Theorem crngpropd 14282

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 2234	. . . . . . 7 |- (mulGrp ` K ) = (mulGrp ` K )
3		eqid 2234	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
4	2, 3	mgpbasg 14165	. . . . . 6 |- ( K e. Ring -> ( Base ` K ) = ( Base ` (mulGrp ` K ) ) )
5	1, 4	sylan9eq 2287	. . . . 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 14281	. . . . . . . 8 |- ( ph -> ( K e. Ring <-> L e. Ring ) )
11	10	biimpa 296	. . . . . . 7 |- ( ( ph /\ K e. Ring ) -> L e. Ring )
12		eqid 2234	. . . . . . . 8 |- (mulGrp ` L ) = (mulGrp ` L )
13		eqid 2234	. . . . . . . 8 |- ( Base ` L ) = ( Base ` L )
14	12, 13	mgpbasg 14165	. . . . . . 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 2267	. . . . 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 2234	. . . . . . . . 9 |- ( .r ` K ) = ( .r ` K )
19	2, 18	mgpplusgg 14163	. . . . . . . 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 6086	. . . . . 6 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( +g ` (mulGrp ` K ) ) y ) )
22		eqid 2234	. . . . . . . . 9 |- ( .r ` L ) = ( .r ` L )
23	12, 22	mgpplusgg 14163	. . . . . . . 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 6086	. . . . . 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 2275	. . . . 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 14048	. . . 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 14246	. 2 |- ( K e. CRing <-> ( K e. Ring /\ (mulGrp ` K ) e. CMnd ) )
32	12	iscrng 14246	. 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 2205   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   .rcmulr 13375  CMndccmn 14037  mulGrpcmgp 14159   Ringcrg 14239   CRingccrg 14240

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-cmn 14039  df-mgp 14160  df-ring 14241  df-cring 14242

This theorem is referenced by:  zncrng  14919