Theorem crngpropd 13535

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 2193	. . . . . . 7 |- (mulGrp ` K ) = (mulGrp ` K )
3		eqid 2193	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
4	2, 3	mgpbasg 13422	. . . . . 6 |- ( K e. Ring -> ( Base ` K ) = ( Base ` (mulGrp ` K ) ) )
5	1, 4	sylan9eq 2246	. . . . 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 13534	. . . . . . . 8 |- ( ph -> ( K e. Ring <-> L e. Ring ) )
11	10	biimpa 296	. . . . . . 7 |- ( ( ph /\ K e. Ring ) -> L e. Ring )
12		eqid 2193	. . . . . . . 8 |- (mulGrp ` L ) = (mulGrp ` L )
13		eqid 2193	. . . . . . . 8 |- ( Base ` L ) = ( Base ` L )
14	12, 13	mgpbasg 13422	. . . . . . 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 2226	. . . . 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 2193	. . . . . . . . 9 |- ( .r ` K ) = ( .r ` K )
19	2, 18	mgpplusgg 13420	. . . . . . . 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 5946	. . . . . 6 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( +g ` (mulGrp ` K ) ) y ) )
22		eqid 2193	. . . . . . . . 9 |- ( .r ` L ) = ( .r ` L )
23	12, 22	mgpplusgg 13420	. . . . . . . 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 5946	. . . . . 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 2234	. . . . 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 13365	. . . 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 13499	. 2 |- ( K e. CRing <-> ( K e. Ring /\ (mulGrp ` K ) e. CMnd ) )
32	12	iscrng 13499	. 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 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   .rcmulr 12696  CMndccmn 13354  mulGrpcmgp 13416   Ringcrg 13492   CRingccrg 13493

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-cmn 13356  df-mgp 13417  df-ring 13494  df-cring 13495

This theorem is referenced by:  zncrng  14133