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Theorem crngpropd 13041
Description: If two structures have the same group components (properties), one is a commutative ring iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
ringpropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
ringpropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
ringpropd.3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
ringpropd.4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
Assertion
Ref Expression
crngpropd  |-  ( ph  ->  ( K  e.  CRing  <->  L  e.  CRing ) )
Distinct variable groups:    x, y, B   
x, K, y    ph, x, y    x, L, y

Proof of Theorem crngpropd
StepHypRef Expression
1 ringpropd.1 . . . . . 6  |-  ( ph  ->  B  =  ( Base `  K ) )
2 eqid 2177 . . . . . . 7  |-  (mulGrp `  K )  =  (mulGrp `  K )
3 eqid 2177 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
42, 3mgpbasg 12960 . . . . . 6  |-  ( K  e.  Ring  ->  ( Base `  K )  =  (
Base `  (mulGrp `  K
) ) )
51, 4sylan9eq 2230 . . . . 5  |-  ( (
ph  /\  K  e.  Ring )  ->  B  =  ( Base `  (mulGrp `  K
) ) )
6 ringpropd.2 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  L ) )
76adantr 276 . . . . . 6  |-  ( (
ph  /\  K  e.  Ring )  ->  B  =  ( Base `  L )
)
8 ringpropd.3 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
9 ringpropd.4 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
101, 6, 8, 9ringpropd 13040 . . . . . . . 8  |-  ( ph  ->  ( K  e.  Ring  <->  L  e.  Ring ) )
1110biimpa 296 . . . . . . 7  |-  ( (
ph  /\  K  e.  Ring )  ->  L  e.  Ring )
12 eqid 2177 . . . . . . . 8  |-  (mulGrp `  L )  =  (mulGrp `  L )
13 eqid 2177 . . . . . . . 8  |-  ( Base `  L )  =  (
Base `  L )
1412, 13mgpbasg 12960 . . . . . . 7  |-  ( L  e.  Ring  ->  ( Base `  L )  =  (
Base `  (mulGrp `  L
) ) )
1511, 14syl 14 . . . . . 6  |-  ( (
ph  /\  K  e.  Ring )  ->  ( Base `  L )  =  (
Base `  (mulGrp `  L
) ) )
167, 15eqtrd 2210 . . . . 5  |-  ( (
ph  /\  K  e.  Ring )  ->  B  =  ( Base `  (mulGrp `  L
) ) )
179adantlr 477 . . . . . 6  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( .r `  K
) y )  =  ( x ( .r
`  L ) y ) )
18 eqid 2177 . . . . . . . . 9  |-  ( .r
`  K )  =  ( .r `  K
)
192, 18mgpplusgg 12958 . . . . . . . 8  |-  ( K  e.  Ring  ->  ( .r
`  K )  =  ( +g  `  (mulGrp `  K ) ) )
2019adantl 277 . . . . . . 7  |-  ( (
ph  /\  K  e.  Ring )  ->  ( .r `  K )  =  ( +g  `  (mulGrp `  K ) ) )
2120oveqdr 5897 . . . . . 6  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( .r `  K
) y )  =  ( x ( +g  `  (mulGrp `  K )
) y ) )
22 eqid 2177 . . . . . . . . 9  |-  ( .r
`  L )  =  ( .r `  L
)
2312, 22mgpplusgg 12958 . . . . . . . 8  |-  ( L  e.  Ring  ->  ( .r
`  L )  =  ( +g  `  (mulGrp `  L ) ) )
2411, 23syl 14 . . . . . . 7  |-  ( (
ph  /\  K  e.  Ring )  ->  ( .r `  L )  =  ( +g  `  (mulGrp `  L ) ) )
2524oveqdr 5897 . . . . . 6  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( .r `  L
) y )  =  ( x ( +g  `  (mulGrp `  L )
) y ) )
2617, 21, 253eqtr3d 2218 . . . . 5  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  (mulGrp `  K ) ) y )  =  ( x ( +g  `  (mulGrp `  L ) ) y ) )
275, 16, 26cmnpropd 12922 . . . 4  |-  ( (
ph  /\  K  e.  Ring )  ->  ( (mulGrp `  K )  e. CMnd  <->  (mulGrp `  L
)  e. CMnd ) )
2827pm5.32da 452 . . 3  |-  ( ph  ->  ( ( K  e. 
Ring  /\  (mulGrp `  K
)  e. CMnd )  <->  ( K  e.  Ring  /\  (mulGrp `  L
)  e. CMnd ) )
)
2910anbi1d 465 . . 3  |-  ( ph  ->  ( ( K  e. 
Ring  /\  (mulGrp `  L
)  e. CMnd )  <->  ( L  e.  Ring  /\  (mulGrp `  L
)  e. CMnd ) )
)
3028, 29bitrd 188 . 2  |-  ( ph  ->  ( ( K  e. 
Ring  /\  (mulGrp `  K
)  e. CMnd )  <->  ( L  e.  Ring  /\  (mulGrp `  L
)  e. CMnd ) )
)
312iscrng 13009 . 2  |-  ( K  e.  CRing 
<->  ( K  e.  Ring  /\  (mulGrp `  K )  e. CMnd ) )
3212iscrng 13009 . 2  |-  ( L  e.  CRing 
<->  ( L  e.  Ring  /\  (mulGrp `  L )  e. CMnd ) )
3330, 31, 323bitr4g 223 1  |-  ( ph  ->  ( K  e.  CRing  <->  L  e.  CRing ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   ` cfv 5212  (class class class)co 5869   Basecbs 12442   +g cplusg 12515   .rcmulr 12516  CMndccmn 12912  mulGrpcmgp 12954   Ringcrg 13002   CRingccrg 13003
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 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7890  ax-resscn 7891  ax-1cn 7892  ax-1re 7893  ax-icn 7894  ax-addcl 7895  ax-addrcl 7896  ax-mulcl 7897  ax-addcom 7899  ax-addass 7901  ax-i2m1 7904  ax-0lt1 7905  ax-0id 7907  ax-rnegex 7908  ax-pre-ltirr 7911  ax-pre-ltadd 7915
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7981  df-mnf 7982  df-ltxr 7984  df-inn 8906  df-2 8964  df-3 8965  df-ndx 12445  df-slot 12446  df-base 12448  df-sets 12449  df-plusg 12528  df-mulr 12529  df-0g 12652  df-mgm 12664  df-sgrp 12697  df-mnd 12707  df-grp 12767  df-cmn 12914  df-mgp 12955  df-ring 13004  df-cring 13005
This theorem is referenced by: (None)
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