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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 32143 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 20954 | . . . . . 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 20805 | . . . . 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 3978 | . . 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 38380 | . . . 4 β’ (π β π β π) |
12 | 6, 11 | sseldd 3978 | . . 3 β’ (π β π β (SubGrpβπ)) |
13 | lsatexch.p | . . . 4 β’ β = (LSSumβπ) | |
14 | 13 | lsmub2 19578 | . . 3 β’ ((π β (SubGrpβπ) β§ π β (SubGrpβπ)) β π β (π β π )) |
15 | 8, 12, 14 | syl2anc 583 | . 2 β’ (π β π β (π β π )) |
16 | eqid 2726 | . . 3 β’ ( βL βπ) = ( βL βπ) | |
17 | 4, 13 | lsmcl 20931 | . . . 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 38380 | . . . 4 β’ (π β π β π) |
21 | 4, 13 | lsmcl 20931 | . . . 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 38423 | . . . . . . 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 38407 | . . . . 5 β’ (π β π β (π β π)) |
28 | 13 | lsmub1 19577 | . . . . . . 7 β’ ((π β (SubGrpβπ) β§ π β (SubGrpβπ)) β π β (π β π )) |
29 | 8, 12, 28 | syl2anc 583 | . . . . . 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 3978 | . . . . . . 7 β’ (π β π β (SubGrpβπ)) |
32 | 6, 18 | sseldd 3978 | . . . . . . 7 β’ (π β (π β π ) β (SubGrpβπ)) |
33 | 13 | lsmlub 19584 | . . . . . . 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 709 | . . . . 5 β’ (π β (π β π) β (π β π )) |
36 | 27, 35 | psssstrd 4104 | . . . 4 β’ (π β π β (π β π )) |
37 | 4, 13, 9, 16, 1, 7, 10 | lcv2 38425 | . . . 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 38410 | . 2 β’ (π β (π β π) = (π β π )) |
40 | 15, 39 | sseqtrrd 4018 | 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 395 = wceq 1533 β wcel 2098 β© cin 3942 β wss 3943 β wpss 3944 {csn 4623 class class class wbr 5141 βcfv 6537 (class class class)co 7405 0gc0g 17394 SubGrpcsubg 19047 LSSumclsm 19554 LModclmod 20706 LSubSpclss 20778 LVecclvec 20950 LSAtomsclsa 38357 βL clcv 38401 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-0g 17396 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cntz 19233 df-oppg 19262 df-lsm 19556 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-drng 20589 df-lmod 20708 df-lss 20779 df-lsp 20819 df-lvec 20951 df-lsatoms 38359 df-lcv 38402 |
This theorem is referenced by: lsatexch1 38429 |
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