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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatexch | Structured version Visualization version GIF version |
Description: The atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem was originally proved by Hermann Grassmann in 1862. (atexch 32219 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsatexch.s | β’ π = (LSubSpβπ) |
lsatexch.p | β’ β = (LSSumβπ) |
lsatexch.o | β’ 0 = (0gβπ) |
lsatexch.a | β’ π΄ = (LSAtomsβπ) |
lsatexch.w | β’ (π β π β LVec) |
lsatexch.u | β’ (π β π β π) |
lsatexch.q | β’ (π β π β π΄) |
lsatexch.r | β’ (π β π β π΄) |
lsatexch.l | β’ (π β π β (π β π )) |
lsatexch.z | β’ (π β (π β© π) = { 0 }) |
Ref | Expression |
---|---|
lsatexch | β’ (π β π β (π β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatexch.w | . . . . . 6 β’ (π β π β LVec) | |
2 | lveclmod 21005 | . . . . . 6 β’ (π β LVec β π β LMod) | |
3 | 1, 2 | syl 17 | . . . . 5 β’ (π β π β LMod) |
4 | lsatexch.s | . . . . . 6 β’ π = (LSubSpβπ) | |
5 | 4 | lsssssubg 20856 | . . . . 5 β’ (π β LMod β π β (SubGrpβπ)) |
6 | 3, 5 | syl 17 | . . . 4 β’ (π β π β (SubGrpβπ)) |
7 | lsatexch.u | . . . 4 β’ (π β π β π) | |
8 | 6, 7 | sseldd 3983 | . . 3 β’ (π β π β (SubGrpβπ)) |
9 | lsatexch.a | . . . . 5 β’ π΄ = (LSAtomsβπ) | |
10 | lsatexch.r | . . . . 5 β’ (π β π β π΄) | |
11 | 4, 9, 3, 10 | lsatlssel 38509 | . . . 4 β’ (π β π β π) |
12 | 6, 11 | sseldd 3983 | . . 3 β’ (π β π β (SubGrpβπ)) |
13 | lsatexch.p | . . . 4 β’ β = (LSSumβπ) | |
14 | 13 | lsmub2 19627 | . . 3 β’ ((π β (SubGrpβπ) β§ π β (SubGrpβπ)) β π β (π β π )) |
15 | 8, 12, 14 | syl2anc 582 | . 2 β’ (π β π β (π β π )) |
16 | eqid 2728 | . . 3 β’ ( βL βπ) = ( βL βπ) | |
17 | 4, 13 | lsmcl 20982 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β (π β π ) β π) |
18 | 3, 7, 11, 17 | syl3anc 1368 | . . 3 β’ (π β (π β π ) β π) |
19 | lsatexch.q | . . . . 5 β’ (π β π β π΄) | |
20 | 4, 9, 3, 19 | lsatlssel 38509 | . . . 4 β’ (π β π β π) |
21 | 4, 13 | lsmcl 20982 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β (π β π) β π) |
22 | 3, 7, 20, 21 | syl3anc 1368 | . . 3 β’ (π β (π β π) β π) |
23 | lsatexch.z | . . . . . . 7 β’ (π β (π β© π) = { 0 }) | |
24 | lsatexch.o | . . . . . . . 8 β’ 0 = (0gβπ) | |
25 | 4, 13, 24, 9, 16, 1, 7, 19 | lcvp 38552 | . . . . . . 7 β’ (π β ((π β© π) = { 0 } β π( βL βπ)(π β π))) |
26 | 23, 25 | mpbid 231 | . . . . . 6 β’ (π β π( βL βπ)(π β π)) |
27 | 4, 16, 1, 7, 22, 26 | lcvpss 38536 | . . . . 5 β’ (π β π β (π β π)) |
28 | 13 | lsmub1 19626 | . . . . . . 7 β’ ((π β (SubGrpβπ) β§ π β (SubGrpβπ)) β π β (π β π )) |
29 | 8, 12, 28 | syl2anc 582 | . . . . . 6 β’ (π β π β (π β π )) |
30 | lsatexch.l | . . . . . 6 β’ (π β π β (π β π )) | |
31 | 6, 20 | sseldd 3983 | . . . . . . 7 β’ (π β π β (SubGrpβπ)) |
32 | 6, 18 | sseldd 3983 | . . . . . . 7 β’ (π β (π β π ) β (SubGrpβπ)) |
33 | 13 | lsmlub 19633 | . . . . . . 7 β’ ((π β (SubGrpβπ) β§ π β (SubGrpβπ) β§ (π β π ) β (SubGrpβπ)) β ((π β (π β π ) β§ π β (π β π )) β (π β π) β (π β π ))) |
34 | 8, 31, 32, 33 | syl3anc 1368 | . . . . . 6 β’ (π β ((π β (π β π ) β§ π β (π β π )) β (π β π) β (π β π ))) |
35 | 29, 30, 34 | mpbi2and 710 | . . . . 5 β’ (π β (π β π) β (π β π )) |
36 | 27, 35 | psssstrd 4109 | . . . 4 β’ (π β π β (π β π )) |
37 | 4, 13, 9, 16, 1, 7, 10 | lcv2 38554 | . . . 4 β’ (π β (π β (π β π ) β π( βL βπ)(π β π ))) |
38 | 36, 37 | mpbid 231 | . . 3 β’ (π β π( βL βπ)(π β π )) |
39 | 4, 16, 1, 7, 18, 22, 38, 27, 35 | lcvnbtwn2 38539 | . 2 β’ (π β (π β π) = (π β π )) |
40 | 15, 39 | sseqtrrd 4023 | 1 β’ (π β π β (π β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β© cin 3948 β wss 3949 β wpss 3950 {csn 4632 class class class wbr 5152 βcfv 6553 (class class class)co 7426 0gc0g 17430 SubGrpcsubg 19089 LSSumclsm 19603 LModclmod 20757 LSubSpclss 20829 LVecclvec 21001 LSAtomsclsa 38486 βL clcv 38530 |
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 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
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 7879 df-1st 8001 df-2nd 8002 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-0g 17432 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19092 df-cntz 19282 df-oppg 19311 df-lsm 19605 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-drng 20640 df-lmod 20759 df-lss 20830 df-lsp 20870 df-lvec 21002 df-lsatoms 38488 df-lcv 38531 |
This theorem is referenced by: lsatexch1 38558 |
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