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Theorem asclpropd 16396
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 452 . . . . . 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 ) )
43proplem 13907 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  P  /\  ( 1r `  K )  e.  W ) )  -> 
( z ( .s
`  K ) ( 1r `  K ) )  =  ( z ( .s `  L
) ( 1r `  K ) ) )
54anassrs 630 . . . . . 6  |-  ( ( ( ph  /\  z  e.  P )  /\  ( 1r `  K )  e.  W )  ->  (
z ( .s `  K ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  K
) ) )
62, 5mpdan 650 . . . . 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 6089 . . . . . 6  |-  ( ph  ->  ( z ( .s
`  L ) ( 1r `  K ) )  =  ( z ( .s `  L
) ( 1r `  L ) ) )
98adantr 452 . . . . 5  |-  ( (
ph  /\  z  e.  P )  ->  (
z ( .s `  L ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  L
) ) )
106, 9eqtrd 2467 . . . 4  |-  ( (
ph  /\  z  e.  P )  ->  (
z ( .s `  K ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  L
) ) )
1110mpteq2dva 4287 . . 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 4282 . . 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 4282 . . 3  |-  ( ph  ->  ( z  e.  P  |->  ( z ( .s
`  L ) ( 1r `  L ) ) )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) ) )
1611, 13, 153eqtr3d 2475 . 2  |-  ( ph  ->  ( z  e.  (
Base `  F )  |->  ( z ( .s
`  K ) ( 1r `  K ) ) )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) ) )
17 eqid 2435 . . 3  |-  (algSc `  K )  =  (algSc `  K )
18 asclpropd.f . . 3  |-  F  =  (Scalar `  K )
19 eqid 2435 . . 3  |-  ( Base `  F )  =  (
Base `  F )
20 eqid 2435 . . 3  |-  ( .s
`  K )  =  ( .s `  K
)
21 eqid 2435 . . 3  |-  ( 1r
`  K )  =  ( 1r `  K
)
2217, 18, 19, 20, 21asclfval 16385 . 2  |-  (algSc `  K )  =  ( z  e.  ( Base `  F )  |->  ( z ( .s `  K
) ( 1r `  K ) ) )
23 eqid 2435 . . 3  |-  (algSc `  L )  =  (algSc `  L )
24 asclpropd.g . . 3  |-  G  =  (Scalar `  L )
25 eqid 2435 . . 3  |-  ( Base `  G )  =  (
Base `  G )
26 eqid 2435 . . 3  |-  ( .s
`  L )  =  ( .s `  L
)
27 eqid 2435 . . 3  |-  ( 1r
`  L )  =  ( 1r `  L
)
2823, 24, 25, 26, 27asclfval 16385 . 2  |-  (algSc `  L )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) )
2916, 22, 283eqtr4g 2492 1  |-  ( ph  ->  (algSc `  K )  =  (algSc `  L )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   Basecbs 13461  Scalarcsca 13524   .scvsca 13525   1rcur 15654  algSccascl 16363
This theorem is referenced by:  ply1ascl  16643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-slot 13465  df-base 13466  df-ascl 16366
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