Theorem cmnpropd 13872

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 13513	. . 3 |- ( ph -> ( K e. Mnd <-> L e. Mnd ) )
5	3	oveqrspc2v 6040	. . . . . 6 |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u ( +g ` K ) v ) = ( u ( +g ` L ) v ) )
6	3	oveqrspc2v 6040	. . . . . . 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 2244	. . . . 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 2552	. . . 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 2734	. . . . 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 2744	. . . 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 2734	. . . . 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 2744	. . . 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 2229	. . 3 |- ( Base ` K ) = ( Base ` K )
17		eqid 2229	. . 3 |- ( +g ` K ) = ( +g ` K )
18	16, 17	iscmn 13870	. 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 2229	. . 3 |- ( Base ` L ) = ( Base ` L )
20		eqid 2229	. . 3 |- ( +g ` L ) = ( +g ` L )
21	19, 20	iscmn 13870	. 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 1395   e. wcel 2200   A.wral 2508   ` cfv 5324  (class class class)co 6013   Basecbs 13072   +g cplusg 13150   Mndcmnd 13489  CMndccmn 13861

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-ov 6016  df-inn 9134  df-2 9192  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-cmn 13863

This theorem is referenced by:  ablpropd  13873  crngpropd  14042