Theorem homco1 28660

Description: Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
homco1	|- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( ( A .op T ) o. U ) = ( A .op ( T o. U ) ) )

Proof of Theorem homco1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvco3 6275	. . . . . 6 |- ( ( U : ~H --> ~H /\ x e. ~H ) -> ( ( ( A .op T ) o. U ) ` x ) = ( ( A .op T ) ` ( U ` x ) ) )
2	1	3ad2antl3 1225	. . . . 5 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A .op T ) o. U ) ` x ) = ( ( A .op T ) ` ( U ` x ) ) )
3		fvco3 6275	. . . . . . . 8 |- ( ( U : ~H --> ~H /\ x e. ~H ) -> ( ( T o. U ) ` x ) = ( T ` ( U ` x ) ) )
4	3	3ad2antl3 1225	. . . . . . 7 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( T o. U ) ` x ) = ( T ` ( U ` x ) ) )
5	4	oveq2d 6666	. . . . . 6 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( ( T o. U ) ` x ) ) = ( A .h ( T ` ( U ` x ) ) ) )
6		ffvelrn 6357	. . . . . . . . . 10 |- ( ( U : ~H --> ~H /\ x e. ~H ) -> ( U ` x ) e. ~H )
7		homval 28600	. . . . . . . . . 10 |- ( ( A e. CC /\ T : ~H --> ~H /\ ( U ` x ) e. ~H ) -> ( ( A .op T ) ` ( U ` x ) ) = ( A .h ( T ` ( U ` x ) ) ) )
8	6, 7	syl3an3 1361	. . . . . . . . 9 |- ( ( A e. CC /\ T : ~H --> ~H /\ ( U : ~H --> ~H /\ x e. ~H ) ) -> ( ( A .op T ) ` ( U ` x ) ) = ( A .h ( T ` ( U ` x ) ) ) )
9	8	3expa 1265	. . . . . . . 8 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( U : ~H --> ~H /\ x e. ~H ) ) -> ( ( A .op T ) ` ( U ` x ) ) = ( A .h ( T ` ( U ` x ) ) ) )
10	9	exp43 640	. . . . . . 7 |- ( A e. CC -> ( T : ~H --> ~H -> ( U : ~H --> ~H -> ( x e. ~H -> ( ( A .op T ) ` ( U ` x ) ) = ( A .h ( T ` ( U ` x ) ) ) ) ) ) )
11	10	3imp1 1280	. . . . . 6 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op T ) ` ( U ` x ) ) = ( A .h ( T ` ( U ` x ) ) ) )
12	5, 11	eqtr4d 2659	. . . . 5 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( ( T o. U ) ` x ) ) = ( ( A .op T ) ` ( U ` x ) ) )
13	2, 12	eqtr4d 2659	. . . 4 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A .op T ) o. U ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) )
14		fco 6058	. . . . . . . 8 |- ( ( T : ~H --> ~H /\ U : ~H --> ~H ) -> ( T o. U ) : ~H --> ~H )
15		homval 28600	. . . . . . . 8 |- ( ( A e. CC /\ ( T o. U ) : ~H --> ~H /\ x e. ~H ) -> ( ( A .op ( T o. U ) ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) )
16	14, 15	syl3an2 1360	. . . . . . 7 |- ( ( A e. CC /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op ( T o. U ) ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) )
17	16	3expia 1267	. . . . . 6 |- ( ( A e. CC /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( x e. ~H -> ( ( A .op ( T o. U ) ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) ) )
18	17	3impb 1260	. . . . 5 |- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( x e. ~H -> ( ( A .op ( T o. U ) ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) ) )
19	18	imp 445	. . . 4 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op ( T o. U ) ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) )
20	13, 19	eqtr4d 2659	. . 3 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A .op T ) o. U ) ` x ) = ( ( A .op ( T o. U ) ) ` x ) )
21	20	ralrimiva 2966	. 2 |- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> A. x e. ~H ( ( ( A .op T ) o. U ) ` x ) = ( ( A .op ( T o. U ) ) ` x ) )
22		homulcl 28618	. . . 4 |- ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) : ~H --> ~H )
23		fco 6058	. . . 4 |- ( ( ( A .op T ) : ~H --> ~H /\ U : ~H --> ~H ) -> ( ( A .op T ) o. U ) : ~H --> ~H )
24	22, 23	stoic3 1701	. . 3 |- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( ( A .op T ) o. U ) : ~H --> ~H )
25		homulcl 28618	. . . . 5 |- ( ( A e. CC /\ ( T o. U ) : ~H --> ~H ) -> ( A .op ( T o. U ) ) : ~H --> ~H )
26	14, 25	sylan2 491	. . . 4 |- ( ( A e. CC /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( A .op ( T o. U ) ) : ~H --> ~H )
27	26	3impb 1260	. . 3 |- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( A .op ( T o. U ) ) : ~H --> ~H )
28		hoeq 28619	. . 3 |- ( ( ( ( A .op T ) o. U ) : ~H --> ~H /\ ( A .op ( T o. U ) ) : ~H --> ~H ) -> ( A. x e. ~H ( ( ( A .op T ) o. U ) ` x ) = ( ( A .op ( T o. U ) ) ` x ) <-> ( ( A .op T ) o. U ) = ( A .op ( T o. U ) ) ) )
29	24, 27, 28	syl2anc 693	. 2 |- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( A. x e. ~H ( ( ( A .op T ) o. U ) ` x ) = ( ( A .op ( T o. U ) ) ` x ) <-> ( ( A .op T ) o. U ) = ( A .op ( T o. U ) ) ) )
30	21, 29	mpbid 222	1 |- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( ( A .op T ) o. U ) = ( A .op ( T o. U ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   ~Hchil 27776   .h csm 27778   .op chot 27796

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-hilex 27856  ax-hfvmul 27862

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-pw 4160  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  df-map 7859  df-homul 28590

This theorem is referenced by:  opsqrlem1  28999