Theorem issgrp 12881

Description: The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)

Hypotheses

Ref	Expression
issgrp.b	|- B = ( Base ` M )
issgrp.o	|- .o. = ( +g ` M )

Assertion

Ref	Expression
issgrp	|- ( M e. Smgrp <-> ( M e. Mgm /\ A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )

Distinct variable groups:   x, B, y, z   x, M, y, z   x, .o. , y, z

Proof of Theorem issgrp

Dummy variables b g o are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		basfn 12573	. . . . 5 |- Base Fn _V
2		vex 2755	. . . . 5 |- g e. _V
3		funfvex 5551	. . . . . 6 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
4	3	funfni 5335	. . . . 5 |- ( ( Base Fn _V /\ g e. _V ) -> ( Base ` g ) e. _V )
5	1, 2, 4	mp2an 426	. . . 4 |- ( Base ` g ) e. _V
6	5	a1i 9	. . 3 |- ( g = M -> ( Base ` g ) e. _V )
7		fveq2 5534	. . . 4 |- ( g = M -> ( Base ` g ) = ( Base ` M ) )
8		issgrp.b	. . . 4 |- B = ( Base ` M )
9	7, 8	eqtr4di 2240	. . 3 |- ( g = M -> ( Base ` g ) = B )
10		plusgslid 12627	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
11	10	slotex 12542	. . . . . 6 |- ( g e. _V -> ( +g ` g ) e. _V )
12	11	elv 2756	. . . . 5 |- ( +g ` g ) e. _V
13	12	a1i 9	. . . 4 |- ( ( g = M /\ b = B ) -> ( +g ` g ) e. _V )
14		fveq2 5534	. . . . . 6 |- ( g = M -> ( +g ` g ) = ( +g ` M ) )
15	14	adantr 276	. . . . 5 |- ( ( g = M /\ b = B ) -> ( +g ` g ) = ( +g ` M ) )
16		issgrp.o	. . . . 5 |- .o. = ( +g ` M )
17	15, 16	eqtr4di 2240	. . . 4 |- ( ( g = M /\ b = B ) -> ( +g ` g ) = .o. )
18		simplr 528	. . . . 5 |- ( ( ( g = M /\ b = B ) /\ o = .o. ) -> b = B )
19		id 19	. . . . . . . . . 10 |- ( o = .o. -> o = .o. )
20		oveq 5903	. . . . . . . . . 10 |- ( o = .o. -> ( x o y ) = ( x .o. y ) )
21		eqidd 2190	. . . . . . . . . 10 |- ( o = .o. -> z = z )
22	19, 20, 21	oveq123d 5918	. . . . . . . . 9 |- ( o = .o. -> ( ( x o y ) o z ) = ( ( x .o. y ) .o. z ) )
23		eqidd 2190	. . . . . . . . . 10 |- ( o = .o. -> x = x )
24		oveq 5903	. . . . . . . . . 10 |- ( o = .o. -> ( y o z ) = ( y .o. z ) )
25	19, 23, 24	oveq123d 5918	. . . . . . . . 9 |- ( o = .o. -> ( x o ( y o z ) ) = ( x .o. ( y .o. z ) ) )
26	22, 25	eqeq12d 2204	. . . . . . . 8 |- ( o = .o. -> ( ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
27	26	adantl 277	. . . . . . 7 |- ( ( ( g = M /\ b = B ) /\ o = .o. ) -> ( ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
28	18, 27	raleqbidv 2698	. . . . . 6 |- ( ( ( g = M /\ b = B ) /\ o = .o. ) -> ( A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
29	18, 28	raleqbidv 2698	. . . . 5 |- ( ( ( g = M /\ b = B ) /\ o = .o. ) -> ( A. y e. b A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
30	18, 29	raleqbidv 2698	. . . 4 |- ( ( ( g = M /\ b = B ) /\ o = .o. ) -> ( A. x e. b A. y e. b A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
31	13, 17, 30	sbcied2 3015	. . 3 |- ( ( g = M /\ b = B ) -> ( [. ( +g ` g ) / o ]. A. x e. b A. y e. b A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
32	6, 9, 31	sbcied2 3015	. 2 |- ( g = M -> ( [. ( Base ` g ) / b ]. [. ( +g ` g ) / o ]. A. x e. b A. y e. b A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
33		df-sgrp 12880	. 2 |- Smgrp = { g e. Mgm | [. ( Base ` g ) / b ]. [. ( +g ` g ) / o ]. A. x e. b A. y e. b A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) }
34	32, 33	elrab2 2911	1 |- ( M e. Smgrp <-> ( M e. Mgm /\ A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   A.wral 2468   _Vcvv 2752   [.wsbc 2977   Fn wfn 5230   ` cfv 5235  (class class class)co 5897   Basecbs 12515   +g cplusg 12592  Mgmcmgm 12833  Smgrpcsgrp 12879

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7933  ax-resscn 7934  ax-1re 7936  ax-addrcl 7939

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-ov 5900  df-inn 8951  df-2 9009  df-ndx 12518  df-slot 12519  df-base 12521  df-plusg 12605  df-sgrp 12880

This theorem is referenced by:  issgrpv  12882  issgrpn0  12883  isnsgrp  12884  sgrpmgm  12885  sgrpass  12886  sgrp0  12888  sgrp1  12889  rnglidlmsgrp  13830