Theorem asclpropd 15040

Description: If two structures have the same components (properties), one is an associative algebra iff the other one is. The last hypotheses on 1r can be discharged either by letting W = _V (if strong equality is known on .s) or assuming K is a ring. (Contributed by Mario Carneiro, 5-Jul-2015.)

Hypotheses

Ref	Expression
asclpropd.f	|- F = (Scalar ` K )
asclpropd.g	|- G = (Scalar ` L )
asclpropd.1	|- ( ph -> P = ( Base ` F ) )
asclpropd.2	|- ( ph -> P = ( Base ` G ) )
asclpropd.3	|- ( ( ph /\ ( x e. P /\ y e. W ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )
asclpropd.4	|- ( ph -> ( 1r ` K ) = ( 1r ` L ) )
asclpropd.5	|- ( ph -> ( 1r ` K ) e. W )

Assertion

Ref	Expression
asclpropd	|- ( ph -> (algSc ` K ) = (algSc ` L ) )

Distinct variable groups:   x, y, K   x, L, y   x, P, y   ph, x, y   x, W, y

Allowed substitution hints:   F( x, y)   G( x, y)

Proof of Theorem asclpropd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		asclpropd.5	. . . . . 6 |- ( ph -> ( 1r ` K ) e. W )
2		asclpropd.3	. . . . . . . 8 |- ( ( ph /\ ( x e. P /\ y e. W ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )
3	2	oveqrspc2v 6112	. . . . . . 7 |- ( ( ph /\ ( z e. P /\ ( 1r ` K ) e. W ) ) -> ( z ( .s ` K ) ( 1r ` K ) ) = ( z ( .s ` L ) ( 1r ` K ) ) )
4	3	anassrs 404	. . . . . 6 |- ( ( ( ph /\ z e. P ) /\ ( 1r ` K ) e. W ) -> ( z ( .s ` K ) ( 1r ` K ) ) = ( z ( .s ` L ) ( 1r ` K ) ) )
5	1, 4	mpidan 427	. . . . 5 |- ( ( ph /\ z e. P ) -> ( z ( .s ` K ) ( 1r ` K ) ) = ( z ( .s ` L ) ( 1r ` K ) ) )
6		asclpropd.4	. . . . . . 7 |- ( ph -> ( 1r ` K ) = ( 1r ` L ) )
7	6	oveq2d 6101	. . . . . 6 |- ( ph -> ( z ( .s ` L ) ( 1r ` K ) ) = ( z ( .s ` L ) ( 1r ` L ) ) )
8	7	adantr 276	. . . . 5 |- ( ( ph /\ z e. P ) -> ( z ( .s ` L ) ( 1r ` K ) ) = ( z ( .s ` L ) ( 1r ` L ) ) )
9	5, 8	eqtrd 2271	. . . 4 |- ( ( ph /\ z e. P ) -> ( z ( .s ` K ) ( 1r ` K ) ) = ( z ( .s ` L ) ( 1r ` L ) ) )
10	9	mpteq2dva 4221	. . 3 |- ( ph -> ( z e. P |-> ( z ( .s ` K ) ( 1r ` K ) ) ) = ( z e. P |-> ( z ( .s ` L ) ( 1r ` L ) ) ) )
11		asclpropd.1	. . . 4 |- ( ph -> P = ( Base ` F ) )
12	11	mpteq1d 4216	. . 3 |- ( ph -> ( z e. P |-> ( z ( .s ` K ) ( 1r ` K ) ) ) = ( z e. ( Base ` F ) |-> ( z ( .s ` K ) ( 1r ` K ) ) ) )
13		asclpropd.2	. . . 4 |- ( ph -> P = ( Base ` G ) )
14	13	mpteq1d 4216	. . 3 |- ( ph -> ( z e. P |-> ( z ( .s ` L ) ( 1r ` L ) ) ) = ( z e. ( Base ` G ) |-> ( z ( .s ` L ) ( 1r ` L ) ) ) )
15	10, 12, 14	3eqtr3d 2279	. 2 |- ( ph -> ( z e. ( Base ` F ) |-> ( z ( .s ` K ) ( 1r ` K ) ) ) = ( z e. ( Base ` G ) |-> ( z ( .s ` L ) ( 1r ` L ) ) ) )
16		eqid 2238	. . 3 |- (algSc ` K ) = (algSc ` K )
17		asclpropd.f	. . 3 |- F = (Scalar ` K )
18		eqid 2238	. . 3 |- ( Base ` F ) = ( Base ` F )
19		eqid 2238	. . 3 |- ( .s ` K ) = ( .s ` K )
20		eqid 2238	. . 3 |- ( 1r ` K ) = ( 1r ` K )
21	16, 17, 18, 19, 20	asclfval 15021	. 2 |- (algSc ` K ) = ( z e. ( Base ` F ) |-> ( z ( .s ` K ) ( 1r ` K ) ) )
22		eqid 2238	. . 3 |- (algSc ` L ) = (algSc ` L )
23		asclpropd.g	. . 3 |- G = (Scalar ` L )
24		eqid 2238	. . 3 |- ( Base ` G ) = ( Base ` G )
25		eqid 2238	. . 3 |- ( .s ` L ) = ( .s ` L )
26		eqid 2238	. . 3 |- ( 1r ` L ) = ( 1r ` L )
27	22, 23, 24, 25, 26	asclfval 15021	. 2 |- (algSc ` L ) = ( z e. ( Base ` G ) |-> ( z ( .s ` L ) ( 1r ` L ) ) )
28	15, 21, 27	3eqtr4g 2296	1 |- ( ph -> (algSc ` K ) = (algSc ` L ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   Basecbs 13352  Scalarcsca 13434   .scvsca 13435   1rcur 14262  algSccascl 14998

This proof depends on 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-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof 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-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-ndx 13355  df-slot 13356  df-base 13358  df-sca 13447  df-ascl 15001

This theorem is used by: (None)