Theorem ismnddef 13365

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 13005	. . . 4 |- Base Fn _V
2		vex 2779	. . . 4 |- g e. _V
3		funfvex 5616	. . . . 5 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
4	3	funfni 5395	. . . 4 |- ( ( Base Fn _V /\ g e. _V ) -> ( Base ` g ) e. _V )
5	1, 2, 4	mp2an 426	. . 3 |- ( Base ` g ) e. _V
6		plusgslid 13059	. . . . 5 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
7	6	slotex 12974	. . . 4 |- ( g e. _V -> ( +g ` g ) e. _V )
8	7	elv 2780	. . 3 |- ( +g ` g ) e. _V
9		fveq2 5599	. . . . . . 7 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
10		ismnddef.b	. . . . . . 7 |- B = ( Base ` G )
11	9, 10	eqtr4di 2258	. . . . . 6 |- ( g = G -> ( Base ` g ) = B )
12	11	eqeq2d 2219	. . . . 5 |- ( g = G -> ( b = ( Base ` g ) <-> b = B ) )
13		fveq2 5599	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
14		ismnddef.p	. . . . . . 7 |- .+ = ( +g ` G )
15	13, 14	eqtr4di 2258	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
16	15	eqeq2d 2219	. . . . 5 |- ( g = G -> ( p = ( +g ` g ) <-> p = .+ ) )
17	12, 16	anbi12d 473	. . . 4 |- ( g = G -> ( ( b = ( Base ` g ) /\ p = ( +g ` g ) ) <-> ( b = B /\ p = .+ ) ) )
18		simpl 109	. . . . 5 |- ( ( b = B /\ p = .+ ) -> b = B )
19		oveq 5973	. . . . . . . . 9 |- ( p = .+ -> ( e p a ) = ( e .+ a ) )
20	19	eqeq1d 2216	. . . . . . . 8 |- ( p = .+ -> ( ( e p a ) = a <-> ( e .+ a ) = a ) )
21		oveq 5973	. . . . . . . . 9 |- ( p = .+ -> ( a p e ) = ( a .+ e ) )
22	21	eqeq1d 2216	. . . . . . . 8 |- ( p = .+ -> ( ( a p e ) = a <-> ( a .+ e ) = a ) )
23	20, 22	anbi12d 473	. . . . . . 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 2721	. . . . 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 2722	. . . 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 3078	. 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 13364	. 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 2939	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 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   _Vcvv 2776   [.wsbc 3005   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024  Smgrpcsgrp 13348   Mndcmnd 13363

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ov 5970  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-mnd 13364

This theorem is referenced by:  ismnd  13366  sgrpidmndm  13367  mndsgrp  13368  mnd1  13402