Theorem crngpropd 13010

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

Step	Hyp	Ref	Expression
1		ringpropd.1	. . . . . 6 |- ( ph -> B = ( Base ` K ) )
2		eqid 2175	. . . . . . 7 |- (mulGrp ` K ) = (mulGrp ` K )
3		eqid 2175	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
4	2, 3	mgpbasg 12930	. . . . . 6 |- ( K e. Ring -> ( Base ` K ) = ( Base ` (mulGrp ` K ) ) )
5	1, 4	sylan9eq 2228	. . . . 5 |- ( ( ph /\ K e. Ring ) -> B = ( Base ` (mulGrp ` K ) ) )
6		ringpropd.2	. . . . . . 7 |- ( ph -> B = ( Base ` L ) )
7	6	adantr 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 ) )
10	1, 6, 8, 9	ringpropd 13009	. . . . . . . 8 |- ( ph -> ( K e. Ring <-> L e. Ring ) )
11	10	biimpa 296	. . . . . . 7 |- ( ( ph /\ K e. Ring ) -> L e. Ring )
12		eqid 2175	. . . . . . . 8 |- (mulGrp ` L ) = (mulGrp ` L )
13		eqid 2175	. . . . . . . 8 |- ( Base ` L ) = ( Base ` L )
14	12, 13	mgpbasg 12930	. . . . . . 7 |- ( L e. Ring -> ( Base ` L ) = ( Base ` (mulGrp ` L ) ) )
15	11, 14	syl 14	. . . . . 6 |- ( ( ph /\ K e. Ring ) -> ( Base ` L ) = ( Base ` (mulGrp ` L ) ) )
16	7, 15	eqtrd 2208	. . . . 5 |- ( ( ph /\ K e. Ring ) -> B = ( Base ` (mulGrp ` L ) ) )
17	9	adantlr 477	. . . . . 6 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
18		eqid 2175	. . . . . . . . 9 |- ( .r ` K ) = ( .r ` K )
19	2, 18	mgpplusgg 12929	. . . . . . . 8 |- ( K e. Ring -> ( .r ` K ) = ( +g ` (mulGrp ` K ) ) )
20	19	adantl 277	. . . . . . 7 |- ( ( ph /\ K e. Ring ) -> ( .r ` K ) = ( +g ` (mulGrp ` K ) ) )
21	20	oveqdr 5893	. . . . . 6 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( +g ` (mulGrp ` K ) ) y ) )
22		eqid 2175	. . . . . . . . 9 |- ( .r ` L ) = ( .r ` L )
23	12, 22	mgpplusgg 12929	. . . . . . . 8 |- ( L e. Ring -> ( .r ` L ) = ( +g ` (mulGrp ` L ) ) )
24	11, 23	syl 14	. . . . . . 7 |- ( ( ph /\ K e. Ring ) -> ( .r ` L ) = ( +g ` (mulGrp ` L ) ) )
25	24	oveqdr 5893	. . . . . 6 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` L ) y ) = ( x ( +g ` (mulGrp ` L ) ) y ) )
26	17, 21, 25	3eqtr3d 2216	. . . . 5 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` (mulGrp ` K ) ) y ) = ( x ( +g ` (mulGrp ` L ) ) y ) )
27	5, 16, 26	cmnpropd 12894	. . . 4 |- ( ( ph /\ K e. Ring ) -> ( (mulGrp ` K ) e. CMnd <-> (mulGrp ` L ) e. CMnd ) )
28	27	pm5.32da 452	. . 3 |- ( ph -> ( ( K e. Ring /\ (mulGrp ` K ) e. CMnd ) <-> ( K e. Ring /\ (mulGrp ` L ) e. CMnd ) ) )
29	10	anbi1d 465	. . 3 |- ( ph -> ( ( K e. Ring /\ (mulGrp ` L ) e. CMnd ) <-> ( L e. Ring /\ (mulGrp ` L ) e. CMnd ) ) )
30	28, 29	bitrd 188	. 2 |- ( ph -> ( ( K e. Ring /\ (mulGrp ` K ) e. CMnd ) <-> ( L e. Ring /\ (mulGrp ` L ) e. CMnd ) ) )
31	2	iscrng 12979	. 2 |- ( K e. CRing <-> ( K e. Ring /\ (mulGrp ` K ) e. CMnd ) )
32	12	iscrng 12979	. 2 |- ( L e. CRing <-> ( L e. Ring /\ (mulGrp ` L ) e. CMnd ) )
33	30, 31, 32	3bitr4g 223	1 |- ( ph -> ( K e. CRing <-> L e. CRing ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2146   ` cfv 5208  (class class class)co 5865   Basecbs 12428   +g cplusg 12492   .rcmulr 12493  CMndccmn 12884  mulGrpcmgp 12925   Ringcrg 12972   CRingccrg 12973

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-pre-ltirr 7898  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-ltxr 7971  df-inn 8891  df-2 8949  df-3 8950  df-ndx 12431  df-slot 12432  df-base 12434  df-sets 12435  df-plusg 12505  df-mulr 12506  df-0g 12628  df-mgm 12640  df-sgrp 12673  df-mnd 12683  df-grp 12741  df-cmn 12886  df-mgp 12926  df-ring 12974  df-cring 12975

This theorem is referenced by: (None)