Theorem ismnddef 13708

Description: The predicate "is a monoid", corresponding 1-to-1 to the definition. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 1-Feb-2020.)

Hypotheses

Ref	Expression
ismnddef.b	|- B = ( Base ` G )
ismnddef.p	|- .+ = ( +g ` G )

Assertion

Ref	Expression
ismnddef	|- ( G e. Mnd <-> ( G e. Smgrp /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )

Distinct variable groups:   B, a, e   .+ , a, e

Allowed substitution hints:   G( e, a)

Proof of Theorem ismnddef

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

Step	Hyp	Ref	Expression
1		basfn 13389	. . . 4 |- Base Fn _V
2		vex 2824	. . . 4 |- g e. _V
3		funfvex 5707	. . . . 5 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
4	3	funfni 5478	. . . 4 |- ( ( Base Fn _V /\ g e. _V ) -> ( Base ` g ) e. _V )
5	1, 2, 4	mp2an 430	. . 3 |- ( Base ` g ) e. _V
6		plusgslid 13443	. . . . 5 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
7	6	slotex 13357	. . . 4 |- ( g e. _V -> ( +g ` g ) e. _V )
8	7	elv 2825	. . 3 |- ( +g ` g ) e. _V
9		fveq2 5690	. . . . . . 7 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
10		ismnddef.b	. . . . . . 7 |- B = ( Base ` G )
11	9, 10	eqtr4di 2289	. . . . . 6 |- ( g = G -> ( Base ` g ) = B )
12	11	eqeq2d 2250	. . . . 5 |- ( g = G -> ( b = ( Base ` g ) <-> b = B ) )
13		fveq2 5690	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
14		ismnddef.p	. . . . . . 7 |- .+ = ( +g ` G )
15	13, 14	eqtr4di 2289	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
16	15	eqeq2d 2250	. . . . 5 |- ( g = G -> ( p = ( +g ` g ) <-> p = .+ ) )
17	12, 16	anbi12d 477	. . . 4 |- ( g = G -> ( ( b = ( Base ` g ) /\ p = ( +g ` g ) ) <-> ( b = B /\ p = .+ ) ) )
18		simpl 109	. . . . 5 |- ( ( b = B /\ p = .+ ) -> b = B )
19		oveq 6081	. . . . . . . . 9 |- ( p = .+ -> ( e p a ) = ( e .+ a ) )
20	19	eqeq1d 2247	. . . . . . . 8 |- ( p = .+ -> ( ( e p a ) = a <-> ( e .+ a ) = a ) )
21		oveq 6081	. . . . . . . . 9 |- ( p = .+ -> ( a p e ) = ( a .+ e ) )
22	21	eqeq1d 2247	. . . . . . . 8 |- ( p = .+ -> ( ( a p e ) = a <-> ( a .+ e ) = a ) )
23	20, 22	anbi12d 477	. . . . . . 7 |- ( p = .+ -> ( ( ( e p a ) = a /\ ( a p e ) = a ) <-> ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
24	23	adantl 277	. . . . . 6 |- ( ( b = B /\ p = .+ ) -> ( ( ( e p a ) = a /\ ( a p e ) = a ) <-> ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
25	18, 24	raleqbidv 2765	. . . . 5 |- ( ( b = B /\ p = .+ ) -> ( A. a e. b ( ( e p a ) = a /\ ( a p e ) = a ) <-> A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
26	18, 25	rexeqbidv 2766	. . . 4 |- ( ( b = B /\ p = .+ ) -> ( E. e e. b A. a e. b ( ( e p a ) = a /\ ( a p e ) = a ) <-> E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
27	17, 26	biimtrdi 163	. . 3 |- ( g = G -> ( ( b = ( Base ` g ) /\ p = ( +g ` g ) ) -> ( E. e e. b A. a e. b ( ( e p a ) = a /\ ( a p e ) = a ) <-> E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) ) )
28	5, 8, 27	sbc2iedv 3124	. 2 |- ( g = G -> ( [. ( Base ` g ) / b ]. [. ( +g ` g ) / p ]. E. e e. b A. a e. b ( ( e p a ) = a /\ ( a p e ) = a ) <-> E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
29		df-mnd 13707	. 2 |- Mnd = { g e. Smgrp | [. ( Base ` g ) / b ]. [. ( +g ` g ) / p ]. E. e e. b A. a e. b ( ( e p a ) = a /\ ( a p e ) = a ) }
30	28, 29	elrab2 2985	1 |- ( G e. Mnd <-> ( G e. Smgrp /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   _Vcvv 2821   [.wsbc 3051   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408  Smgrpcsgrp 13693   Mndcmnd 13706

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-mnd 13707

This theorem is referenced by:  ismnd  13709  sgrpidmndm  13710  mndsgrp  13711  mnd1  13739