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Theorem homco2 28836
Description: Move a scalar product out of a composition of operators. The operator T must be linear, unlike homco1 28660 that works for any operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
homco2 |- ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) -> ( T o. ( A .op U ) ) = ( A .op ( T o. U ) ) )
Proof of Theorem homco2
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 simpl1 1064 . . . . . 6 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> A e. CC )
2 simpl3 1066 . . . . . 6 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> U : ~H --> ~H )
3 simpr 477 . . . . . 6 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> x e. ~H )
4 homval 28600 . . . . . 6 |- ( ( A e. CC /\ U : ~H --> ~H /\ x e. ~H ) -> ( ( A .op U ) ` x ) = ( A .h ( U ` x ) ) )
51, 2, 3, 4syl3anc 1326 . . . . 5 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op U ) ` x ) = ( A .h ( U ` x ) ) )
65fveq2d 6195 . . . 4 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( T ` ( ( A .op U ) ` x ) ) = ( T ` ( A .h ( U ` x ) ) ) )
7 homulcl 28618 . . . . . 6 |- ( ( A e. CC /\ U : ~H --> ~H ) -> ( A .op U ) : ~H --> ~H )
873adant2 1080 . . . . 5 |- ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) -> ( A .op U ) : ~H --> ~H )
9 fvco3 6275 . . . . 5 |- ( ( ( A .op U ) : ~H --> ~H /\ x e. ~H ) -> ( ( T o. ( A .op U ) ) ` x ) = ( T ` ( ( A .op U ) ` x ) ) )
108, 9sylan 488 . . . 4 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( T o. ( A .op U ) ) ` x ) = ( T ` ( ( A .op U ) ` x ) ) )
11 fvco3 6275 . . . . . . 7 |- ( ( U : ~H --> ~H /\ x e. ~H ) -> ( ( T o. U ) ` x ) = ( T ` ( U ` x ) ) )
122, 3, 11syl2anc 693 . . . . . 6 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( T o. U ) ` x ) = ( T ` ( U ` x ) ) )
1312oveq2d 6666 . . . . 5 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( ( T o. U ) ` x ) ) = ( A .h ( T ` ( U ` x ) ) ) )
14 lnopf 28718 . . . . . . . . 9 |- ( T e. LinOp -> T : ~H --> ~H )
15143ad2ant2 1083 . . . . . . . 8 |- ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) -> T : ~H --> ~H )
16 simp3 1063 . . . . . . . 8 |- ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) -> U : ~H --> ~H )
17 fco 6058 . . . . . . . 8 |- ( ( T : ~H --> ~H /\ U : ~H --> ~H ) -> ( T o. U ) : ~H --> ~H )
1815, 16, 17syl2anc 693 . . . . . . 7 |- ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) -> ( T o. U ) : ~H --> ~H )
1918adantr 481 . . . . . 6 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( T o. U ) : ~H --> ~H )
20 homval 28600 . . . . . 6 |- ( ( A e. CC /\ ( T o. U ) : ~H --> ~H /\ x e. ~H ) -> ( ( A .op ( T o. U ) ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) )
211, 19, 3, 20syl3anc 1326 . . . . 5 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op ( T o. U ) ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) )
22 simpl2 1065 . . . . . 6 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> T e. LinOp )
2316ffvelrnda 6359 . . . . . 6 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( U ` x ) e. ~H )
24 lnopmul 28826 . . . . . 6 |- ( ( T e. LinOp /\ A e. CC /\ ( U ` x ) e. ~H ) -> ( T ` ( A .h ( U ` x ) ) ) = ( A .h ( T ` ( U ` x ) ) ) )
2522, 1, 23, 24syl3anc 1326 . . . . 5 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( T ` ( A .h ( U ` x ) ) ) = ( A .h ( T ` ( U ` x ) ) ) )
2613, 21, 253eqtr4d 2666 . . . 4 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op ( T o. U ) ) ` x ) = ( T ` ( A .h ( U ` x ) ) ) )
276, 10, 263eqtr4d 2666 . . 3 |- ( ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( T o. ( A .op U ) ) ` x ) = ( ( A .op ( T o. U ) ) ` x ) )
2827ralrimiva 2966 . 2 |- ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) -> A. x e. ~H ( ( T o. ( A .op U ) ) ` x ) = ( ( A .op ( T o. U ) ) ` x ) )
29 fco 6058 . . . 4 |- ( ( T : ~H --> ~H /\ ( A .op U ) : ~H --> ~H ) -> ( T o. ( A .op U ) ) : ~H --> ~H )
3015, 8, 29syl2anc 693 . . 3 |- ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) -> ( T o. ( A .op U ) ) : ~H --> ~H )
31 simp1 1061 . . . 4 |- ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) -> A e. CC )
32 homulcl 28618 . . . 4 |- ( ( A e. CC /\ ( T o. U ) : ~H --> ~H ) -> ( A .op ( T o. U ) ) : ~H --> ~H )
3331, 18, 32syl2anc 693 . . 3 |- ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) -> ( A .op ( T o. U ) ) : ~H --> ~H )
34 hoeq 28619 . . 3 |- ( ( ( T o. ( A .op U ) ) : ~H --> ~H /\ ( A .op ( T o. U ) ) : ~H --> ~H ) -> ( A. x e. ~H ( ( T o. ( A .op U ) ) ` x ) = ( ( A .op ( T o. U ) ) ` x ) <-> ( T o. ( A .op U ) ) = ( A .op ( T o. U ) ) ) )
3530, 33, 34syl2anc 693 . 2 |- ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) -> ( A. x e. ~H ( ( T o. ( A .op U ) ) ` x ) = ( ( A .op ( T o. U ) ) ` x ) <-> ( T o. ( A .op U ) ) = ( A .op ( T o. U ) ) ) )
3628, 35mpbid 222 1 |- ( ( A e. CC /\ T e. LinOp /\ U : ~H --> ~H ) -> ( T o. ( A .op U ) ) = ( A .op ( T o. U ) ) )
Colors of variables: wff setvar class
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   LinOpclo 27804
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-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-hilex 27856  ax-hfvadd 27857  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvdistr2 27866  ax-hvmul0 27867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-po 5035  df-so 5036  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269  df-hvsub 27828  df-homul 28590  df-lnop 28700
This theorem is referenced by:  opsqrlem1  28999
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