Theorem asclpropd 19346

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 6673	. . . . . . 7 |- ( ( ph /\ ( z e. P /\ ( 1r ` K ) e. W ) ) -> ( z ( .s ` K ) ( 1r ` K ) ) = ( z ( .s ` L ) ( 1r ` K ) ) )
4	3	anassrs 680	. . . . . 6 |- ( ( ( ph /\ z e. P ) /\ ( 1r ` K ) e. W ) -> ( z ( .s ` K ) ( 1r ` K ) ) = ( z ( .s ` L ) ( 1r ` K ) ) )
5	1, 4	mpidan 704	. . . . 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 6666	. . . . . 6 |- ( ph -> ( z ( .s ` L ) ( 1r ` K ) ) = ( z ( .s ` L ) ( 1r ` L ) ) )
8	7	adantr 481	. . . . 5 |- ( ( ph /\ z e. P ) -> ( z ( .s ` L ) ( 1r ` K ) ) = ( z ( .s ` L ) ( 1r ` L ) ) )
9	5, 8	eqtrd 2656	. . . 4 |- ( ( ph /\ z e. P ) -> ( z ( .s ` K ) ( 1r ` K ) ) = ( z ( .s ` L ) ( 1r ` L ) ) )
10	9	mpteq2dva 4744	. . 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 4738	. . 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 4738	. . 3 |- ( ph -> ( z e. P |-> ( z ( .s ` L ) ( 1r ` L ) ) ) = ( z e. ( Base ` G ) |-> ( z ( .s ` L ) ( 1r ` L ) ) ) )
15	10, 12, 14	3eqtr3d 2664	. 2 |- ( ph -> ( z e. ( Base ` F ) |-> ( z ( .s ` K ) ( 1r ` K ) ) ) = ( z e. ( Base ` G ) |-> ( z ( .s ` L ) ( 1r ` L ) ) ) )
16		eqid 2622	. . 3 |- (algSc ` K ) = (algSc ` K )
17		asclpropd.f	. . 3 |- F = (Scalar ` K )
18		eqid 2622	. . 3 |- ( Base ` F ) = ( Base ` F )
19		eqid 2622	. . 3 |- ( .s ` K ) = ( .s ` K )
20		eqid 2622	. . 3 |- ( 1r ` K ) = ( 1r ` K )
21	16, 17, 18, 19, 20	asclfval 19334	. 2 |- (algSc ` K ) = ( z e. ( Base ` F ) |-> ( z ( .s ` K ) ( 1r ` K ) ) )
22		eqid 2622	. . 3 |- (algSc ` L ) = (algSc ` L )
23		asclpropd.g	. . 3 |- G = (Scalar ` L )
24		eqid 2622	. . 3 |- ( Base ` G ) = ( Base ` G )
25		eqid 2622	. . 3 |- ( .s ` L ) = ( .s ` L )
26		eqid 2622	. . 3 |- ( 1r ` L ) = ( 1r ` L )
27	22, 23, 24, 25, 26	asclfval 19334	. 2 |- (algSc ` L ) = ( z e. ( Base ` G ) |-> ( z ( .s ` L ) ( 1r ` L ) ) )
28	15, 21, 27	3eqtr4g 2681	1 |- ( ph -> (algSc ` K ) = (algSc ` L ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   1rcur 18501  algSccascl 19311

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-slot 15861  df-base 15863  df-ascl 19314

This theorem is referenced by:  ply1ascl  19628