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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > asclmulg | Structured version Visualization version GIF version | ||
| Description: Apply group multiplication to the algebra scalars. (Contributed by Thierry Arnoux, 24-Jul-2024.) |
| Ref | Expression |
|---|---|
| asclmulg.a | β’ π΄ = (algScβπ) |
| asclmulg.f | β’ πΉ = (Scalarβπ) |
| asclmulg.k | β’ πΎ = (BaseβπΉ) |
| asclmulg.m | β’ β = (.gβπ) |
| asclmulg.t | β’ β = (.gβπΉ) |
| Ref | Expression |
|---|---|
| asclmulg | β’ ((π β AssAlg β§ π β β0 β§ π β πΎ) β (π΄β(π β π)) = (π β (π΄βπ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | assalmod 21635 | . . . 4 β’ (π β AssAlg β π β LMod) | |
| 2 | 1 | 3ad2ant1 1132 | . . 3 β’ ((π β AssAlg β§ π β β0 β§ π β πΎ) β π β LMod) |
| 3 | simp3 1137 | . . 3 β’ ((π β AssAlg β§ π β β0 β§ π β πΎ) β π β πΎ) | |
| 4 | simp2 1136 | . . 3 β’ ((π β AssAlg β§ π β β0 β§ π β πΎ) β π β β0) | |
| 5 | assaring 21636 | . . . . 5 β’ (π β AssAlg β π β Ring) | |
| 6 | eqid 2731 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
| 7 | eqid 2731 | . . . . . 6 β’ (1rβπ) = (1rβπ) | |
| 8 | 6, 7 | ringidcl 20155 | . . . . 5 β’ (π β Ring β (1rβπ) β (Baseβπ)) |
| 9 | 5, 8 | syl 17 | . . . 4 β’ (π β AssAlg β (1rβπ) β (Baseβπ)) |
| 10 | 9 | 3ad2ant1 1132 | . . 3 β’ ((π β AssAlg β§ π β β0 β§ π β πΎ) β (1rβπ) β (Baseβπ)) |
| 11 | asclmulg.f | . . . 4 β’ πΉ = (Scalarβπ) | |
| 12 | eqid 2731 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 13 | asclmulg.k | . . . 4 β’ πΎ = (BaseβπΉ) | |
| 14 | asclmulg.m | . . . 4 β’ β = (.gβπ) | |
| 15 | asclmulg.t | . . . 4 β’ β = (.gβπΉ) | |
| 16 | 6, 11, 12, 13, 14, 15 | lmodvsmmulgdi 20652 | . . 3 β’ ((π β LMod β§ (π β πΎ β§ π β β0 β§ (1rβπ) β (Baseβπ))) β (π β (π( Β·π βπ)(1rβπ))) = ((π β π)( Β·π βπ)(1rβπ))) |
| 17 | 2, 3, 4, 10, 16 | syl13anc 1371 | . 2 β’ ((π β AssAlg β§ π β β0 β§ π β πΎ) β (π β (π( Β·π βπ)(1rβπ))) = ((π β π)( Β·π βπ)(1rβπ))) |
| 18 | asclmulg.a | . . . . 5 β’ π΄ = (algScβπ) | |
| 19 | 18, 11, 13, 12, 7 | asclval 21654 | . . . 4 β’ (π β πΎ β (π΄βπ) = (π( Β·π βπ)(1rβπ))) |
| 20 | 3, 19 | syl 17 | . . 3 β’ ((π β AssAlg β§ π β β0 β§ π β πΎ) β (π΄βπ) = (π( Β·π βπ)(1rβπ))) |
| 21 | 20 | oveq2d 7428 | . 2 β’ ((π β AssAlg β§ π β β0 β§ π β πΎ) β (π β (π΄βπ)) = (π β (π( Β·π βπ)(1rβπ)))) |
| 22 | 11 | assasca 21637 | . . . . . 6 β’ (π β AssAlg β πΉ β Ring) |
| 23 | 22 | ringgrpd 20137 | . . . . 5 β’ (π β AssAlg β πΉ β Grp) |
| 24 | 23 | 3ad2ant1 1132 | . . . 4 β’ ((π β AssAlg β§ π β β0 β§ π β πΎ) β πΉ β Grp) |
| 25 | 4 | nn0zd 12589 | . . . 4 β’ ((π β AssAlg β§ π β β0 β§ π β πΎ) β π β β€) |
| 26 | 13, 15, 24, 25, 3 | mulgcld 19013 | . . 3 β’ ((π β AssAlg β§ π β β0 β§ π β πΎ) β (π β π) β πΎ) |
| 27 | 18, 11, 13, 12, 7 | asclval 21654 | . . 3 β’ ((π β π) β πΎ β (π΄β(π β π)) = ((π β π)( Β·π βπ)(1rβπ))) |
| 28 | 26, 27 | syl 17 | . 2 β’ ((π β AssAlg β§ π β β0 β§ π β πΎ) β (π΄β(π β π)) = ((π β π)( Β·π βπ)(1rβπ))) |
| 29 | 17, 21, 28 | 3eqtr4rd 2782 | 1 β’ ((π β AssAlg β§ π β β0 β§ π β πΎ) β (π΄β(π β π)) = (π β (π΄βπ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1086 = wceq 1540 β wcel 2105 βcfv 6543 (class class class)co 7412 β0cn0 12477 Basecbs 17149 Scalarcsca 17205 Β·π cvsca 17206 Grpcgrp 18856 .gcmg 18987 1rcur 20076 Ringcrg 20128 LModclmod 20615 AssAlgcasa 21625 algSccascl 21627 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-seq 13972 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-mulg 18988 df-mgp 20030 df-ur 20077 df-ring 20130 df-lmod 20617 df-assa 21628 df-ascl 21630 |
| This theorem is referenced by: ply1chr 32936 |
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