Theorem lnopm 9840

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 9809	. . . 4 |- T:H~-->H~
3		homulclt 9602	. . . 4 |- ((A e. CC /\ T:H~-->H~) -> (A .op T):H~-->H~)
4	2, 3	mpan2 694	. . 3 |- (A e. CC -> (A .op T):H~-->H~)
5	1	lnopl 9808	. . . . . . . . . . 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 831	. . . . . . . . . 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 3961	. . . . . . . . 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 8799	. . . . . . . . . 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 8804	. . . . . . . . . . 11 |- ((x e. CC /\ (T` y) e. H~) -> (x .h (T` y)) e. H~)
12	2	ffvelrni 3800	. . . . . . . . . . 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 3800	. . . . . . . . . 10 |- (z e. H~ -> (T` z) e. H~)
15	9, 10, 13, 14	syl3an 866	. . . . . . . . 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 832	. . . . . . . 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 1499	. . . . . . 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 9435	. . . . . . . . 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 901	. . . . . . . 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 8803	. . . . . . . . 9 |- (((x .h y) e. H~ /\ z e. H~) -> ((x .h y) +h z) e. H~)
21		hvmulclt 8804	. . . . . . . . 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 9435	. . . . . . . . . . . . 13 |- ((A e. CC /\ T:H~-->H~ /\ y e. H~) -> ((A .op T)` y) = (A .h (T` y)))
25	2, 24	mp3an2 901	. . . . . . . . . . . 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 3961	. . . . . . . . . 10 |- ((A e. CC /\ (x e. CC /\ y e. H~)) -> (x .h ((A .op T)` y)) = (x .h (A .h (T` y))))
28		hvmulcomt 8833	. . . . . . . . . . . 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 859	. . . . . . . . . . 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 832	. . . . . . . . . 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 1502	. . . . . . . . 9 |- ((A e. CC /\ (x e. CC /\ y e. H~)) -> (x .h ((A .op T)` y)) = (A .h (x .h (T` y))))
32		homvalt 9435	. . . . . . . . . 10 |- ((A e. CC /\ T:H~-->H~ /\ z e. H~) -> ((A .op T)` z) = (A .h (T` z)))
33	2, 32	mp3an2 901	. . . . . . . . 9 |- ((A e. CC /\ z e. H~) -> ((A .op T)` z) = (A .h (T` z)))
34	31, 33	opreqan12d 3964	. . . . . . . 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 511	. . . . . . 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 1509	. . . . . 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 1710	. . . 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 1712	. . 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 9701	. 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 953   e. wcel 955  A.wral 1637  -->wf 3168  ` cfv 3172  (class class class)co 3948  CCcc 5204  H~chil 8727   +h cva 8728   .h csm 8729   .op chot 8747  LinOpclo 8755

This theorem is referenced by:  lnophd 9842  bdophm 9877

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-hilex 8790  ax-hfvadd 8791  ax-hfvmul 8796  ax-hvmulass 8798  ax-hvdistr1 8799

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3