Theorem tendoplcbv 36063

Description: Define sum operation for trace-perserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)

Hypothesis

Ref	Expression
tendoplcbv.p	|- P = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )

Assertion

Ref	Expression
tendoplcbv	|- P = ( u e. E , v e. E |-> ( g e. T |-> ( ( u ` g ) o. ( v ` g ) ) ) )

Distinct variable groups:   t, s, u, v, E   f, g, s, t, u, v, T

Allowed substitution hints:   P( v, u, t, f, g, s)   E( f, g)

Proof of Theorem tendoplcbv

Step	Hyp	Ref	Expression
1		tendoplcbv.p	. 2 |- P = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
2		fveq1 6190	. . . . 5 |- ( s = u -> ( s ` f ) = ( u ` f ) )
3	2	coeq1d 5283	. . . 4 |- ( s = u -> ( ( s ` f ) o. ( t ` f ) ) = ( ( u ` f ) o. ( t ` f ) ) )
4	3	mpteq2dv 4745	. . 3 |- ( s = u -> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) = ( f e. T |-> ( ( u ` f ) o. ( t ` f ) ) ) )
5		fveq1 6190	. . . . . 6 |- ( t = v -> ( t ` f ) = ( v ` f ) )
6	5	coeq2d 5284	. . . . 5 |- ( t = v -> ( ( u ` f ) o. ( t ` f ) ) = ( ( u ` f ) o. ( v ` f ) ) )
7	6	mpteq2dv 4745	. . . 4 |- ( t = v -> ( f e. T |-> ( ( u ` f ) o. ( t ` f ) ) ) = ( f e. T |-> ( ( u ` f ) o. ( v ` f ) ) ) )
8		fveq2 6191	. . . . . 6 |- ( f = g -> ( u ` f ) = ( u ` g ) )
9		fveq2 6191	. . . . . 6 |- ( f = g -> ( v ` f ) = ( v ` g ) )
10	8, 9	coeq12d 5286	. . . . 5 |- ( f = g -> ( ( u ` f ) o. ( v ` f ) ) = ( ( u ` g ) o. ( v ` g ) ) )
11	10	cbvmptv 4750	. . . 4 |- ( f e. T |-> ( ( u ` f ) o. ( v ` f ) ) ) = ( g e. T |-> ( ( u ` g ) o. ( v ` g ) ) )
12	7, 11	syl6eq 2672	. . 3 |- ( t = v -> ( f e. T |-> ( ( u ` f ) o. ( t ` f ) ) ) = ( g e. T |-> ( ( u ` g ) o. ( v ` g ) ) ) )
13	4, 12	cbvmpt2v 6735	. 2 |- ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) = ( u e. E , v e. E |-> ( g e. T |-> ( ( u ` g ) o. ( v ` g ) ) ) )
14	1, 13	eqtri 2644	1 |- P = ( u e. E , v e. E |-> ( g e. T |-> ( ( u ` g ) o. ( v ` g ) ) ) )

Syntax hints:   = wceq 1483   |-> cmpt 4729   o. ccom 5118   ` cfv 5888   |-> cmpt2 6652

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-opab 4713  df-mpt 4730  df-co 5123  df-iota 5851  df-fv 5896  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  tendopl  36064