Theorem isassad 14994

Description: Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by SN, 2-Mar-2025.)

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.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	1, 2	jca 306	. 2 |- ( ph -> ( W e. LMod /\ W e. Ring ) )
4		isassad.4	. . . . 5 |- ( ( ph /\ ( r e. B /\ x e. V /\ y e. V ) ) -> ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) )
5		isassad.5	. . . . 5 |- ( ( ph /\ ( r e. B /\ x e. V /\ y e. V ) ) -> ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) )
6	4, 5	jca 306	. . . 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 ) ) ) )
7	6	ralrimivvva 2633	. . 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 ) ) ) )
8		isassad.b	. . . . 5 |- ( ph -> B = ( Base ` F ) )
9		isassad.f	. . . . . 6 |- ( ph -> F = (Scalar ` W ) )
10	9	fveq2d 5697	. . . . 5 |- ( ph -> ( Base ` F ) = ( Base ` (Scalar ` W ) ) )
11	8, 10	eqtrd 2271	. . . 4 |- ( ph -> B = ( Base ` (Scalar ` W ) ) )
12		isassad.v	. . . . 5 |- ( ph -> V = ( Base ` W ) )
13		isassad.t	. . . . . . . . 9 |- ( ph -> .X. = ( .r ` W ) )
14		isassad.s	. . . . . . . . . 10 |- ( ph -> .x. = ( .s ` W ) )
15	14	oveqd 6096	. . . . . . . . 9 |- ( ph -> ( r .x. x ) = ( r ( .s ` W ) x ) )
16		eqidd 2239	. . . . . . . . 9 |- ( ph -> y = y )
17	13, 15, 16	oveq123d 6100	. . . . . . . 8 |- ( ph -> ( ( r .x. x ) .X. y ) = ( ( r ( .s ` W ) x ) ( .r ` W ) y ) )
18		eqidd 2239	. . . . . . . . 9 |- ( ph -> r = r )
19	13	oveqd 6096	. . . . . . . . 9 |- ( ph -> ( x .X. y ) = ( x ( .r ` W ) y ) )
20	14, 18, 19	oveq123d 6100	. . . . . . . 8 |- ( ph -> ( r .x. ( x .X. y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) )
21	17, 20	eqeq12d 2253	. . . . . . 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 ) ) ) )
22		eqidd 2239	. . . . . . . . 9 |- ( ph -> x = x )
23	14	oveqd 6096	. . . . . . . . 9 |- ( ph -> ( r .x. y ) = ( r ( .s ` W ) y ) )
24	13, 22, 23	oveq123d 6100	. . . . . . . 8 |- ( ph -> ( x .X. ( r .x. y ) ) = ( x ( .r ` W ) ( r ( .s ` W ) y ) ) )
25	24, 20	eqeq12d 2253	. . . . . . 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 ) ) ) )
26	21, 25	anbi12d 477	. . . . . 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 ) ) ) ) )
27	12, 26	raleqbidv 2765	. . . . 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 ) ) ) ) )
28	12, 27	raleqbidv 2765	. . . 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 ) ) ) ) )
29	11, 28	raleqbidv 2765	. . 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 ) ) ) ) )
30	7, 29	mpbid 147	. 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 ) ) ) )
31		eqid 2238	. . 3 |- ( Base ` W ) = ( Base ` W )
32		eqid 2238	. . 3 |- (Scalar ` W ) = (Scalar ` W )
33		eqid 2238	. . 3 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
34		eqid 2238	. . 3 |- ( .s ` W ) = ( .s ` W )
35		eqid 2238	. . 3 |- ( .r ` W ) = ( .r ` W )
36	31, 32, 33, 34, 35	isassa 14985	. 2 |- ( W e. AssAlg <-> ( ( W e. LMod /\ W e. Ring ) /\ 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 ) ) ) ) )
37	3, 30, 36	sylanbrc 421	1 |- ( ph -> W e. AssAlg )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5375  (class class class)co 6079   Basecbs 13335   .rcmulr 13415  Scalarcsca 13417   .scvsca 13418   Ringcrg 14283   LModclmod 14606  AssAlgcasa 14979

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 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-ov 6082  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-ndx 13338  df-slot 13339  df-mulr 13428  df-sca 13430  df-vsca 13431  df-assa 14982

This theorem is referenced by:  issubassa3  14995