Theorem toycom 34260

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

Step	Hyp	Ref	Expression
1		toycom.1	. . . . . 6 |- C = { g e. Abel | ( Base ` g ) = CC }
2		ssrab2 3687	. . . . . 6 |- { g e. Abel | ( Base ` g ) = CC } C_ Abel
3	1, 2	eqsstri 3635	. . . . 5 |- C C_ Abel
4	3	sseli 3599	. . . 4 |- ( K e. C -> K e. Abel )
5	4	3ad2ant1 1082	. . 3 |- ( ( K e. C /\ A e. CC /\ B e. CC ) -> K e. Abel )
6		simp2 1062	. . . 4 |- ( ( K e. C /\ A e. CC /\ B e. CC ) -> A e. CC )
7		fveq2 6191	. . . . . . . 8 |- ( g = K -> ( Base ` g ) = ( Base ` K ) )
8	7	eqeq1d 2624	. . . . . . 7 |- ( g = K -> ( ( Base ` g ) = CC <-> ( Base ` K ) = CC ) )
9	8, 1	elrab2 3366	. . . . . 6 |- ( K e. C <-> ( K e. Abel /\ ( Base ` K ) = CC ) )
10	9	simprbi 480	. . . . 5 |- ( K e. C -> ( Base ` K ) = CC )
11	10	3ad2ant1 1082	. . . 4 |- ( ( K e. C /\ A e. CC /\ B e. CC ) -> ( Base ` K ) = CC )
12	6, 11	eleqtrrd 2704	. . 3 |- ( ( K e. C /\ A e. CC /\ B e. CC ) -> A e. ( Base ` K ) )
13		simp3 1063	. . . 4 |- ( ( K e. C /\ A e. CC /\ B e. CC ) -> B e. CC )
14	13, 11	eleqtrrd 2704	. . 3 |- ( ( K e. C /\ A e. CC /\ B e. CC ) -> B e. ( Base ` K ) )
15		eqid 2622	. . . 4 |- ( Base ` K ) = ( Base ` K )
16		eqid 2622	. . . 4 |- ( +g ` K ) = ( +g ` K )
17	15, 16	ablcom 18210	. . 3 |- ( ( K e. Abel /\ A e. ( Base ` K ) /\ B e. ( Base ` K ) ) -> ( A ( +g ` K ) B ) = ( B ( +g ` K ) A ) )
18	5, 12, 14, 17	syl3anc 1326	. 2 |- ( ( K e. C /\ A e. CC /\ B e. CC ) -> ( A ( +g ` K ) B ) = ( B ( +g ` K ) A ) )
19		toycom.2	. . 3 |- .+ = ( +g ` K )
20	19	oveqi 6663	. 2 |- ( A .+ B ) = ( A ( +g ` K ) B )
21	19	oveqi 6663	. 2 |- ( B .+ A ) = ( B ( +g ` K ) A )
22	18, 20, 21	3eqtr4g 2681	1 |- ( ( K e. C /\ A e. CC /\ B e. CC ) -> ( A .+ B ) = ( B .+ A ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   ` cfv 5888  (class class class)co 6650   CCcc 9934   Basecbs 15857   +g cplusg 15941   Abelcabl 18194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-cmn 18195  df-abl 18196

This theorem is referenced by: (None)