Theorem issgrp 13431

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 13086	. . . . 5 |- Base Fn _V
2		vex 2802	. . . . 5 |- g e. _V
3		funfvex 5643	. . . . . 6 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
4	3	funfni 5422	. . . . 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 5626	. . . 4 |- ( g = M -> ( Base ` g ) = ( Base ` M ) )
8		issgrp.b	. . . 4 |- B = ( Base ` M )
9	7, 8	eqtr4di 2280	. . 3 |- ( g = M -> ( Base ` g ) = B )
10		plusgslid 13140	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
11	10	slotex 13054	. . . . . 6 |- ( g e. _V -> ( +g ` g ) e. _V )
12	11	elv 2803	. . . . 5 |- ( +g ` g ) e. _V
13	12	a1i 9	. . . 4 |- ( ( g = M /\ b = B ) -> ( +g ` g ) e. _V )
14		fveq2 5626	. . . . . 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 2280	. . . 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 6006	. . . . . . . . . 10 |- ( o = .o. -> ( x o y ) = ( x .o. y ) )
21		eqidd 2230	. . . . . . . . . 10 |- ( o = .o. -> z = z )
22	19, 20, 21	oveq123d 6021	. . . . . . . . 9 |- ( o = .o. -> ( ( x o y ) o z ) = ( ( x .o. y ) .o. z ) )
23		eqidd 2230	. . . . . . . . . 10 |- ( o = .o. -> x = x )
24		oveq 6006	. . . . . . . . . 10 |- ( o = .o. -> ( y o z ) = ( y .o. z ) )
25	19, 23, 24	oveq123d 6021	. . . . . . . . 9 |- ( o = .o. -> ( x o ( y o z ) ) = ( x .o. ( y .o. z ) ) )
26	22, 25	eqeq12d 2244	. . . . . . . 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 2744	. . . . . 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 2744	. . . . 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 2744	. . . 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 3066	. . 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 3066	. 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 13430	. 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 2962	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 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   [.wsbc 3028   Fn wfn 5312   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105  Mgmcmgm 13382  Smgrpcsgrp 13429

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-ov 6003  df-inn 9107  df-2 9165  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-sgrp 13430

This theorem is referenced by:  issgrpv  13432  issgrpn0  13433  isnsgrp  13434  sgrpmgm  13435  sgrpass  13436  sgrp0  13438  sgrp1  13439  rnglidlmsgrp  14455