Theorem cmnpropd 18202

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 17316	. . 3 |- ( ph -> ( K e. Mnd <-> L e. Mnd ) )
5	3	oveqrspc2v 6673	. . . . . 6 |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u ( +g ` K ) v ) = ( u ( +g ` L ) v ) )
6	3	oveqrspc2v 6673	. . . . . . 7 |- ( ( ph /\ ( v e. B /\ u e. B ) ) -> ( v ( +g ` K ) u ) = ( v ( +g ` L ) u ) )
7	6	ancom2s 844	. . . . . 6 |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( v ( +g ` K ) u ) = ( v ( +g ` L ) u ) )
8	5, 7	eqeq12d 2637	. . . . 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 2988	. . . 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 3144	. . . . 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 3152	. . . 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 3144	. . . . 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 3152	. . . 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 298	. . 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 747	. 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 2622	. . 3 |- ( Base ` K ) = ( Base ` K )
17		eqid 2622	. . 3 |- ( +g ` K ) = ( +g ` K )
18	16, 17	iscmn 18200	. 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 2622	. . 3 |- ( Base ` L ) = ( Base ` L )
20		eqid 2622	. . 3 |- ( +g ` L ) = ( +g ` L )
21	19, 20	iscmn 18200	. 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 303	1 |- ( ph -> ( K e. CMnd <-> L e. CMnd ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294  CMndccmn 18193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cmn 18195

This theorem is referenced by:  ablpropd  18203  crngpropd  18583  prdscrngd  18613  resvcmn  29838