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Theorem asclpropd 17865
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 465 . . . . . 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 14962 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  P  /\  ( 1r `  K )  e.  W ) )  -> 
( z ( .s
`  K ) ( 1r `  K ) )  =  ( z ( .s `  L
) ( 1r `  K ) ) )
54anassrs 648 . . . . . 6  |-  ( ( ( ph  /\  z  e.  P )  /\  ( 1r `  K )  e.  W )  ->  (
z ( .s `  K ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  K
) ) )
62, 5mpdan 668 . . . . 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 6311 . . . . . 6  |-  ( ph  ->  ( z ( .s
`  L ) ( 1r `  K ) )  =  ( z ( .s `  L
) ( 1r `  L ) ) )
98adantr 465 . . . . 5  |-  ( (
ph  /\  z  e.  P )  ->  (
z ( .s `  L ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  L
) ) )
106, 9eqtrd 2508 . . . 4  |-  ( (
ph  /\  z  e.  P )  ->  (
z ( .s `  K ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  L
) ) )
1110mpteq2dva 4539 . . 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 4534 . . 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 4534 . . 3  |-  ( ph  ->  ( z  e.  P  |->  ( z ( .s
`  L ) ( 1r `  L ) ) )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) ) )
1611, 13, 153eqtr3d 2516 . 2  |-  ( ph  ->  ( z  e.  (
Base `  F )  |->  ( z ( .s
`  K ) ( 1r `  K ) ) )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) ) )
17 eqid 2467 . . 3  |-  (algSc `  K )  =  (algSc `  K )
18 asclpropd.f . . 3  |-  F  =  (Scalar `  K )
19 eqid 2467 . . 3  |-  ( Base `  F )  =  (
Base `  F )
20 eqid 2467 . . 3  |-  ( .s
`  K )  =  ( .s `  K
)
21 eqid 2467 . . 3  |-  ( 1r
`  K )  =  ( 1r `  K
)
2217, 18, 19, 20, 21asclfval 17853 . 2  |-  (algSc `  K )  =  ( z  e.  ( Base `  F )  |->  ( z ( .s `  K
) ( 1r `  K ) ) )
23 eqid 2467 . . 3  |-  (algSc `  L )  =  (algSc `  L )
24 asclpropd.g . . 3  |-  G  =  (Scalar `  L )
25 eqid 2467 . . 3  |-  ( Base `  G )  =  (
Base `  G )
26 eqid 2467 . . 3  |-  ( .s
`  L )  =  ( .s `  L
)
27 eqid 2467 . . 3  |-  ( 1r
`  L )  =  ( 1r `  L
)
2823, 24, 25, 26, 27asclfval 17853 . 2  |-  (algSc `  L )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) )
2916, 22, 283eqtr4g 2533 1  |-  ( ph  ->  (algSc `  K )  =  (algSc `  L )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295   Basecbs 14507  Scalarcsca 14575   .scvsca 14576   1rcur 17025  algSccascl 17830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-slot 14511  df-base 14512  df-ascl 17833
This theorem is referenced by:  ply1ascl  18169
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