Theorem isassad 19323

Description: Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.)

Hypotheses

Ref	Expression
isassad.v	|- ( ph -> V = ( Base ` W ) )
isassad.f	|- ( ph -> F = (Scalar ` W ) )
isassad.b	|- ( ph -> B = ( Base ` F ) )
isassad.s	|- ( ph -> .x. = ( .s ` W ) )
isassad.t	|- ( ph -> .X. = ( .r ` W ) )
isassad.1	|- ( ph -> W e. LMod )
isassad.2	|- ( ph -> W e. Ring )
isassad.3	|- ( ph -> F e. CRing )
isassad.4	|- ( ( ph /\ ( r e. B /\ x e. V /\ y e. V ) ) -> ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) )
isassad.5	|- ( ( ph /\ ( r e. B /\ x e. V /\ y e. V ) ) -> ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) )

Assertion

Ref	Expression
isassad	|- ( ph -> W e. AssAlg )

Distinct variable groups:   x, r, y, B   ph, r, x, y   x, V, y   W, r, x, y

Allowed substitution hints:   .x. ( x, y, r)   .X. ( x, y, r)   F( x, y, r)   V( r)

Proof of Theorem isassad

Step	Hyp	Ref	Expression
1		isassad.1	. . 3 |- ( ph -> W e. LMod )
2		isassad.2	. . 3 |- ( ph -> W e. Ring )
3		isassad.f	. . . 4 |- ( ph -> F = (Scalar ` W ) )
4		isassad.3	. . . 4 |- ( ph -> F e. CRing )
5	3, 4	eqeltrrd 2702	. . 3 |- ( ph -> (Scalar ` W ) e. CRing )
6	1, 2, 5	3jca 1242	. 2 |- ( ph -> ( W e. LMod /\ W e. Ring /\ (Scalar ` W ) e. CRing ) )
7		isassad.4	. . . . 5 |- ( ( ph /\ ( r e. B /\ x e. V /\ y e. V ) ) -> ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) )
8		isassad.5	. . . . 5 |- ( ( ph /\ ( r e. B /\ x e. V /\ y e. V ) ) -> ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) )
9	7, 8	jca 554	. . . 4 |- ( ( ph /\ ( r e. B /\ x e. V /\ y e. V ) ) -> ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) /\ ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) )
10	9	ralrimivvva 2972	. . 3 |- ( ph -> A. r e. B A. x e. V A. y e. V ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) /\ ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) )
11		isassad.b	. . . . 5 |- ( ph -> B = ( Base ` F ) )
12	3	fveq2d 6195	. . . . 5 |- ( ph -> ( Base ` F ) = ( Base ` (Scalar ` W ) ) )
13	11, 12	eqtrd 2656	. . . 4 |- ( ph -> B = ( Base ` (Scalar ` W ) ) )
14		isassad.v	. . . . 5 |- ( ph -> V = ( Base ` W ) )
15		isassad.t	. . . . . . . . 9 |- ( ph -> .X. = ( .r ` W ) )
16		isassad.s	. . . . . . . . . 10 |- ( ph -> .x. = ( .s ` W ) )
17	16	oveqd 6667	. . . . . . . . 9 |- ( ph -> ( r .x. x ) = ( r ( .s ` W ) x ) )
18		eqidd 2623	. . . . . . . . 9 |- ( ph -> y = y )
19	15, 17, 18	oveq123d 6671	. . . . . . . 8 |- ( ph -> ( ( r .x. x ) .X. y ) = ( ( r ( .s ` W ) x ) ( .r ` W ) y ) )
20		eqidd 2623	. . . . . . . . 9 |- ( ph -> r = r )
21	15	oveqd 6667	. . . . . . . . 9 |- ( ph -> ( x .X. y ) = ( x ( .r ` W ) y ) )
22	16, 20, 21	oveq123d 6671	. . . . . . . 8 |- ( ph -> ( r .x. ( x .X. y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) )
23	19, 22	eqeq12d 2637	. . . . . . 7 |- ( ph -> ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) <-> ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) )
24		eqidd 2623	. . . . . . . . 9 |- ( ph -> x = x )
25	16	oveqd 6667	. . . . . . . . 9 |- ( ph -> ( r .x. y ) = ( r ( .s ` W ) y ) )
26	15, 24, 25	oveq123d 6671	. . . . . . . 8 |- ( ph -> ( x .X. ( r .x. y ) ) = ( x ( .r ` W ) ( r ( .s ` W ) y ) ) )
27	26, 22	eqeq12d 2637	. . . . . . 7 |- ( ph -> ( ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) <-> ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) )
28	23, 27	anbi12d 747	. . . . . 6 |- ( ph -> ( ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) /\ ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) <-> ( ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) /\ ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) ) )
29	14, 28	raleqbidv 3152	. . . . 5 |- ( ph -> ( A. y e. V ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) /\ ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) <-> A. y e. ( Base ` W ) ( ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) /\ ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) ) )
30	14, 29	raleqbidv 3152	. . . 4 |- ( ph -> ( A. x e. V A. y e. V ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) /\ ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) <-> A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) /\ ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) ) )
31	13, 30	raleqbidv 3152	. . 3 |- ( ph -> ( A. r e. B A. x e. V A. y e. V ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) /\ ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) <-> A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) /\ ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) ) )
32	10, 31	mpbid 222	. 2 |- ( ph -> A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) /\ ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) )
33		eqid 2622	. . 3 |- ( Base ` W ) = ( Base ` W )
34		eqid 2622	. . 3 |- (Scalar ` W ) = (Scalar ` W )
35		eqid 2622	. . 3 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
36		eqid 2622	. . 3 |- ( .s ` W ) = ( .s ` W )
37		eqid 2622	. . 3 |- ( .r ` W ) = ( .r ` W )
38	33, 34, 35, 36, 37	isassa 19315	. 2 |- ( W e. AssAlg <-> ( ( W e. LMod /\ W e. Ring /\ (Scalar ` W ) e. CRing ) /\ A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) /\ ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) ) )
39	6, 32, 38	sylanbrc 698	1 |- ( ph -> W e. AssAlg )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   Ringcrg 18547   CRingccrg 18548   LModclmod 18863  AssAlgcasa 19309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-assa 19312

This theorem is referenced by:  issubassa  19324  sraassa  19325  psrassa  19414  zlmassa  19872  matassa  20250  mendassa  37764