Theorem hoadddirt 9725

Description: Scalar product reverse distributive law for Hilbert space operators.

Assertion

Ref	Expression
hoadddirt	|- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A + B) .op T) = ((A .op T) +op (B .op T)))

Proof of Theorem hoadddirt

Step	Hyp	Ref	Expression
1		ax-hvdistr2 8874	. . . . . . . . 9 |- ((A e. CC /\ B e. CC /\ (T` x) e. H~) -> ((A + B) .h (T` x)) = ((A .h (T` x)) +h (B .h (T` x))))
2		ffvelrn 3820	. . . . . . . . 9 |- ((T:H~-->H~ /\ x e. H~) -> (T` x) e. H~)
3	1, 2	syl3an3 863	. . . . . . . 8 |- ((A e. CC /\ B e. CC /\ (T:H~-->H~ /\ x e. H~)) -> ((A + B) .h (T` x)) = ((A .h (T` x)) +h (B .h (T` x))))
4	3	3exp 834	. . . . . . 7 |- (A e. CC -> (B e. CC -> ((T:H~-->H~ /\ x e. H~) -> ((A + B) .h (T` x)) = ((A .h (T` x)) +h (B .h (T` x))))))
5	4	exp4a 380	. . . . . 6 |- (A e. CC -> (B e. CC -> (T:H~-->H~ -> (x e. H~ -> ((A + B) .h (T` x)) = ((A .h (T` x)) +h (B .h (T` x)))))))
6	5	3imp1 848	. . . . 5 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> ((A + B) .h (T` x)) = ((A .h (T` x)) +h (B .h (T` x))))
7		homvalt 9513	. . . . . . 7 |- (((A + B) e. CC /\ T:H~-->H~ /\ x e. H~) -> (((A + B) .op T)` x) = ((A + B) .h (T` x)))
8	7	3expa 835	. . . . . 6 |- ((((A + B) e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A + B) .op T)` x) = ((A + B) .h (T` x)))
9		axaddcl 5283	. . . . . . . 8 |- ((A e. CC /\ B e. CC) -> (A + B) e. CC)
10	9	anim1i 334	. . . . . . 7 |- (((A e. CC /\ B e. CC) /\ T:H~-->H~) -> ((A + B) e. CC /\ T:H~-->H~))
11	10	3impa 830	. . . . . 6 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A + B) e. CC /\ T:H~-->H~))
12	8, 11	sylan 450	. . . . 5 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A + B) .op T)` x) = ((A + B) .h (T` x)))
13		homvalt 9513	. . . . . . . 8 |- ((A e. CC /\ T:H~-->H~ /\ x e. H~) -> ((A .op T)` x) = (A .h (T` x)))
14	13	3expa 835	. . . . . . 7 |- (((A e. CC /\ T:H~-->H~) /\ x e. H~) -> ((A .op T)` x) = (A .h (T` x)))
15	14	3adantl2 806	. . . . . 6 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> ((A .op T)` x) = (A .h (T` x)))
16		homvalt 9513	. . . . . . . 8 |- ((B e. CC /\ T:H~-->H~ /\ x e. H~) -> ((B .op T)` x) = (B .h (T` x)))
17	16	3expa 835	. . . . . . 7 |- (((B e. CC /\ T:H~-->H~) /\ x e. H~) -> ((B .op T)` x) = (B .h (T` x)))
18	17	3adantl1 805	. . . . . 6 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> ((B .op T)` x) = (B .h (T` x)))
19	15, 18	opreq12d 3984	. . . . 5 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A .op T)` x) +h ((B .op T)` x)) = ((A .h (T` x)) +h (B .h (T` x))))
20	6, 12, 19	3eqtr4d 1520	. . . 4 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A + B) .op T)` x) = (((A .op T)` x) +h ((B .op T)` x)))
21		hosvalt 9511	. . . . . 6 |- (((A .op T):H~-->H~ /\ (B .op T):H~-->H~ /\ x e. H~) -> (((A .op T) +op (B .op T))` x) = (((A .op T)` x) +h ((B .op T)` x)))
22	21	3expa 835	. . . . 5 |- ((((A .op T):H~-->H~ /\ (B .op T):H~-->H~) /\ x e. H~) -> (((A .op T) +op (B .op T))` x) = (((A .op T)` x) +h ((B .op T)` x)))
23		homulclt 9680	. . . . . . 7 |- ((A e. CC /\ T:H~-->H~) -> (A .op T):H~-->H~)
24		homulclt 9680	. . . . . . 7 |- ((B e. CC /\ T:H~-->H~) -> (B .op T):H~-->H~)
25	23, 24	anim12i 333	. . . . . 6 |- (((A e. CC /\ T:H~-->H~) /\ (B e. CC /\ T:H~-->H~)) -> ((A .op T):H~-->H~ /\ (B .op T):H~-->H~))
26	25	3impdir 883	. . . . 5 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A .op T):H~-->H~ /\ (B .op T):H~-->H~))
27	22, 26	sylan 450	. . . 4 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A .op T) +op (B .op T))` x) = (((A .op T)` x) +h ((B .op T)` x)))
28	20, 27	eqtr4d 1513	. . 3 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A + B) .op T)` x) = (((A .op T) +op (B .op T))` x))
29	28	r19.21aiva 1717	. 2 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> A.x e. H~ (((A + B) .op T)` x) = (((A .op T) +op (B .op T))` x))
30		hoeqt 9681	. . 3 |- ((((A + B) .op T):H~-->H~ /\ ((A .op T) +op (B .op T)):H~-->H~) -> (A.x e. H~ (((A + B) .op T)` x) = (((A .op T) +op (B .op T))` x) <-> ((A + B) .op T) = ((A .op T) +op (B .op T))))
31		homulclt 9680	. . . . 5 |- (((A + B) e. CC /\ T:H~-->H~) -> ((A + B) .op T):H~-->H~)
32	31, 9	sylan 450	. . . 4 |- (((A e. CC /\ B e. CC) /\ T:H~-->H~) -> ((A + B) .op T):H~-->H~)
33	32	3impa 830	. . 3 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A + B) .op T):H~-->H~)
34		hoaddclt 9679	. . . . 5 |- (((A .op T):H~-->H~ /\ (B .op T):H~-->H~) -> ((A .op T) +op (B .op T)):H~-->H~)
35	34, 23, 24	syl2an 456	. . . 4 |- (((A e. CC /\ T:H~-->H~) /\ (B e. CC /\ T:H~-->H~)) -> ((A .op T) +op (B .op T)):H~-->H~)
36	35	3impdir 883	. . 3 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A .op T) +op (B .op T)):H~-->H~)
37	30, 33, 36	sylanc 473	. 2 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> (A.x e. H~ (((A + B) .op T)` x) = (((A .op T) +op (B .op T))` x) <-> ((A + B) .op T) = ((A .op T) +op (B .op T))))
38	29, 37	mpbid 195	1 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A + B) .op T) = ((A .op T) +op (B .op T)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  -->wf 3184  ` cfv 3188  (class class class)co 3969  CCcc 5244   + caddc 5249  H~chil 8783   +h cva 8784   .h csm 8785   +op chos 8802   .op chot 8803

This theorem is referenced by:  ho2timest 9740

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634  ax-hilex 8864  ax-hfvadd 8865  ax-hfvmul 8870  ax-hvdistr2 8874

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-map 4330