Theorem hoadddirt 9670

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 8818	. . . . . . . . 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 3805	. . . . . . . . 9 |- ((T:H~-->H~ /\ x e. H~) -> (T` x) e. H~)
3	1, 2	syl3an3 860	. . . . . . . 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 831	. . . . . . 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 378	. . . . . 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 845	. . . . 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 9458	. . . . . . 7 |- (((A + B) e. CC /\ T:H~-->H~ /\ x e. H~) -> (((A + B) .op T)` x) = ((A + B) .h (T` x)))
8	7	3expa 832	. . . . . 6 |- ((((A + B) e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A + B) .op T)` x) = ((A + B) .h (T` x)))
9		axaddcl 5251	. . . . . . . 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 827	. . . . . 6 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A + B) e. CC /\ T:H~-->H~))
12	8, 11	sylan 448	. . . . 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 9458	. . . . . . . 8 |- ((A e. CC /\ T:H~-->H~ /\ x e. H~) -> ((A .op T)` x) = (A .h (T` x)))
14	13	3expa 832	. . . . . . 7 |- (((A e. CC /\ T:H~-->H~) /\ x e. H~) -> ((A .op T)` x) = (A .h (T` x)))
15	14	3adantl2 803	. . . . . 6 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> ((A .op T)` x) = (A .h (T` x)))
16		homvalt 9458	. . . . . . . 8 |- ((B e. CC /\ T:H~-->H~ /\ x e. H~) -> ((B .op T)` x) = (B .h (T` x)))
17	16	3expa 832	. . . . . . 7 |- (((B e. CC /\ T:H~-->H~) /\ x e. H~) -> ((B .op T)` x) = (B .h (T` x)))
18	17	3adantl1 802	. . . . . 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 3969	. . . . 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 1514	. . . 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 9456	. . . . . 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 832	. . . . 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 9625	. . . . . . 7 |- ((A e. CC /\ T:H~-->H~) -> (A .op T):H~-->H~)
24		homulclt 9625	. . . . . . 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 879	. . . . 5 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A .op T):H~-->H~ /\ (B .op T):H~-->H~))
27	22, 26	sylan 448	. . . 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 1507	. . 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 1711	. 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 9626	. . 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 9625	. . . . 5 |- (((A + B) e. CC /\ T:H~-->H~) -> ((A + B) .op T):H~-->H~)
32	31, 9	sylan 448	. . . 4 |- (((A e. CC /\ B e. CC) /\ T:H~-->H~) -> ((A + B) .op T):H~-->H~)
33	32	3impa 827	. . 3 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A + B) .op T):H~-->H~)
34		hoaddclt 9624	. . . . 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 454	. . . 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 879	. . 3 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A .op T) +op (B .op T)):H~-->H~)
37	30, 33, 36	sylanc 471	. 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 774   = wceq 954   e. wcel 956  A.wral 1642  -->wf 3173  ` cfv 3177  (class class class)co 3954  CCcc 5212   + caddc 5217  H~chil 8727   +h cva 8728   .h csm 8729   +op chos 8746   .op chot 8747

This theorem is referenced by:  ho2timest 9685

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605  ax-hilex 8808  ax-hfvadd 8809  ax-hfvmul 8814  ax-hvdistr2 8818

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-map 4314  df-ni