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Theorem asclpropd 16101
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 451 . . . . . 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 13608 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  P  /\  ( 1r `  K )  e.  W ) )  -> 
( z ( .s
`  K ) ( 1r `  K ) )  =  ( z ( .s `  L
) ( 1r `  K ) ) )
54anassrs 629 . . . . . 6  |-  ( ( ( ph  /\  z  e.  P )  /\  ( 1r `  K )  e.  W )  ->  (
z ( .s `  K ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  K
) ) )
62, 5mpdan 649 . . . . 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 5890 . . . . . 6  |-  ( ph  ->  ( z ( .s
`  L ) ( 1r `  K ) )  =  ( z ( .s `  L
) ( 1r `  L ) ) )
98adantr 451 . . . . 5  |-  ( (
ph  /\  z  e.  P )  ->  (
z ( .s `  L ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  L
) ) )
106, 9eqtrd 2328 . . . 4  |-  ( (
ph  /\  z  e.  P )  ->  (
z ( .s `  K ) ( 1r
`  K ) )  =  ( z ( .s `  L ) ( 1r `  L
) ) )
1110mpteq2dva 4122 . . 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 ) )
13 mpteq1 4116 . . . 4  |-  ( P  =  ( Base `  F
)  ->  ( z  e.  P  |->  ( z ( .s `  K
) ( 1r `  K ) ) )  =  ( z  e.  ( Base `  F
)  |->  ( z ( .s `  K ) ( 1r `  K
) ) ) )
1412, 13syl 15 . . 3  |-  ( ph  ->  ( z  e.  P  |->  ( z ( .s
`  K ) ( 1r `  K ) ) )  =  ( z  e.  ( Base `  F )  |->  ( z ( .s `  K
) ( 1r `  K ) ) ) )
15 asclpropd.2 . . . 4  |-  ( ph  ->  P  =  ( Base `  G ) )
16 mpteq1 4116 . . . 4  |-  ( P  =  ( Base `  G
)  ->  ( z  e.  P  |->  ( z ( .s `  L
) ( 1r `  L ) ) )  =  ( z  e.  ( Base `  G
)  |->  ( z ( .s `  L ) ( 1r `  L
) ) ) )
1715, 16syl 15 . . 3  |-  ( ph  ->  ( z  e.  P  |->  ( z ( .s
`  L ) ( 1r `  L ) ) )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) ) )
1811, 14, 173eqtr3d 2336 . 2  |-  ( ph  ->  ( z  e.  (
Base `  F )  |->  ( z ( .s
`  K ) ( 1r `  K ) ) )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) ) )
19 eqid 2296 . . 3  |-  (algSc `  K )  =  (algSc `  K )
20 asclpropd.f . . 3  |-  F  =  (Scalar `  K )
21 eqid 2296 . . 3  |-  ( Base `  F )  =  (
Base `  F )
22 eqid 2296 . . 3  |-  ( .s
`  K )  =  ( .s `  K
)
23 eqid 2296 . . 3  |-  ( 1r
`  K )  =  ( 1r `  K
)
2419, 20, 21, 22, 23asclfval 16090 . 2  |-  (algSc `  K )  =  ( z  e.  ( Base `  F )  |->  ( z ( .s `  K
) ( 1r `  K ) ) )
25 eqid 2296 . . 3  |-  (algSc `  L )  =  (algSc `  L )
26 asclpropd.g . . 3  |-  G  =  (Scalar `  L )
27 eqid 2296 . . 3  |-  ( Base `  G )  =  (
Base `  G )
28 eqid 2296 . . 3  |-  ( .s
`  L )  =  ( .s `  L
)
29 eqid 2296 . . 3  |-  ( 1r
`  L )  =  ( 1r `  L
)
3025, 26, 27, 28, 29asclfval 16090 . 2  |-  (algSc `  L )  =  ( z  e.  ( Base `  G )  |->  ( z ( .s `  L
) ( 1r `  L ) ) )
3118, 24, 303eqtr4g 2353 1  |-  ( ph  ->  (algSc `  K )  =  (algSc `  L )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   1rcur 15355  algSccascl 16068
This theorem is referenced by:  ply1ascl  16351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-slot 13168  df-base 13169  df-ascl 16071
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