lvec 20993 df-lsatoms 38452 df-lcv 38495 |
This theorem is referenced by: lsatcvat2 38527 |
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