Theorem lnopm 9881

Description: The scalar product of a linear operator is a linear operator.

Hypothesis

Ref	Expression
lnopm.1	|- T e. LinOp

Assertion

Ref	Expression
lnopm	|- (A e. CC -> (A .op T) e. LinOp)

Proof of Theorem lnopm

Step	Hyp	Ref	Expression
1		lnopm.1	. . . . 5 |- T e. LinOp
2	1	lnopf 9850	. . . 4 |- T:H~-->H~
3		homulclt 9642	. . . 4 |- ((A e. CC /\ T:H~-->H~) -> (A .op T):H~-->H~)
4	2, 3	mpan2 695	. . 3 |- (A e. CC -> (A .op T):H~-->H~)
5	1	lnopl 9849	. . . . . . . . . . 11 |- ((x e. CC /\ y e. H~ /\ z e. H~) -> (T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z)))
6	5	3expa 832	. . . . . . . . . 10 |- (((x e. CC /\ y e. H~) /\ z e. H~) -> (T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z)))
7	6	opreq2d 3971	. . . . . . . . 9 |- (((x e. CC /\ y e. H~) /\ z e. H~) -> (A .h (T` ((x .h y) +h z))) = (A .h ((x .h (T` y)) +h (T` z))))
8	7	adantl 388	. . . . . . . 8 |- ((A e. CC /\ ((x e. CC /\ y e. H~) /\ z e. H~)) -> (A .h (T` ((x .h y) +h z))) = (A .h ((x .h (T` y)) +h (T` z))))
9		ax-hvdistr1 8833	. . . . . . . . . 10 |- ((A e. CC /\ (x .h (T` y)) e. H~ /\ (T` z) e. H~) -> (A .h ((x .h (T` y)) +h (T` z))) = ((A .h (x .h (T` y))) +h (A .h (T` z))))
10		id 59	. . . . . . . . . 10 |- (A e. CC -> A e. CC)
11		hvmulclt 8838	. . . . . . . . . . 11 |- ((x e. CC /\ (T` y) e. H~) -> (x .h (T` y)) e. H~)
12	2	ffvelrni 3810	. . . . . . . . . . 11 |- (y e. H~ -> (T` y) e. H~)
13	11, 12	sylan2 451	. . . . . . . . . 10 |- ((x e. CC /\ y e. H~) -> (x .h (T` y)) e. H~)
14	2	ffvelrni 3810	. . . . . . . . . 10 |- (z e. H~ -> (T` z) e. H~)
15	9, 10, 13, 14	syl3an 867	. . . . . . . . 9 |- ((A e. CC /\ (x e. CC /\ y e. H~) /\ z e. H~) -> (A .h ((x .h (T` y)) +h (T` z))) = ((A .h (x .h (T` y))) +h (A .h (T` z))))
16	15	3expb 833	. . . . . . . 8 |- ((A e. CC /\ ((x e. CC /\ y e. H~) /\ z e. H~)) -> (A .h ((x .h (T` y)) +h (T` z))) = ((A .h (x .h (T` y))) +h (A .h (T` z))))
17	8, 16	eqtrd 1505	. . . . . . 7 |- ((A e. CC /\ ((x e. CC /\ y e. H~) /\ z e. H~)) -> (A .h (T` ((x .h y) +h z))) = ((A .h (x .h (T` y))) +h (A .h (T` z))))
18		homvalt 9475	. . . . . . . . 9 |- ((A e. CC /\ T:H~-->H~ /\ ((x .h y) +h z) e. H~) -> ((A .op T)` ((x .h y) +h z)) = (A .h (T` ((x .h y) +h z))))
19	2, 18	mp3an2 903	. . . . . . . 8 |- ((A e. CC /\ ((x .h y) +h z) e. H~) -> ((A .op T)` ((x .h y) +h z)) = (A .h (T` ((x .h y) +h z))))
20		hvaddclt 8837	. . . . . . . . 9 |- (((x .h y) e. H~ /\ z e. H~) -> ((x .h y) +h z) e. H~)
21		hvmulclt 8838	. . . . . . . . 9 |- ((x e. CC /\ y e. H~) -> (x .h y) e. H~)
22	20, 21	sylan 448	. . . . . . . 8 |- (((x e. CC /\ y e. H~) /\ z e. H~) -> ((x .h y) +h z) e. H~)
23	19, 22	sylan2 451	. . . . . . 7 |- ((A e. CC /\ ((x e. CC /\ y e. H~) /\ z e. H~)) -> ((A .op T)` ((x .h y) +h z)) = (A .h (T` ((x .h y) +h z))))
24		homvalt 9475	. . . . . . . . . . . . 13 |- ((A e. CC /\ T:H~-->H~ /\ y e. H~) -> ((A .op T)` y) = (A .h (T` y)))
25	2, 24	mp3an2 903	. . . . . . . . . . . 12 |- ((A e. CC /\ y e. H~) -> ((A .op T)` y) = (A .h (T` y)))
26	25	adantrl 394	. . . . . . . . . . 11 |- ((A e. CC /\ (x e. CC /\ y e. H~)) -> ((A .op T)` y) = (A .h (T` y)))
27	26	opreq2d 3971	. . . . . . . . . 10 |- ((A e. CC /\ (x e. CC /\ y e. H~)) -> (x .h ((A .op T)` y)) = (x .h (A .h (T` y))))
28		hvmulcomt 8867	. . . . . . . . . . . 12 |- ((A e. CC /\ x e. CC /\ (T` y) e. H~) -> (A .h (x .h (T` y))) = (x .h (A .h (T` y))))
29	28, 12	syl3an3 860	. . . . . . . . . . 11 |- ((A e. CC /\ x e. CC /\ y e. H~) -> (A .h (x .h (T` y))) = (x .h (A .h (T` y))))
30	29	3expb 833	. . . . . . . . . 10 |- ((A e. CC /\ (x e. CC /\ y e. H~)) -> (A .h (x .h (T` y))) = (x .h (A .h (T` y))))
31	27, 30	eqtr4d 1508	. . . . . . . . 9 |- ((A e. CC /\ (x e. CC /\ y e. H~)) -> (x .h ((A .op T)` y)) = (A .h (x .h (T` y))))
32		homvalt 9475	. . . . . . . . . 10 |- ((A e. CC /\ T:H~-->H~ /\ z e. H~) -> ((A .op T)` z) = (A .h (T` z)))
33	2, 32	mp3an2 903	. . . . . . . . 9 |- ((A e. CC /\ z e. H~) -> ((A .op T)` z) = (A .h (T` z)))
34	31, 33	opreqan12d 3974	. . . . . . . 8 |- (((A e. CC /\ (x e. CC /\ y e. H~)) /\ (A e. CC /\ z e. H~)) -> ((x .h ((A .op T)` y)) +h ((A .op T)` z)) = ((A .h (x .h (T` y))) +h (A .h (T` z))))
35	34	anandis 512	. . . . . . 7 |- ((A e. CC /\ ((x e. CC /\ y e. H~) /\ z e. H~)) -> ((x .h ((A .op T)` y)) +h ((A .op T)` z)) = ((A .h (x .h (T` y))) +h (A .h (T` z))))
36	17, 23, 35	3eqtr4d 1515	. . . . . 6 |- ((A e. CC /\ ((x e. CC /\ y e. H~) /\ z e. H~)) -> ((A .op T)` ((x .h y) +h z)) = ((x .h ((A .op T)` y)) +h ((A .op T)` z)))
37	36	exp32 377	. . . . 5 |- (A e. CC -> ((x e. CC /\ y e. H~) -> (z e. H~ -> ((A .op T)` ((x .h y) +h z)) = ((x .h ((A .op T)` y)) +h ((A .op T)` z)))))
38	37	r19.21adv 1716	. . . 4 |- (A e. CC -> ((x e. CC /\ y e. H~) -> A.z e. H~ ((A .op T)` ((x .h y) +h z)) = ((x .h ((A .op T)` y)) +h ((A .op T)` z))))
39	38	r19.21aivv 1718	. . 3 |- (A e. CC -> A.x e. CC A.y e. H~ A.z e. H~ ((A .op T)` ((x .h y) +h z)) = ((x .h ((A .op T)` y)) +h ((A .op T)` z)))
40	4, 39	jca 288	. 2 |- (A e. CC -> ((A .op T):H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ ((A .op T)` ((x .h y) +h z)) = ((x .h ((A .op T)` y)) +h ((A .op T)` z))))
41		ellnopt 9741	. 2 |- ((A .op T) e. LinOp <-> ((A .op T):H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ ((A .op T)` ((x .h y) +h z)) = ((x .h ((A .op T)` y)) +h ((A .op T)` z))))
42	40, 41	sylibr 200	1 |- (A e. CC -> (A .op T) e. LinOp)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1643  -->wf 3174  ` cfv 3178  (class class class)co 3958  CCcc 5215  H~chil 8743   +h cva 8744   .h csm 8745   .op chot 8763  LinOpclo 8771

This theorem is referenced by:  lnophd 9883  bdophm 9918

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608  ax-hilex 8824  ax-hfvadd 8825  ax-hfvmul 8830  ax-hvmulass 8832  ax-hvdistr1 8833

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3