Theorem tendopl 36064

Description: Value of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Hypotheses

Ref	Expression
tendoplcbv.p	|- P = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
tendopl2.t	|- T = ( ( LTrn ` K ) ` W )

Assertion

Ref	Expression
tendopl	|- ( ( U e. E /\ V e. E ) -> ( U P V ) = ( g e. T |-> ( ( U ` g ) o. ( V ` g ) ) ) )

Distinct variable groups:   t, s, E   f, g, s, t, T   f, W, g, s, t   U, g   g, V

Allowed substitution hints:   P( t, f, g, s)   U( t, f, s)   E( f, g)   K( t, f, g, s)   V( t, f, s)

Proof of Theorem tendopl

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6190	. . . 4 |- ( u = U -> ( u ` g ) = ( U ` g ) )
2	1	coeq1d 5283	. . 3 |- ( u = U -> ( ( u ` g ) o. ( v ` g ) ) = ( ( U ` g ) o. ( v ` g ) ) )
3	2	mpteq2dv 4745	. 2 |- ( u = U -> ( g e. T |-> ( ( u ` g ) o. ( v ` g ) ) ) = ( g e. T |-> ( ( U ` g ) o. ( v ` g ) ) ) )
4		fveq1 6190	. . . 4 |- ( v = V -> ( v ` g ) = ( V ` g ) )
5	4	coeq2d 5284	. . 3 |- ( v = V -> ( ( U ` g ) o. ( v ` g ) ) = ( ( U ` g ) o. ( V ` g ) ) )
6	5	mpteq2dv 4745	. 2 |- ( v = V -> ( g e. T |-> ( ( U ` g ) o. ( v ` g ) ) ) = ( g e. T |-> ( ( U ` g ) o. ( V ` g ) ) ) )
7		tendoplcbv.p	. . 3 |- P = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
8	7	tendoplcbv 36063	. 2 |- P = ( u e. E , v e. E |-> ( g e. T |-> ( ( u ` g ) o. ( v ` g ) ) ) )
9		tendopl2.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
10		fvex 6201	. . . 4 |- ( ( LTrn ` K ) ` W ) e. _V
11	9, 10	eqeltri 2697	. . 3 |- T e. _V
12	11	mptex 6486	. 2 |- ( g e. T |-> ( ( U ` g ) o. ( V ` g ) ) ) e. _V
13	3, 6, 8, 12	ovmpt2 6796	1 |- ( ( U e. E /\ V e. E ) -> ( U P V ) = ( g e. T |-> ( ( U ` g ) o. ( V ` g ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   o. ccom 5118   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   LTrncltrn 35387

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-rep 4771  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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  tendopl2  36065  tendoplcl  36069  erngplus  36091  erngplus-rN  36099  dvaplusg  36297