Theorem crngpropd 14133

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 2231	. . . . . . 7 |- (mulGrp ` K ) = (mulGrp ` K )
3		eqid 2231	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
4	2, 3	mgpbasg 14020	. . . . . 6 |- ( K e. Ring -> ( Base ` K ) = ( Base ` (mulGrp ` K ) ) )
5	1, 4	sylan9eq 2284	. . . . 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 14132	. . . . . . . 8 |- ( ph -> ( K e. Ring <-> L e. Ring ) )
11	10	biimpa 296	. . . . . . 7 |- ( ( ph /\ K e. Ring ) -> L e. Ring )
12		eqid 2231	. . . . . . . 8 |- (mulGrp ` L ) = (mulGrp ` L )
13		eqid 2231	. . . . . . . 8 |- ( Base ` L ) = ( Base ` L )
14	12, 13	mgpbasg 14020	. . . . . . 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 2264	. . . . 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 2231	. . . . . . . . 9 |- ( .r ` K ) = ( .r ` K )
19	2, 18	mgpplusgg 14018	. . . . . . . 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 6056	. . . . . 6 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( +g ` (mulGrp ` K ) ) y ) )
22		eqid 2231	. . . . . . . . 9 |- ( .r ` L ) = ( .r ` L )
23	12, 22	mgpplusgg 14018	. . . . . . . 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 6056	. . . . . 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 2272	. . . . 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 13962	. . . 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 14097	. 2 |- ( K e. CRing <-> ( K e. Ring /\ (mulGrp ` K ) e. CMnd ) )
32	12	iscrng 14097	. 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 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   .rcmulr 13241  CMndccmn 13951  mulGrpcmgp 14014   Ringcrg 14090   CRingccrg 14091

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-cmn 13953  df-mgp 14015  df-ring 14092  df-cring 14093

This theorem is referenced by:  zncrng  14741