Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > assamulgscmlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for assamulgscm 21838 (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 21798 | . . . 4 β’ (π β AssAlg β π β LMod) | |
| 2 | assaring 21799 | . . . . 5 β’ (π β AssAlg β π β Ring) | |
| 3 | assamulgscm.v | . . . . . 6 β’ π = (Baseβπ) | |
| 4 | eqid 2725 | . . . . . 6 β’ (1rβπ) = (1rβπ) | |
| 5 | 3, 4 | ringidcl 20206 | . . . . 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 2725 | . . . . . 6 β’ (1rβπΉ) = (1rβπΉ) | |
| 10 | 3, 7, 8, 9 | lmodvs1 20777 | . . . . 5 β’ ((π β LMod β§ (1rβπ) β π) β ((1rβπΉ) Β· (1rβπ)) = (1rβπ)) |
| 11 | 10 | eqcomd 2731 | . . . 4 β’ ((π β LMod β§ (1rβπ) β π) β (1rβπ) = ((1rβπΉ) Β· (1rβπ))) |
| 12 | 1, 6, 11 | syl2anc 582 | . . 3 β’ (π β AssAlg β (1rβπ) = ((1rβπΉ) Β· (1rβπ))) |
| 13 | 12 | adantl 480 | . 2 β’ (((π΄ β π΅ β§ π β π) β§ π β AssAlg) β (1rβπ) = ((1rβπΉ) Β· (1rβπ))) |
| 14 | 1 | adantl 480 | . . . 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 20765 | . . . 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 1368 | . . 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 20084 | . . . 4 β’ π = (Baseβπ») |
| 22 | 20, 4 | ringidval 20127 | . . . 4 β’ (1rβπ) = (0gβπ») |
| 23 | assamulgscm.e | . . . 4 β’ πΈ = (.gβπ») | |
| 24 | 21, 22, 23 | mulg0 19034 | . . 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 20084 | . . . . 5 β’ π΅ = (BaseβπΊ) |
| 28 | 26, 9 | ringidval 20127 | . . . . 5 β’ (1rβπΉ) = (0gβπΊ) |
| 29 | assamulgscm.p | . . . . 5 β’ β = (.gβπΊ) | |
| 30 | 27, 28, 29 | mulg0 19034 | . . . 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 19034 | . . . 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 7435 | . 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 2775 | 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 394 = wceq 1533 β wcel 2098 βcfv 6547 (class class class)co 7417 0cc0 11138 Basecbs 17179 Scalarcsca 17235 Β·π cvsca 17236 .gcmg 19027 mulGrpcmgp 20078 1rcur 20125 Ringcrg 20177 LModclmod 20747 AssAlgcasa 21788 |
| 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-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 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 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7870 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 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-n0 12503 df-z 12589 df-uz 12853 df-seq 13999 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mulg 19028 df-mgp 20079 df-ur 20126 df-ring 20179 df-lmod 20749 df-assa 21791 |
| This theorem is referenced by: assamulgscm 21838 |
| Copyright terms: Public domain | W3C validator |