Theorem cmnpropd 13501

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 13142	. . 3 |- ( ph -> ( K e. Mnd <-> L e. Mnd ) )
5	3	oveqrspc2v 5952	. . . . . 6 |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u ( +g ` K ) v ) = ( u ( +g ` L ) v ) )
6	3	oveqrspc2v 5952	. . . . . . 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 2211	. . . . 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 2519	. . . 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 2699	. . . . 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 2709	. . . 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 2699	. . . . 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 2709	. . . 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 2196	. . 3 |- ( Base ` K ) = ( Base ` K )
17		eqid 2196	. . 3 |- ( +g ` K ) = ( +g ` K )
18	16, 17	iscmn 13499	. 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 2196	. . 3 |- ( Base ` L ) = ( Base ` L )
20		eqid 2196	. . 3 |- ( +g ` L ) = ( +g ` L )
21	19, 20	iscmn 13499	. 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 1364   e. wcel 2167   A.wral 2475   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   Mndcmnd 13118  CMndccmn 13490

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5928  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-cmn 13492

This theorem is referenced by:  ablpropd  13502  crngpropd  13671