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Theorem asclpropd 18205
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.
StepHypRef Expression
1 asclpropd.5 . . . . . . 7  |-  ( ph  ->  ( 1r `  K
)  e.  W )
21adantr 463 . . . . . 6  |-  ( (
ph  /\  z  e.  P )  ->  ( 1r `  K )  e.  W )
3 asclpropd.3 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  W ) )  -> 
( x ( .s
`  K ) y )  =  ( x ( .s `  L
) y ) )
43oveqrspc2v 6255 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  P  /\  ( 1r `  K )  e.  W ) )  -> 
( z ( .s
`  K ) ( 1r `  K ) )  =  ( z ( .s `  L
) ( 1r `  K ) ) )
54anassrs 646 . . . . . 6  |-  ( ( ( ph  /\  z  e.  P )  /\  ( 1r `  K )  e.  W )  ->  (
z ( .s `  K ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  K
) ) )
62, 5mpdan 666 . . . . 5  |-  ( (
ph  /\  z  e.  P )  ->  (
z ( .s `  K ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  K
) ) )
7 asclpropd.4 . . . . . . 7  |-  ( ph  ->  ( 1r `  K
)  =  ( 1r
`  L ) )
87oveq2d 6248 . . . . . 6  |-  ( ph  ->  ( z ( .s
`  L ) ( 1r `  K ) )  =  ( z ( .s `  L
) ( 1r `  L ) ) )
98adantr 463 . . . . 5  |-  ( (
ph  /\  z  e.  P )  ->  (
z ( .s `  L ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  L
) ) )
106, 9eqtrd 2441 . . . 4  |-  ( (
ph  /\  z  e.  P )  ->  (
z ( .s `  K ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  L
) ) )
1110mpteq2dva 4478 . . 3  |-  ( ph  ->  ( z  e.  P  |->  ( z ( .s
`  K ) ( 1r `  K ) ) )  =  ( z  e.  P  |->  ( z ( .s `  L ) ( 1r
`  L ) ) ) )
12 asclpropd.1 . . . 4  |-  ( ph  ->  P  =  ( Base `  F ) )
1312mpteq1d 4473 . . 3  |-  ( ph  ->  ( z  e.  P  |->  ( z ( .s
`  K ) ( 1r `  K ) ) )  =  ( z  e.  ( Base `  F )  |->  ( z ( .s `  K
) ( 1r `  K ) ) ) )
14 asclpropd.2 . . . 4  |-  ( ph  ->  P  =  ( Base `  G ) )
1514mpteq1d 4473 . . 3  |-  ( ph  ->  ( z  e.  P  |->  ( z ( .s
`  L ) ( 1r `  L ) ) )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) ) )
1611, 13, 153eqtr3d 2449 . 2  |-  ( ph  ->  ( z  e.  (
Base `  F )  |->  ( z ( .s
`  K ) ( 1r `  K ) ) )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) ) )
17 eqid 2400 . . 3  |-  (algSc `  K )  =  (algSc `  K )
18 asclpropd.f . . 3  |-  F  =  (Scalar `  K )
19 eqid 2400 . . 3  |-  ( Base `  F )  =  (
Base `  F )
20 eqid 2400 . . 3  |-  ( .s
`  K )  =  ( .s `  K
)
21 eqid 2400 . . 3  |-  ( 1r
`  K )  =  ( 1r `  K
)
2217, 18, 19, 20, 21asclfval 18193 . 2  |-  (algSc `  K )  =  ( z  e.  ( Base `  F )  |->  ( z ( .s `  K
) ( 1r `  K ) ) )
23 eqid 2400 . . 3  |-  (algSc `  L )  =  (algSc `  L )
24 asclpropd.g . . 3  |-  G  =  (Scalar `  L )
25 eqid 2400 . . 3  |-  ( Base `  G )  =  (
Base `  G )
26 eqid 2400 . . 3  |-  ( .s
`  L )  =  ( .s `  L
)
27 eqid 2400 . . 3  |-  ( 1r
`  L )  =  ( 1r `  L
)
2823, 24, 25, 26, 27asclfval 18193 . 2  |-  (algSc `  L )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) )
2916, 22, 283eqtr4g 2466 1  |-  ( ph  ->  (algSc `  K )  =  (algSc `  L )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840    |-> cmpt 4450   ` cfv 5523  (class class class)co 6232   Basecbs 14731  Scalarcsca 14802   .scvsca 14803   1rcur 17363  algSccascl 18170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-slot 14735  df-base 14736  df-ascl 18173
This theorem is referenced by:  ply1ascl  18509
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