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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 12605 | . . . . 5 β’ (.r = Slot (.rβndx) β§ (.rβndx) β β) | |
3 | 2 | slotex 12503 | . . . 4 β’ (π β π β (.rβπ ) β V) |
4 | baseslid 12533 | . . . . 5 β’ (Base = Slot (Baseβndx) β§ (Baseβndx) β β) | |
5 | basendxnplusgndx 12598 | . . . . 5 β’ (Baseβndx) β (+gβndx) | |
6 | plusgslid 12586 | . . . . . 6 β’ (+g = Slot (+gβndx) β§ (+gβndx) β β) | |
7 | 6 | simpri 113 | . . . . 5 β’ (+gβndx) β β |
8 | 4, 5, 7 | setsslnid 12528 | . . . 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 2187 | . . . . 5 β’ (.rβπ ) = (.rβπ ) | |
12 | 10, 11 | mgpvalg 13232 | . . . 4 β’ (π β π β π = (π sSet β¨(+gβndx), (.rβπ )β©)) |
13 | 12 | fveq2d 5531 | . . 3 β’ (π β π β (Baseβπ) = (Baseβ(π sSet β¨(+gβndx), (.rβπ )β©))) |
14 | 9, 13 | eqtr4d 2223 | . 2 β’ (π β π β (Baseβπ ) = (Baseβπ)) |
15 | 1, 14 | eqtrid 2232 | 1 β’ (π β π β π΅ = (Baseβπ)) |
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 βcn 8933 ndxcnx 12473 sSet csts 12474 Slot cslot 12475 Basecbs 12476 +gcplusg 12551 .rcmulr 12552 mulGrpcmgp 13229 |
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 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-i2m1 7930 ax-0lt1 7931 ax-0id 7933 ax-rnegex 7934 ax-pre-ltirr 7937 ax-pre-ltadd 7941 |
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-nel 2453 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-pnf 8008 df-mnf 8009 df-ltxr 8011 df-inn 8934 df-2 8992 df-3 8993 df-ndx 12479 df-slot 12480 df-base 12482 df-sets 12483 df-plusg 12564 df-mulr 12565 df-mgp 13230 |
This theorem is referenced by: mgptopng 13238 mgpress 13240 rngass 13248 rngcl 13253 isrngd 13262 rngpropd 13264 dfur2g 13271 srgcl 13279 srgass 13280 srgideu 13281 srgidcl 13285 srgidmlem 13287 issrgid 13290 srg1zr 13296 srgpcomp 13299 srgpcompp 13300 srgpcomppsc 13301 ringcl 13322 crngcom 13323 iscrng2 13324 ringass 13325 ringideu 13326 ringidcl 13329 ringidmlem 13331 isringid 13334 ringidss 13338 ringpropd 13347 crngpropd 13348 isringd 13350 iscrngd 13351 ring1 13366 oppr1g 13387 unitgrpbasd 13420 unitsubm 13424 rngidpropdg 13451 subrgsubm 13511 issubrg3 13524 rnglidlmmgm 13742 rnglidlmsgrp 13743 cnfldexp 13810 |
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