Theorem crngpropd 14317

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 2238	. . . . . . 7 |- (mulGrp ` K ) = (mulGrp ` K )
3		eqid 2238	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
4	2, 3	mgpbasg 14200	. . . . . 6 |- ( K e. Ring -> ( Base ` K ) = ( Base ` (mulGrp ` K ) ) )
5	1, 4	sylan9eq 2291	. . . . 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 14316	. . . . . . . 8 |- ( ph -> ( K e. Ring <-> L e. Ring ) )
11	10	biimpa 296	. . . . . . 7 |- ( ( ph /\ K e. Ring ) -> L e. Ring )
12		eqid 2238	. . . . . . . 8 |- (mulGrp ` L ) = (mulGrp ` L )
13		eqid 2238	. . . . . . . 8 |- ( Base ` L ) = ( Base ` L )
14	12, 13	mgpbasg 14200	. . . . . . 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 2271	. . . . 5 |- ( ( ph /\ K e. Ring ) -> B = ( Base ` (mulGrp ` L ) ) )
17	9	adantlr 481	. . . . . 6 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
18		eqid 2238	. . . . . . . . 9 |- ( .r ` K ) = ( .r ` K )
19	2, 18	mgpplusgg 14198	. . . . . . . 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 6103	. . . . . 6 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( +g ` (mulGrp ` K ) ) y ) )
22		eqid 2238	. . . . . . . . 9 |- ( .r ` L ) = ( .r ` L )
23	12, 22	mgpplusgg 14198	. . . . . . . 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 6103	. . . . . 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 2279	. . . . 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 14075	. . . 4 |- ( ( ph /\ K e. Ring ) -> ( (mulGrp ` K ) e. CMnd <-> (mulGrp ` L ) e. CMnd ) )
28	27	pm5.32da 456	. . 3 |- ( ph -> ( ( K e. Ring /\ (mulGrp ` K ) e. CMnd ) <-> ( K e. Ring /\ (mulGrp ` L ) e. CMnd ) ) )
29	10	anbi1d 469	. . 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 14281	. 2 |- ( K e. CRing <-> ( K e. Ring /\ (mulGrp ` K ) e. CMnd ) )
32	12	iscrng 14281	. 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 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   .rcmulr 13409  CMndccmn 14064  mulGrpcmgp 14194   Ringcrg 14274   CRingccrg 14275

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-cmn 14066  df-mgp 14195  df-ring 14276  df-cring 14277

This theorem is referenced by:  zncrng  14952