Theorem ho2times 28678

Description: Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
ho2times	|- ( T : ~H --> ~H -> ( 2 .op T ) = ( T +op T ) )

Proof of Theorem ho2times

Step	Hyp	Ref	Expression
1		df-2 11079	. . . 4 |- 2 = ( 1 + 1 )
2	1	oveq1i 6660	. . 3 |- ( 2 .op T ) = ( ( 1 + 1 ) .op T )
3		ax-1cn 9994	. . . 4 |- 1 e. CC
4		hoadddir 28663	. . . 4 |- ( ( 1 e. CC /\ 1 e. CC /\ T : ~H --> ~H ) -> ( ( 1 + 1 ) .op T ) = ( ( 1 .op T ) +op ( 1 .op T ) ) )
5	3, 3, 4	mp3an12 1414	. . 3 |- ( T : ~H --> ~H -> ( ( 1 + 1 ) .op T ) = ( ( 1 .op T ) +op ( 1 .op T ) ) )
6	2, 5	syl5eq 2668	. 2 |- ( T : ~H --> ~H -> ( 2 .op T ) = ( ( 1 .op T ) +op ( 1 .op T ) ) )
7		hoadddi 28662	. . . 4 |- ( ( 1 e. CC /\ T : ~H --> ~H /\ T : ~H --> ~H ) -> ( 1 .op ( T +op T ) ) = ( ( 1 .op T ) +op ( 1 .op T ) ) )
8	3, 7	mp3an1 1411	. . 3 |- ( ( T : ~H --> ~H /\ T : ~H --> ~H ) -> ( 1 .op ( T +op T ) ) = ( ( 1 .op T ) +op ( 1 .op T ) ) )
9	8	anidms 677	. 2 |- ( T : ~H --> ~H -> ( 1 .op ( T +op T ) ) = ( ( 1 .op T ) +op ( 1 .op T ) ) )
10		hoaddcl 28617	. . . 4 |- ( ( T : ~H --> ~H /\ T : ~H --> ~H ) -> ( T +op T ) : ~H --> ~H )
11	10	anidms 677	. . 3 |- ( T : ~H --> ~H -> ( T +op T ) : ~H --> ~H )
12		homulid2 28659	. . 3 |- ( ( T +op T ) : ~H --> ~H -> ( 1 .op ( T +op T ) ) = ( T +op T ) )
13	11, 12	syl 17	. 2 |- ( T : ~H --> ~H -> ( 1 .op ( T +op T ) ) = ( T +op T ) )
14	6, 9, 13	3eqtr2d 2662	1 |- ( T : ~H --> ~H -> ( 2 .op T ) = ( T +op T ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   -->wf 5884  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   2c2 11070   ~Hchil 27776   +op chos 27795   .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-1cn 9994  ax-addcl 9996  ax-hilex 27856  ax-hfvadd 27857  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvdistr1 27865  ax-hvdistr2 27866

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-2 11079  df-hosum 28589  df-homul 28590

This theorem is referenced by:  opsqrlem6  29004