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Mirrors > Home > ILE Home > Th. List > mgpbasg | GIF version |
Description: Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
mgpbas.1 | β’ π = (mulGrpβπ ) |
mgpbas.2 | β’ π΅ = (Baseβπ ) |
Ref | Expression |
---|---|
mgpbasg | β’ (π β π β π΅ = (Baseβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpbas.2 | . 2 β’ π΅ = (Baseβπ ) | |
2 | mulrslid 12589 | . . . . 5 β’ (.r = Slot (.rβndx) β§ (.rβndx) β β) | |
3 | 2 | slotex 12488 | . . . 4 β’ (π β π β (.rβπ ) β V) |
4 | baseslid 12518 | . . . . 5 β’ (Base = Slot (Baseβndx) β§ (Baseβndx) β β) | |
5 | basendxnplusgndx 12582 | . . . . 5 β’ (Baseβndx) β (+gβndx) | |
6 | plusgslid 12570 | . . . . . 6 β’ (+g = Slot (+gβndx) β§ (+gβndx) β β) | |
7 | 6 | simpri 113 | . . . . 5 β’ (+gβndx) β β |
8 | 4, 5, 7 | setsslnid 12513 | . . . 4 β’ ((π β π β§ (.rβπ ) β V) β (Baseβπ ) = (Baseβ(π sSet β¨(+gβndx), (.rβπ )β©))) |
9 | 3, 8 | mpdan 421 | . . 3 β’ (π β π β (Baseβπ ) = (Baseβ(π sSet β¨(+gβndx), (.rβπ )β©))) |
10 | mgpbas.1 | . . . . 5 β’ π = (mulGrpβπ ) | |
11 | eqid 2177 | . . . . 5 β’ (.rβπ ) = (.rβπ ) | |
12 | 10, 11 | mgpvalg 13131 | . . . 4 β’ (π β π β π = (π sSet β¨(+gβndx), (.rβπ )β©)) |
13 | 12 | fveq2d 5519 | . . 3 β’ (π β π β (Baseβπ) = (Baseβ(π sSet β¨(+gβndx), (.rβπ )β©))) |
14 | 9, 13 | eqtr4d 2213 | . 2 β’ (π β π β (Baseβπ ) = (Baseβπ)) |
15 | 1, 14 | eqtrid 2222 | 1 β’ (π β π β π΅ = (Baseβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 Vcvv 2737 β¨cop 3595 βcfv 5216 (class class class)co 5874 βcn 8918 ndxcnx 12458 sSet csts 12459 Slot cslot 12460 Basecbs 12461 +gcplusg 12535 .rcmulr 12536 mulGrpcmgp 13128 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-ltadd 7926 |
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 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7993 df-mnf 7994 df-ltxr 7996 df-inn 8919 df-2 8977 df-3 8978 df-ndx 12464 df-slot 12465 df-base 12467 df-sets 12468 df-plusg 12548 df-mulr 12549 df-mgp 13129 |
This theorem is referenced by: mgptopng 13137 mgpress 13139 dfur2g 13143 srgcl 13151 srgass 13152 srgideu 13153 srgidcl 13157 srgidmlem 13159 issrgid 13162 srg1zr 13168 srgpcomp 13171 srgpcompp 13172 srgpcomppsc 13173 ringcl 13194 crngcom 13195 iscrng2 13196 ringass 13197 ringideu 13198 ringidcl 13201 ringidmlem 13203 isringid 13206 ringidss 13210 ringpropd 13215 crngpropd 13216 isringd 13218 iscrngd 13219 ring1 13234 oppr1g 13250 unitgrpbasd 13282 unitsubm 13286 rngidpropdg 13313 subrgsubm 13353 issubrg3 13366 cnfldexp 13407 |
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