MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  asclpropd Unicode version

Theorem asclpropd 16324
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 13835 . . . . . . 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 6029 . . . . . 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 2412 . . . 4  |-  ( (
ph  /\  z  e.  P )  ->  (
z ( .s `  K ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  L
) ) )
1110mpteq2dva 4229 . . 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 4224 . . 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 4224 . . 3  |-  ( ph  ->  ( z  e.  P  |->  ( z ( .s
`  L ) ( 1r `  L ) ) )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) ) )
1611, 13, 153eqtr3d 2420 . 2  |-  ( ph  ->  ( z  e.  (
Base `  F )  |->  ( z ( .s
`  K ) ( 1r `  K ) ) )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) ) )
17 eqid 2380 . . 3  |-  (algSc `  K )  =  (algSc `  K )
18 asclpropd.f . . 3  |-  F  =  (Scalar `  K )
19 eqid 2380 . . 3  |-  ( Base `  F )  =  (
Base `  F )
20 eqid 2380 . . 3  |-  ( .s
`  K )  =  ( .s `  K
)
21 eqid 2380 . . 3  |-  ( 1r
`  K )  =  ( 1r `  K
)
2217, 18, 19, 20, 21asclfval 16313 . 2  |-  (algSc `  K )  =  ( z  e.  ( Base `  F )  |->  ( z ( .s `  K
) ( 1r `  K ) ) )
23 eqid 2380 . . 3  |-  (algSc `  L )  =  (algSc `  L )
24 asclpropd.g . . 3  |-  G  =  (Scalar `  L )
25 eqid 2380 . . 3  |-  ( Base `  G )  =  (
Base `  G )
26 eqid 2380 . . 3  |-  ( .s
`  L )  =  ( .s `  L
)
27 eqid 2380 . . 3  |-  ( 1r
`  L )  =  ( 1r `  L
)
2823, 24, 25, 26, 27asclfval 16313 . 2  |-  (algSc `  L )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) )
2916, 22, 283eqtr4g 2437 1  |-  ( ph  ->  (algSc `  K )  =  (algSc `  L )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    e. cmpt 4200   ` cfv 5387  (class class class)co 6013   Basecbs 13389  Scalarcsca 13452   .scvsca 13453   1rcur 15582  algSccascl 16291
This theorem is referenced by:  ply1ascl  16571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-slot 13393  df-base 13394  df-ascl 16294
  Copyright terms: Public domain W3C validator