Theorem issgrp 13695

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 13389	. . . . 5 |- Base Fn _V
2		vex 2824	. . . . 5 |- g e. _V
3		funfvex 5707	. . . . . 6 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
4	3	funfni 5478	. . . . 5 |- ( ( Base Fn _V /\ g e. _V ) -> ( Base ` g ) e. _V )
5	1, 2, 4	mp2an 430	. . . 4 |- ( Base ` g ) e. _V
6	5	a1i 9	. . 3 |- ( g = M -> ( Base ` g ) e. _V )
7		fveq2 5690	. . . 4 |- ( g = M -> ( Base ` g ) = ( Base ` M ) )
8		issgrp.b	. . . 4 |- B = ( Base ` M )
9	7, 8	eqtr4di 2289	. . 3 |- ( g = M -> ( Base ` g ) = B )
10		plusgslid 13443	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
11	10	slotex 13357	. . . . . 6 |- ( g e. _V -> ( +g ` g ) e. _V )
12	11	elv 2825	. . . . 5 |- ( +g ` g ) e. _V
13	12	a1i 9	. . . 4 |- ( ( g = M /\ b = B ) -> ( +g ` g ) e. _V )
14		fveq2 5690	. . . . . 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 2289	. . . 4 |- ( ( g = M /\ b = B ) -> ( +g ` g ) = .o. )
18		simplr 533	. . . . 5 |- ( ( ( g = M /\ b = B ) /\ o = .o. ) -> b = B )
19		id 19	. . . . . . . . . 10 |- ( o = .o. -> o = .o. )
20		oveq 6081	. . . . . . . . . 10 |- ( o = .o. -> ( x o y ) = ( x .o. y ) )
21		eqidd 2239	. . . . . . . . . 10 |- ( o = .o. -> z = z )
22	19, 20, 21	oveq123d 6096	. . . . . . . . 9 |- ( o = .o. -> ( ( x o y ) o z ) = ( ( x .o. y ) .o. z ) )
23		eqidd 2239	. . . . . . . . . 10 |- ( o = .o. -> x = x )
24		oveq 6081	. . . . . . . . . 10 |- ( o = .o. -> ( y o z ) = ( y .o. z ) )
25	19, 23, 24	oveq123d 6096	. . . . . . . . 9 |- ( o = .o. -> ( x o ( y o z ) ) = ( x .o. ( y .o. z ) ) )
26	22, 25	eqeq12d 2253	. . . . . . . 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 2765	. . . . . 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 2765	. . . . 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 2765	. . . 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 3089	. . 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 3089	. 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 13694	. 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 2985	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 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   [.wsbc 3051   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408  Mgmcmgm 13651  Smgrpcsgrp 13693

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-sgrp 13694

This theorem is referenced by:  issgrpv  13696  issgrpn0  13697  isnsgrp  13698  sgrpmgm  13699  sgrpass  13700  sgrp0  13702  sgrp1  13703  rnglidlmsgrp  14806