Theorem cmnpropd 12894

Description: If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
ablpropd.1	|- ( ph -> B = ( Base ` K ) )
ablpropd.2	|- ( ph -> B = ( Base ` L ) )
ablpropd.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )

Assertion

Ref	Expression
cmnpropd	|- ( ph -> ( K e. CMnd <-> L e. CMnd ) )

Distinct variable groups:   x, y, B   x, K, y   x, L, y   ph, x, y

Proof of Theorem cmnpropd

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablpropd.1	. . . 4 |- ( ph -> B = ( Base ` K ) )
2		ablpropd.2	. . . 4 |- ( ph -> B = ( Base ` L ) )
3		ablpropd.3	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
4	1, 2, 3	mndpropd 12706	. . 3 |- ( ph -> ( K e. Mnd <-> L e. Mnd ) )
5	3	oveqrspc2v 5892	. . . . . 6 |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u ( +g ` K ) v ) = ( u ( +g ` L ) v ) )
6	3	oveqrspc2v 5892	. . . . . . 7 |- ( ( ph /\ ( v e. B /\ u e. B ) ) -> ( v ( +g ` K ) u ) = ( v ( +g ` L ) u ) )
7	6	ancom2s 566	. . . . . 6 |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( v ( +g ` K ) u ) = ( v ( +g ` L ) u ) )
8	5, 7	eqeq12d 2190	. . . . 5 |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( ( u ( +g ` K ) v ) = ( v ( +g ` K ) u ) <-> ( u ( +g ` L ) v ) = ( v ( +g ` L ) u ) ) )
9	8	2ralbidva 2497	. . . 4 |- ( ph -> ( A. u e. B A. v e. B ( u ( +g ` K ) v ) = ( v ( +g ` K ) u ) <-> A. u e. B A. v e. B ( u ( +g ` L ) v ) = ( v ( +g ` L ) u ) ) )
10	1	raleqdv 2676	. . . . 5 |- ( ph -> ( A. v e. B ( u ( +g ` K ) v ) = ( v ( +g ` K ) u ) <-> A. v e. ( Base ` K ) ( u ( +g ` K ) v ) = ( v ( +g ` K ) u ) ) )
11	1, 10	raleqbidv 2682	. . . 4 |- ( ph -> ( A. u e. B A. v e. B ( u ( +g ` K ) v ) = ( v ( +g ` K ) u ) <-> A. u e. ( Base ` K ) A. v e. ( Base ` K ) ( u ( +g ` K ) v ) = ( v ( +g ` K ) u ) ) )
12	2	raleqdv 2676	. . . . 5 |- ( ph -> ( A. v e. B ( u ( +g ` L ) v ) = ( v ( +g ` L ) u ) <-> A. v e. ( Base ` L ) ( u ( +g ` L ) v ) = ( v ( +g ` L ) u ) ) )
13	2, 12	raleqbidv 2682	. . . 4 |- ( ph -> ( A. u e. B A. v e. B ( u ( +g ` L ) v ) = ( v ( +g ` L ) u ) <-> A. u e. ( Base ` L ) A. v e. ( Base ` L ) ( u ( +g ` L ) v ) = ( v ( +g ` L ) u ) ) )
14	9, 11, 13	3bitr3d 218	. . 3 |- ( ph -> ( A. u e. ( Base ` K ) A. v e. ( Base ` K ) ( u ( +g ` K ) v ) = ( v ( +g ` K ) u ) <-> A. u e. ( Base ` L ) A. v e. ( Base ` L ) ( u ( +g ` L ) v ) = ( v ( +g ` L ) u ) ) )
15	4, 14	anbi12d 473	. 2 |- ( ph -> ( ( K e. Mnd /\ A. u e. ( Base ` K ) A. v e. ( Base ` K ) ( u ( +g ` K ) v ) = ( v ( +g ` K ) u ) ) <-> ( L e. Mnd /\ A. u e. ( Base ` L ) A. v e. ( Base ` L ) ( u ( +g ` L ) v ) = ( v ( +g ` L ) u ) ) ) )
16		eqid 2175	. . 3 |- ( Base ` K ) = ( Base ` K )
17		eqid 2175	. . 3 |- ( +g ` K ) = ( +g ` K )
18	16, 17	iscmn 12892	. 2 |- ( K e. CMnd <-> ( K e. Mnd /\ A. u e. ( Base ` K ) A. v e. ( Base ` K ) ( u ( +g ` K ) v ) = ( v ( +g ` K ) u ) ) )
19		eqid 2175	. . 3 |- ( Base ` L ) = ( Base ` L )
20		eqid 2175	. . 3 |- ( +g ` L ) = ( +g ` L )
21	19, 20	iscmn 12892	. 2 |- ( L e. CMnd <-> ( L e. Mnd /\ A. u e. ( Base ` L ) A. v e. ( Base ` L ) ( u ( +g ` L ) v ) = ( v ( +g ` L ) u ) ) )
22	15, 18, 21	3bitr4g 223	1 |- ( ph -> ( K e. CMnd <-> L e. CMnd ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2146   A.wral 2453   ` cfv 5208  (class class class)co 5865   Basecbs 12428   +g cplusg 12492   Mndcmnd 12682  CMndccmn 12884

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-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-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  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-ov 5868  df-inn 8891  df-2 8949  df-ndx 12431  df-slot 12432  df-base 12434  df-plusg 12505  df-mgm 12640  df-sgrp 12673  df-mnd 12683  df-cmn 12886

This theorem is referenced by:  ablpropd  12895  crngpropd  13010