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Mathbox for Norm Megill |
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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 32210 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsatcv1.o | β’ 0 = (0gβπ) |
lsatcv1.p | β’ β = (LSSumβπ) |
lsatcv1.s | β’ π = (LSubSpβπ) |
lsatcv1.a | β’ π΄ = (LSAtomsβπ) |
lsatcv1.c | β’ πΆ = ( βL βπ) |
lsatcv1.w | β’ (π β π β LVec) |
lsatcv1.u | β’ (π β π β π) |
lsatcv1.q | β’ (π β π β π΄) |
lsatcv1.r | β’ (π β π β π΄) |
lsatcv1.l | β’ (π β ππΆ(π β π )) |
Ref | Expression |
---|---|
lsatcv1 | β’ (π β (π = { 0 } β π = π )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatcv1.l | . . . 4 β’ (π β ππΆ(π β π )) | |
2 | breq1 5155 | . . . 4 β’ (π = { 0 } β (ππΆ(π β π ) β { 0 }πΆ(π β π ))) | |
3 | 1, 2 | syl5ibcom 244 | . . 3 β’ (π β (π = { 0 } β { 0 }πΆ(π β π ))) |
4 | lsatcv1.o | . . . 4 β’ 0 = (0gβπ) | |
5 | lsatcv1.p | . . . 4 β’ β = (LSSumβπ) | |
6 | lsatcv1.a | . . . 4 β’ π΄ = (LSAtomsβπ) | |
7 | lsatcv1.c | . . . 4 β’ πΆ = ( βL βπ) | |
8 | lsatcv1.w | . . . 4 β’ (π β π β LVec) | |
9 | lsatcv1.q | . . . 4 β’ (π β π β π΄) | |
10 | lsatcv1.r | . . . 4 β’ (π β π β π΄) | |
11 | 4, 5, 6, 7, 8, 9, 10 | lsatcv0eq 38551 | . . 3 β’ (π β ({ 0 }πΆ(π β π ) β π = π )) |
12 | 3, 11 | sylibd 238 | . 2 β’ (π β (π = { 0 } β π = π )) |
13 | 1 | adantr 479 | . . . 4 β’ ((π β§ π = π ) β ππΆ(π β π )) |
14 | lsatcv1.s | . . . . 5 β’ π = (LSubSpβπ) | |
15 | 8 | adantr 479 | . . . . 5 β’ ((π β§ π = π ) β π β LVec) |
16 | lsatcv1.u | . . . . . 6 β’ (π β π β π) | |
17 | 16 | adantr 479 | . . . . 5 β’ ((π β§ π = π ) β π β π) |
18 | oveq1 7433 | . . . . . . 7 β’ (π = π β (π β π ) = (π β π )) | |
19 | lveclmod 20998 | . . . . . . . . . 10 β’ (π β LVec β π β LMod) | |
20 | 8, 19 | syl 17 | . . . . . . . . 9 β’ (π β π β LMod) |
21 | 14, 6, 20, 10 | lsatlssel 38501 | . . . . . . . . 9 β’ (π β π β π) |
22 | 14 | lsssubg 20848 | . . . . . . . . 9 β’ ((π β LMod β§ π β π) β π β (SubGrpβπ)) |
23 | 20, 21, 22 | syl2anc 582 | . . . . . . . 8 β’ (π β π β (SubGrpβπ)) |
24 | 5 | lsmidm 19625 | . . . . . . . 8 β’ (π β (SubGrpβπ) β (π β π ) = π ) |
25 | 23, 24 | syl 17 | . . . . . . 7 β’ (π β (π β π ) = π ) |
26 | 18, 25 | sylan9eqr 2790 | . . . . . 6 β’ ((π β§ π = π ) β (π β π ) = π ) |
27 | 10 | adantr 479 | . . . . . 6 β’ ((π β§ π = π ) β π β π΄) |
28 | 26, 27 | eqeltrd 2829 | . . . . 5 β’ ((π β§ π = π ) β (π β π ) β π΄) |
29 | 4, 14, 6, 7, 15, 17, 28 | lsatcveq0 38536 | . . . 4 β’ ((π β§ π = π ) β (ππΆ(π β π ) β π = { 0 })) |
30 | 13, 29 | mpbid 231 | . . 3 β’ ((π β§ π = π ) β π = { 0 }) |
31 | 30 | ex 411 | . 2 β’ (π β (π = π β π = { 0 })) |
32 | 12, 31 | impbid 211 | 1 β’ (π β (π = { 0 } β π = π )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 {csn 4632 class class class wbr 5152 βcfv 6553 (class class class)co 7426 0gc0g 17428 SubGrpcsubg 19082 LSSumclsm 19596 LModclmod 20750 LSubSpclss 20822 LVecclvec 20994 LSAtomsclsa 38478 βL clcv 38522 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 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 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-0g 17430 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cntz 19275 df-oppg 19304 df-lsm 19598 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-drng 20633 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lvec 20995 df-lsatoms 38480 df-lcv 38523 |
This theorem is referenced by: lsatcvat2 38555 |
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