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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 20345 | . . . . 5 β’ (π β LMod β πΉ β Grp) |
4 | 1, 3 | syl 17 | . . . 4 β’ (π β πΉ β Grp) |
5 | eqid 2733 | . . . . 5 β’ (BaseβπΉ) = (BaseβπΉ) | |
6 | eqid 2733 | . . . . 5 β’ (0gβπΉ) = (0gβπΉ) | |
7 | 5, 6 | grpidcl 18783 | . . . 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 2733 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
11 | eqid 2733 | . . . 4 β’ (1rβπ) = (1rβπ) | |
12 | 9, 2, 5, 10, 11 | asclval 21299 | . . 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 2733 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
16 | 15, 11 | ringidcl 19994 | . . . 4 β’ (π β Ring β (1rβπ) β (Baseβπ)) |
17 | 14, 16 | syl 17 | . . 3 β’ (π β (1rβπ) β (Baseβπ)) |
18 | eqid 2733 | . . . 4 β’ (0gβπ) = (0gβπ) | |
19 | 15, 2, 10, 6, 18 | lmod0vs 20370 | . . 3 β’ ((π β LMod β§ (1rβπ) β (Baseβπ)) β ((0gβπΉ)( Β·π βπ)(1rβπ)) = (0gβπ)) |
20 | 1, 17, 19 | syl2anc 585 | . 2 β’ (π β ((0gβπΉ)( Β·π βπ)(1rβπ)) = (0gβπ)) |
21 | 13, 20 | eqtrd 2773 | 1 β’ (π β (π΄β(0gβπΉ)) = (0gβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17088 Scalarcsca 17141 Β·π cvsca 17142 0gc0g 17326 Grpcgrp 18753 1rcur 19918 Ringcrg 19969 LModclmod 20336 algSccascl 21274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-ascl 21277 |
This theorem is referenced by: ply1ascl0 32325 mplascl0 40785 assaascl0 46546 |
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