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Theorem toycom 29707
Description: Show the commutative law for an operation  O on a toy structure class  C of commuatitive operations on  CC. This illustrates how a structure class can be partially specialized. In practice, we would ordinarily define a new constant such as "CAbel" in place of  C. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
toycom.1  |-  C  =  { g  e.  Abel  |  ( Base `  g
)  =  CC }
toycom.2  |-  .+  =  ( +g  `  K )
Assertion
Ref Expression
toycom  |-  ( ( K  e.  C  /\  A  e.  CC  /\  B  e.  CC )  ->  ( A  .+  B )  =  ( B  .+  A
) )
Distinct variable group:    g, K
Allowed substitution hints:    A( g)    B( g)    C( g)    .+ ( g)

Proof of Theorem toycom
StepHypRef Expression
1 toycom.1 . . . . . 6  |-  C  =  { g  e.  Abel  |  ( Base `  g
)  =  CC }
2 ssrab2 3420 . . . . . 6  |-  { g  e.  Abel  |  ( Base `  g )  =  CC }  C_  Abel
31, 2eqsstri 3370 . . . . 5  |-  C  C_  Abel
43sseli 3336 . . . 4  |-  ( K  e.  C  ->  K  e.  Abel )
543ad2ant1 978 . . 3  |-  ( ( K  e.  C  /\  A  e.  CC  /\  B  e.  CC )  ->  K  e.  Abel )
6 simp2 958 . . . 4  |-  ( ( K  e.  C  /\  A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
7 fveq2 5720 . . . . . . . 8  |-  ( g  =  K  ->  ( Base `  g )  =  ( Base `  K
) )
87eqeq1d 2443 . . . . . . 7  |-  ( g  =  K  ->  (
( Base `  g )  =  CC  <->  ( Base `  K
)  =  CC ) )
98, 1elrab2 3086 . . . . . 6  |-  ( K  e.  C  <->  ( K  e.  Abel  /\  ( Base `  K )  =  CC ) )
109simprbi 451 . . . . 5  |-  ( K  e.  C  ->  ( Base `  K )  =  CC )
11103ad2ant1 978 . . . 4  |-  ( ( K  e.  C  /\  A  e.  CC  /\  B  e.  CC )  ->  ( Base `  K )  =  CC )
126, 11eleqtrrd 2512 . . 3  |-  ( ( K  e.  C  /\  A  e.  CC  /\  B  e.  CC )  ->  A  e.  ( Base `  K
) )
13 simp3 959 . . . 4  |-  ( ( K  e.  C  /\  A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
1413, 11eleqtrrd 2512 . . 3  |-  ( ( K  e.  C  /\  A  e.  CC  /\  B  e.  CC )  ->  B  e.  ( Base `  K
) )
15 eqid 2435 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
16 eqid 2435 . . . 4  |-  ( +g  `  K )  =  ( +g  `  K )
1715, 16ablcom 15421 . . 3  |-  ( ( K  e.  Abel  /\  A  e.  ( Base `  K
)  /\  B  e.  ( Base `  K )
)  ->  ( A
( +g  `  K ) B )  =  ( B ( +g  `  K
) A ) )
185, 12, 14, 17syl3anc 1184 . 2  |-  ( ( K  e.  C  /\  A  e.  CC  /\  B  e.  CC )  ->  ( A ( +g  `  K
) B )  =  ( B ( +g  `  K ) A ) )
19 toycom.2 . . 3  |-  .+  =  ( +g  `  K )
2019oveqi 6086 . 2  |-  ( A 
.+  B )  =  ( A ( +g  `  K ) B )
2119oveqi 6086 . 2  |-  ( B 
.+  A )  =  ( B ( +g  `  K ) A )
2218, 20, 213eqtr4g 2492 1  |-  ( ( K  e.  C  /\  A  e.  CC  /\  B  e.  CC )  ->  ( A  .+  B )  =  ( B  .+  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2701   ` cfv 5446  (class class class)co 6073   CCcc 8980   Basecbs 13461   +g cplusg 13521   Abelcabel 15405
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  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-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-cmn 15406  df-abl 15407
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