Theorem issgrp 12989

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 12679	. . . . 5 |- Base Fn _V
2		vex 2763	. . . . 5 |- g e. _V
3		funfvex 5572	. . . . . 6 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
4	3	funfni 5355	. . . . 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 5555	. . . 4 |- ( g = M -> ( Base ` g ) = ( Base ` M ) )
8		issgrp.b	. . . 4 |- B = ( Base ` M )
9	7, 8	eqtr4di 2244	. . 3 |- ( g = M -> ( Base ` g ) = B )
10		plusgslid 12733	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
11	10	slotex 12648	. . . . . 6 |- ( g e. _V -> ( +g ` g ) e. _V )
12	11	elv 2764	. . . . 5 |- ( +g ` g ) e. _V
13	12	a1i 9	. . . 4 |- ( ( g = M /\ b = B ) -> ( +g ` g ) e. _V )
14		fveq2 5555	. . . . . 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 2244	. . . 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 5925	. . . . . . . . . 10 |- ( o = .o. -> ( x o y ) = ( x .o. y ) )
21		eqidd 2194	. . . . . . . . . 10 |- ( o = .o. -> z = z )
22	19, 20, 21	oveq123d 5940	. . . . . . . . 9 |- ( o = .o. -> ( ( x o y ) o z ) = ( ( x .o. y ) .o. z ) )
23		eqidd 2194	. . . . . . . . . 10 |- ( o = .o. -> x = x )
24		oveq 5925	. . . . . . . . . 10 |- ( o = .o. -> ( y o z ) = ( y .o. z ) )
25	19, 23, 24	oveq123d 5940	. . . . . . . . 9 |- ( o = .o. -> ( x o ( y o z ) ) = ( x .o. ( y .o. z ) ) )
26	22, 25	eqeq12d 2208	. . . . . . . 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 2706	. . . . . 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 2706	. . . . 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 2706	. . . 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 3024	. . 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 3024	. 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 12988	. 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 2920	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 2164   A.wral 2472   _Vcvv 2760   [.wsbc 2986   Fn wfn 5250   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698  Mgmcmgm 12940  Smgrpcsgrp 12987

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-sgrp 12988

This theorem is referenced by:  issgrpv  12990  issgrpn0  12991  isnsgrp  12992  sgrpmgm  12993  sgrpass  12994  sgrp0  12996  sgrp1  12997  rnglidlmsgrp  13996