Theorem cmnpropd 13881

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 13522	. . 3 |- ( ph -> ( K e. Mnd <-> L e. Mnd ) )
5	3	oveqrspc2v 6044	. . . . . 6 |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u ( +g ` K ) v ) = ( u ( +g ` L ) v ) )
6	3	oveqrspc2v 6044	. . . . . . 7 |- ( ( ph /\ ( v e. B /\ u e. B ) ) -> ( v ( +g ` K ) u ) = ( v ( +g ` L ) u ) )
7	6	ancom2s 568	. . . . . 6 |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( v ( +g ` K ) u ) = ( v ( +g ` L ) u ) )
8	5, 7	eqeq12d 2246	. . . . 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 2554	. . . 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 2736	. . . . 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 2746	. . . 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 2736	. . . . 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 2746	. . . 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 2231	. . 3 |- ( Base ` K ) = ( Base ` K )
17		eqid 2231	. . 3 |- ( +g ` K ) = ( +g ` K )
18	16, 17	iscmn 13879	. 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 2231	. . 3 |- ( Base ` L ) = ( Base ` L )
20		eqid 2231	. . 3 |- ( +g ` L ) = ( +g ` L )
21	19, 20	iscmn 13879	. 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 1397   e. wcel 2202   A.wral 2510   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   Mndcmnd 13498  CMndccmn 13870

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-cmn 13872

This theorem is referenced by:  ablpropd  13882  crngpropd  14051