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Mirrors > Home > ILE Home > Th. List > mgpscag | GIF version |
Description: The multiplication monoid has the same (if any) scalars as the original ring. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
Ref | Expression |
---|---|
mgpbas.1 | β’ π = (mulGrpβπ ) |
mgpsca.s | β’ π = (Scalarβπ ) |
Ref | Expression |
---|---|
mgpscag | β’ (π β π β π = (Scalarβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpsca.s | . 2 β’ π = (Scalarβπ ) | |
2 | mulrslid 12593 | . . . . 5 β’ (.r = Slot (.rβndx) β§ (.rβndx) β β) | |
3 | 2 | slotex 12492 | . . . 4 β’ (π β π β (.rβπ ) β V) |
4 | scaslid 12614 | . . . . 5 β’ (Scalar = Slot (Scalarβndx) β§ (Scalarβndx) β β) | |
5 | scandxnplusgndx 12616 | . . . . 5 β’ (Scalarβndx) β (+gβndx) | |
6 | plusgslid 12574 | . . . . . 6 β’ (+g = Slot (+gβndx) β§ (+gβndx) β β) | |
7 | 6 | simpri 113 | . . . . 5 β’ (+gβndx) β β |
8 | 4, 5, 7 | setsslnid 12517 | . . . 4 β’ ((π β π β§ (.rβπ ) β V) β (Scalarβπ ) = (Scalarβ(π sSet β¨(+gβndx), (.rβπ )β©))) |
9 | 3, 8 | mpdan 421 | . . 3 β’ (π β π β (Scalarβπ ) = (Scalarβ(π sSet β¨(+gβndx), (.rβπ )β©))) |
10 | mgpbas.1 | . . . . 5 β’ π = (mulGrpβπ ) | |
11 | eqid 2177 | . . . . 5 β’ (.rβπ ) = (.rβπ ) | |
12 | 10, 11 | mgpvalg 13149 | . . . 4 β’ (π β π β π = (π sSet β¨(+gβndx), (.rβπ )β©)) |
13 | 12 | fveq2d 5521 | . . 3 β’ (π β π β (Scalarβπ) = (Scalarβ(π sSet β¨(+gβndx), (.rβπ )β©))) |
14 | 9, 13 | eqtr4d 2213 | . 2 β’ (π β π β (Scalarβπ ) = (Scalarβπ)) |
15 | 1, 14 | eqtrid 2222 | 1 β’ (π β π β π = (Scalarβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 Vcvv 2739 β¨cop 3597 βcfv 5218 (class class class)co 5878 βcn 8922 ndxcnx 12462 sSet csts 12463 Slot cslot 12464 +gcplusg 12539 .rcmulr 12540 Scalarcsca 12542 mulGrpcmgp 13146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-pre-ltirr 7926 ax-pre-lttrn 7928 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-iota 5180 df-fun 5220 df-fn 5221 df-fv 5226 df-ov 5881 df-oprab 5882 df-mpo 5883 df-pnf 7997 df-mnf 7998 df-ltxr 8000 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-5 8984 df-ndx 12468 df-slot 12469 df-sets 12472 df-plusg 12552 df-mulr 12553 df-sca 12555 df-mgp 13147 |
This theorem is referenced by: (None) |
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