![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ascl0 | Structured version Visualization version GIF version |
Description: The scalar 0 embedded into a left module corresponds to the 0 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.) |
Ref | Expression |
---|---|
ascl0.a | β’ π΄ = (algScβπ) |
ascl0.f | β’ πΉ = (Scalarβπ) |
ascl0.l | β’ (π β π β LMod) |
ascl0.r | β’ (π β π β Ring) |
Ref | Expression |
---|---|
ascl0 | β’ (π β (π΄β(0gβπΉ)) = (0gβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ascl0.l | . . . . 5 β’ (π β π β LMod) | |
2 | ascl0.f | . . . . . 6 β’ πΉ = (Scalarβπ) | |
3 | 2 | lmodfgrp 20745 | . . . . 5 β’ (π β LMod β πΉ β Grp) |
4 | 1, 3 | syl 17 | . . . 4 β’ (π β πΉ β Grp) |
5 | eqid 2728 | . . . . 5 β’ (BaseβπΉ) = (BaseβπΉ) | |
6 | eqid 2728 | . . . . 5 β’ (0gβπΉ) = (0gβπΉ) | |
7 | 5, 6 | grpidcl 18915 | . . . 4 β’ (πΉ β Grp β (0gβπΉ) β (BaseβπΉ)) |
8 | 4, 7 | syl 17 | . . 3 β’ (π β (0gβπΉ) β (BaseβπΉ)) |
9 | ascl0.a | . . . 4 β’ π΄ = (algScβπ) | |
10 | eqid 2728 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
11 | eqid 2728 | . . . 4 β’ (1rβπ) = (1rβπ) | |
12 | 9, 2, 5, 10, 11 | asclval 21806 | . . 3 β’ ((0gβπΉ) β (BaseβπΉ) β (π΄β(0gβπΉ)) = ((0gβπΉ)( Β·π βπ)(1rβπ))) |
13 | 8, 12 | syl 17 | . 2 β’ (π β (π΄β(0gβπΉ)) = ((0gβπΉ)( Β·π βπ)(1rβπ))) |
14 | ascl0.r | . . . 4 β’ (π β π β Ring) | |
15 | eqid 2728 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
16 | 15, 11 | ringidcl 20195 | . . . 4 β’ (π β Ring β (1rβπ) β (Baseβπ)) |
17 | 14, 16 | syl 17 | . . 3 β’ (π β (1rβπ) β (Baseβπ)) |
18 | eqid 2728 | . . . 4 β’ (0gβπ) = (0gβπ) | |
19 | 15, 2, 10, 6, 18 | lmod0vs 20771 | . . 3 β’ ((π β LMod β§ (1rβπ) β (Baseβπ)) β ((0gβπΉ)( Β·π βπ)(1rβπ)) = (0gβπ)) |
20 | 1, 17, 19 | syl2anc 583 | . 2 β’ (π β ((0gβπΉ)( Β·π βπ)(1rβπ)) = (0gβπ)) |
21 | 13, 20 | eqtrd 2768 | 1 β’ (π β (π΄β(0gβπΉ)) = (0gβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 βcfv 6542 (class class class)co 7414 Basecbs 17173 Scalarcsca 17229 Β·π cvsca 17230 0gc0g 17414 Grpcgrp 18883 1rcur 20114 Ringcrg 20166 LModclmod 20736 algSccascl 21779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-plusg 17239 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-mgp 20068 df-ur 20115 df-ring 20168 df-lmod 20738 df-ascl 21782 |
This theorem is referenced by: ply1ascl0 22166 ply1scl0 22202 mplascl0 41781 assaascl0 47442 |
Copyright terms: Public domain | W3C validator |