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Mirrors > Home > ILE Home > Th. List > mgpplusgg | GIF version |
Description: Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) |
Ref | Expression |
---|---|
mgpval.1 | β’ π = (mulGrpβπ ) |
mgpval.2 | β’ Β· = (.rβπ ) |
Ref | Expression |
---|---|
mgpplusgg | β’ (π β π β Β· = (+gβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpval.2 | . . . 4 β’ Β· = (.rβπ ) | |
2 | mulrslid 12604 | . . . . 5 β’ (.r = Slot (.rβndx) β§ (.rβndx) β β) | |
3 | 2 | slotex 12502 | . . . 4 β’ (π β π β (.rβπ ) β V) |
4 | 1, 3 | eqeltrid 2274 | . . 3 β’ (π β π β Β· β V) |
5 | plusgslid 12585 | . . . 4 β’ (+g = Slot (+gβndx) β§ (+gβndx) β β) | |
6 | 5 | setsslid 12526 | . . 3 β’ ((π β π β§ Β· β V) β Β· = (+gβ(π sSet β¨(+gβndx), Β· β©))) |
7 | 4, 6 | mpdan 421 | . 2 β’ (π β π β Β· = (+gβ(π sSet β¨(+gβndx), Β· β©))) |
8 | mgpval.1 | . . . 4 β’ π = (mulGrpβπ ) | |
9 | 8, 1 | mgpvalg 13165 | . . 3 β’ (π β π β π = (π sSet β¨(+gβndx), Β· β©)) |
10 | 9 | fveq2d 5531 | . 2 β’ (π β π β (+gβπ) = (+gβ(π sSet β¨(+gβndx), Β· β©))) |
11 | 7, 10 | eqtr4d 2223 | 1 β’ (π β π β Β· = (+gβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1363 β wcel 2158 Vcvv 2749 β¨cop 3607 βcfv 5228 (class class class)co 5888 ndxcnx 12472 sSet csts 12473 +gcplusg 12550 .rcmulr 12551 mulGrpcmgp 13162 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1re 7918 ax-addrcl 7921 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-iota 5190 df-fun 5230 df-fn 5231 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-inn 8933 df-2 8991 df-3 8992 df-ndx 12478 df-slot 12479 df-sets 12482 df-plusg 12563 df-mulr 12564 df-mgp 13163 |
This theorem is referenced by: rngass 13181 rngcl 13186 isrngd 13195 dfur2g 13199 srgcl 13207 srgass 13208 srgideu 13209 srgidmlem 13215 issrgid 13218 srg1zr 13224 srgpcomp 13227 srgpcompp 13228 ringcl 13250 crngcom 13251 iscrng2 13252 ringass 13253 ringideu 13254 ringidmlem 13259 isringid 13262 ringidss 13266 ringpropd 13275 crngpropd 13276 isringd 13278 iscrngd 13279 ring1 13294 oppr1g 13313 unitgrp 13347 unitlinv 13357 unitrinv 13358 rdivmuldivd 13375 rngidpropdg 13377 invrpropdg 13380 subrgugrp 13423 issubrg3 13430 cnfldexp 13684 |
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