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Mathbox for Norm Megill |
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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 32233 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsatcv0eq.o | β’ 0 = (0gβπ) |
lsatcv0eq.p | β’ β = (LSSumβπ) |
lsatcv0eq.a | β’ π΄ = (LSAtomsβπ) |
lsatcv0eq.c | β’ πΆ = ( βL βπ) |
lsatcv0eq.w | β’ (π β π β LVec) |
lsatcv0eq.q | β’ (π β π β π΄) |
lsatcv0eq.r | β’ (π β π β π΄) |
Ref | Expression |
---|---|
lsatcv0eq | β’ (π β ({ 0 }πΆ(π β π ) β π = π )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatcv0eq.o | . . . . . 6 β’ 0 = (0gβπ) | |
2 | lsatcv0eq.a | . . . . . 6 β’ π΄ = (LSAtomsβπ) | |
3 | lsatcv0eq.w | . . . . . 6 β’ (π β π β LVec) | |
4 | lsatcv0eq.q | . . . . . 6 β’ (π β π β π΄) | |
5 | lsatcv0eq.r | . . . . . 6 β’ (π β π β π΄) | |
6 | 1, 2, 3, 4, 5 | lsatnem0 38573 | . . . . 5 β’ (π β (π β π β (π β© π ) = { 0 })) |
7 | eqid 2725 | . . . . . 6 β’ (LSubSpβπ) = (LSubSpβπ) | |
8 | lsatcv0eq.p | . . . . . 6 β’ β = (LSSumβπ) | |
9 | lsatcv0eq.c | . . . . . 6 β’ πΆ = ( βL βπ) | |
10 | lveclmod 20995 | . . . . . . . 8 β’ (π β LVec β π β LMod) | |
11 | 3, 10 | syl 17 | . . . . . . 7 β’ (π β π β LMod) |
12 | 7, 2, 11, 4 | lsatlssel 38525 | . . . . . 6 β’ (π β π β (LSubSpβπ)) |
13 | 7, 8, 1, 2, 9, 3, 12, 5 | lcvp 38568 | . . . . 5 β’ (π β ((π β© π ) = { 0 } β ππΆ(π β π ))) |
14 | 1, 2, 9, 3, 4 | lsatcv0 38559 | . . . . . 6 β’ (π β { 0 }πΆπ) |
15 | 14 | biantrurd 531 | . . . . 5 β’ (π β (ππΆ(π β π ) β ({ 0 }πΆπ β§ ππΆ(π β π )))) |
16 | 6, 13, 15 | 3bitrd 304 | . . . 4 β’ (π β (π β π β ({ 0 }πΆπ β§ ππΆ(π β π )))) |
17 | 3 | adantr 479 | . . . . . 6 β’ ((π β§ ({ 0 }πΆπ β§ ππΆ(π β π ))) β π β LVec) |
18 | 1, 7 | lsssn0 20836 | . . . . . . . 8 β’ (π β LMod β { 0 } β (LSubSpβπ)) |
19 | 11, 18 | syl 17 | . . . . . . 7 β’ (π β { 0 } β (LSubSpβπ)) |
20 | 19 | adantr 479 | . . . . . 6 β’ ((π β§ ({ 0 }πΆπ β§ ππΆ(π β π ))) β { 0 } β (LSubSpβπ)) |
21 | 12 | adantr 479 | . . . . . 6 β’ ((π β§ ({ 0 }πΆπ β§ ππΆ(π β π ))) β π β (LSubSpβπ)) |
22 | 7, 2, 11, 5 | lsatlssel 38525 | . . . . . . . 8 β’ (π β π β (LSubSpβπ)) |
23 | 7, 8 | lsmcl 20972 | . . . . . . . 8 β’ ((π β LMod β§ π β (LSubSpβπ) β§ π β (LSubSpβπ)) β (π β π ) β (LSubSpβπ)) |
24 | 11, 12, 22, 23 | syl3anc 1368 | . . . . . . 7 β’ (π β (π β π ) β (LSubSpβπ)) |
25 | 24 | adantr 479 | . . . . . 6 β’ ((π β§ ({ 0 }πΆπ β§ ππΆ(π β π ))) β (π β π ) β (LSubSpβπ)) |
26 | simprl 769 | . . . . . 6 β’ ((π β§ ({ 0 }πΆπ β§ ππΆ(π β π ))) β { 0 }πΆπ) | |
27 | simprr 771 | . . . . . 6 β’ ((π β§ ({ 0 }πΆπ β§ ππΆ(π β π ))) β ππΆ(π β π )) | |
28 | 7, 9, 17, 20, 21, 25, 26, 27 | lcvntr 38554 | . . . . 5 β’ ((π β§ ({ 0 }πΆπ β§ ππΆ(π β π ))) β Β¬ { 0 }πΆ(π β π )) |
29 | 28 | ex 411 | . . . 4 β’ (π β (({ 0 }πΆπ β§ ππΆ(π β π )) β Β¬ { 0 }πΆ(π β π ))) |
30 | 16, 29 | sylbid 239 | . . 3 β’ (π β (π β π β Β¬ { 0 }πΆ(π β π ))) |
31 | 30 | necon4ad 2949 | . 2 β’ (π β ({ 0 }πΆ(π β π ) β π = π )) |
32 | 7 | lsssssubg 20846 | . . . . . . 7 β’ (π β LMod β (LSubSpβπ) β (SubGrpβπ)) |
33 | 11, 32 | syl 17 | . . . . . 6 β’ (π β (LSubSpβπ) β (SubGrpβπ)) |
34 | 33, 12 | sseldd 3973 | . . . . 5 β’ (π β π β (SubGrpβπ)) |
35 | 8 | lsmidm 19622 | . . . . 5 β’ (π β (SubGrpβπ) β (π β π) = π) |
36 | 34, 35 | syl 17 | . . . 4 β’ (π β (π β π) = π) |
37 | 14, 36 | breqtrrd 5171 | . . 3 β’ (π β { 0 }πΆ(π β π)) |
38 | oveq2 7424 | . . . 4 β’ (π = π β (π β π) = (π β π )) | |
39 | 38 | breq2d 5155 | . . 3 β’ (π = π β ({ 0 }πΆ(π β π) β { 0 }πΆ(π β π ))) |
40 | 37, 39 | syl5ibcom 244 | . 2 β’ (π β (π = π β { 0 }πΆ(π β π ))) |
41 | 31, 40 | impbid 211 | 1 β’ (π β ({ 0 }πΆ(π β π ) β π = π )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 β© cin 3938 β wss 3939 {csn 4624 class class class wbr 5143 βcfv 6543 (class class class)co 7416 0gc0g 17420 SubGrpcsubg 19079 LSSumclsm 19593 LModclmod 20747 LSubSpclss 20819 LVecclvec 20991 LSAtomsclsa 38502 βL clcv 38546 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-0g 17422 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-subg 19082 df-cntz 19272 df-oppg 19301 df-lsm 19595 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-drng 20630 df-lmod 20749 df-lss 20820 df-lsp 20860 df-lvec 20992 df-lsatoms 38504 df-lcv 38547 |
This theorem is referenced by: lsatcv1 38576 |
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