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| Mirrors > Home > MPE Home > Th. List > assamulgscmlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for assamulgscm 21446 (induction base). (Contributed by AV, 26-Aug-2019.) |
| Ref | Expression |
|---|---|
| assamulgscm.v | β’ π = (Baseβπ) |
| assamulgscm.f | β’ πΉ = (Scalarβπ) |
| assamulgscm.b | β’ π΅ = (BaseβπΉ) |
| assamulgscm.s | β’ Β· = ( Β·π βπ) |
| assamulgscm.g | β’ πΊ = (mulGrpβπΉ) |
| assamulgscm.p | β’ β = (.gβπΊ) |
| assamulgscm.h | β’ π» = (mulGrpβπ) |
| assamulgscm.e | β’ πΈ = (.gβπ») |
| Ref | Expression |
|---|---|
| assamulgscmlem1 | β’ (((π΄ β π΅ β§ π β π) β§ π β AssAlg) β (0πΈ(π΄ Β· π)) = ((0 β π΄) Β· (0πΈπ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | assalmod 21406 | . . . 4 β’ (π β AssAlg β π β LMod) | |
| 2 | assaring 21407 | . . . . 5 β’ (π β AssAlg β π β Ring) | |
| 3 | assamulgscm.v | . . . . . 6 β’ π = (Baseβπ) | |
| 4 | eqid 2732 | . . . . . 6 β’ (1rβπ) = (1rβπ) | |
| 5 | 3, 4 | ringidcl 20076 | . . . . 5 β’ (π β Ring β (1rβπ) β π) |
| 6 | 2, 5 | syl 17 | . . . 4 β’ (π β AssAlg β (1rβπ) β π) |
| 7 | assamulgscm.f | . . . . . 6 β’ πΉ = (Scalarβπ) | |
| 8 | assamulgscm.s | . . . . . 6 β’ Β· = ( Β·π βπ) | |
| 9 | eqid 2732 | . . . . . 6 β’ (1rβπΉ) = (1rβπΉ) | |
| 10 | 3, 7, 8, 9 | lmodvs1 20492 | . . . . 5 β’ ((π β LMod β§ (1rβπ) β π) β ((1rβπΉ) Β· (1rβπ)) = (1rβπ)) |
| 11 | 10 | eqcomd 2738 | . . . 4 β’ ((π β LMod β§ (1rβπ) β π) β (1rβπ) = ((1rβπΉ) Β· (1rβπ))) |
| 12 | 1, 6, 11 | syl2anc 584 | . . 3 β’ (π β AssAlg β (1rβπ) = ((1rβπΉ) Β· (1rβπ))) |
| 13 | 12 | adantl 482 | . 2 β’ (((π΄ β π΅ β§ π β π) β§ π β AssAlg) β (1rβπ) = ((1rβπΉ) Β· (1rβπ))) |
| 14 | 1 | adantl 482 | . . . 4 β’ (((π΄ β π΅ β§ π β π) β§ π β AssAlg) β π β LMod) |
| 15 | simpll 765 | . . . 4 β’ (((π΄ β π΅ β§ π β π) β§ π β AssAlg) β π΄ β π΅) | |
| 16 | simplr 767 | . . . 4 β’ (((π΄ β π΅ β§ π β π) β§ π β AssAlg) β π β π) | |
| 17 | assamulgscm.b | . . . . 5 β’ π΅ = (BaseβπΉ) | |
| 18 | 3, 7, 8, 17 | lmodvscl 20481 | . . . 4 β’ ((π β LMod β§ π΄ β π΅ β§ π β π) β (π΄ Β· π) β π) |
| 19 | 14, 15, 16, 18 | syl3anc 1371 | . . 3 β’ (((π΄ β π΅ β§ π β π) β§ π β AssAlg) β (π΄ Β· π) β π) |
| 20 | assamulgscm.h | . . . . 5 β’ π» = (mulGrpβπ) | |
| 21 | 20, 3 | mgpbas 19987 | . . . 4 β’ π = (Baseβπ») |
| 22 | 20, 4 | ringidval 20000 | . . . 4 β’ (1rβπ) = (0gβπ») |
| 23 | assamulgscm.e | . . . 4 β’ πΈ = (.gβπ») | |
| 24 | 21, 22, 23 | mulg0 18951 | . . 3 β’ ((π΄ Β· π) β π β (0πΈ(π΄ Β· π)) = (1rβπ)) |
| 25 | 19, 24 | syl 17 | . 2 β’ (((π΄ β π΅ β§ π β π) β§ π β AssAlg) β (0πΈ(π΄ Β· π)) = (1rβπ)) |
| 26 | assamulgscm.g | . . . . . 6 β’ πΊ = (mulGrpβπΉ) | |
| 27 | 26, 17 | mgpbas 19987 | . . . . 5 β’ π΅ = (BaseβπΊ) |
| 28 | 26, 9 | ringidval 20000 | . . . . 5 β’ (1rβπΉ) = (0gβπΊ) |
| 29 | assamulgscm.p | . . . . 5 β’ β = (.gβπΊ) | |
| 30 | 27, 28, 29 | mulg0 18951 | . . . 4 β’ (π΄ β π΅ β (0 β π΄) = (1rβπΉ)) |
| 31 | 15, 30 | syl 17 | . . 3 β’ (((π΄ β π΅ β§ π β π) β§ π β AssAlg) β (0 β π΄) = (1rβπΉ)) |
| 32 | 21, 22, 23 | mulg0 18951 | . . . 4 β’ (π β π β (0πΈπ) = (1rβπ)) |
| 33 | 16, 32 | syl 17 | . . 3 β’ (((π΄ β π΅ β§ π β π) β§ π β AssAlg) β (0πΈπ) = (1rβπ)) |
| 34 | 31, 33 | oveq12d 7423 | . 2 β’ (((π΄ β π΅ β§ π β π) β§ π β AssAlg) β ((0 β π΄) Β· (0πΈπ)) = ((1rβπΉ) Β· (1rβπ))) |
| 35 | 13, 25, 34 | 3eqtr4d 2782 | 1 β’ (((π΄ β π΅ β§ π β π) β§ π β AssAlg) β (0πΈ(π΄ Β· π)) = ((0 β π΄) Β· (0πΈπ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βcfv 6540 (class class class)co 7405 0cc0 11106 Basecbs 17140 Scalarcsca 17196 Β·π cvsca 17197 .gcmg 18944 mulGrpcmgp 19981 1rcur 19998 Ringcrg 20049 LModclmod 20463 AssAlgcasa 21396 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-seq 13963 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mulg 18945 df-mgp 19982 df-ur 19999 df-ring 20051 df-lmod 20465 df-assa 21399 |
| This theorem is referenced by: assamulgscm 21446 |
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