Theorem lnoplt 9833

Description: Basic linearity property of a linear Hilbert space operator.

Assertion

Ref	Expression
lnoplt	|- (((T e. LinOp /\ A e. CC) /\ (B e. H~ /\ C e. H~)) -> (T` ((A .h B) +h C)) = ((A .h (T` B)) +h (T` C)))

Proof of Theorem lnoplt

Step	Hyp	Ref	Expression
1		opreq1 3974	. . . . . . . . 9 |- (x = A -> (x .h y) = (A .h y))
2	1	opreq1d 3981	. . . . . . . 8 |- (x = A -> ((x .h y) +h z) = ((A .h y) +h z))
3	2	fveq2d 3734	. . . . . . 7 |- (x = A -> (T` ((x .h y) +h z)) = (T` ((A .h y) +h z)))
4		opreq1 3974	. . . . . . . 8 |- (x = A -> (x .h (T` y)) = (A .h (T` y)))
5	4	opreq1d 3981	. . . . . . 7 |- (x = A -> ((x .h (T` y)) +h (T` z)) = ((A .h (T` y)) +h (T` z)))
6	3, 5	eqeq12d 1492	. . . . . 6 |- (x = A -> ((T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z)) <-> (T` ((A .h y) +h z)) = ((A .h (T` y)) +h (T` z))))
7		opreq2 3975	. . . . . . . . 9 |- (y = B -> (A .h y) = (A .h B))
8	7	opreq1d 3981	. . . . . . . 8 |- (y = B -> ((A .h y) +h z) = ((A .h B) +h z))
9	8	fveq2d 3734	. . . . . . 7 |- (y = B -> (T` ((A .h y) +h z)) = (T` ((A .h B) +h z)))
10		fveq2 3730	. . . . . . . . 9 |- (y = B -> (T` y) = (T` B))
11	10	opreq2d 3982	. . . . . . . 8 |- (y = B -> (A .h (T` y)) = (A .h (T` B)))
12	11	opreq1d 3981	. . . . . . 7 |- (y = B -> ((A .h (T` y)) +h (T` z)) = ((A .h (T` B)) +h (T` z)))
13	9, 12	eqeq12d 1492	. . . . . 6 |- (y = B -> ((T` ((A .h y) +h z)) = ((A .h (T` y)) +h (T` z)) <-> (T` ((A .h B) +h z)) = ((A .h (T` B)) +h (T` z))))
14		opreq2 3975	. . . . . . . 8 |- (z = C -> ((A .h B) +h z) = ((A .h B) +h C))
15	14	fveq2d 3734	. . . . . . 7 |- (z = C -> (T` ((A .h B) +h z)) = (T` ((A .h B) +h C)))
16		fveq2 3730	. . . . . . . 8 |- (z = C -> (T` z) = (T` C))
17	16	opreq2d 3982	. . . . . . 7 |- (z = C -> ((A .h (T` B)) +h (T` z)) = ((A .h (T` B)) +h (T` C)))
18	15, 17	eqeq12d 1492	. . . . . 6 |- (z = C -> ((T` ((A .h B) +h z)) = ((A .h (T` B)) +h (T` z)) <-> (T` ((A .h B) +h C)) = ((A .h (T` B)) +h (T` C))))
19	6, 13, 18	rcla43v 1885	. . . . 5 |- ((A e. CC /\ B e. H~ /\ C e. H~) -> (A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z)) -> (T` ((A .h B) +h C)) = ((A .h (T` B)) +h (T` C))))
20		ellnopt 9779	. . . . . 6 |- (T e. LinOp <-> (T:H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z))))
21	20	pm3.27bi 326	. . . . 5 |- (T e. LinOp -> A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z)))
22	19, 21	syl5 21	. . . 4 |- ((A e. CC /\ B e. H~ /\ C e. H~) -> (T e. LinOp -> (T` ((A .h B) +h C)) = ((A .h (T` B)) +h (T` C))))
23	22	3expb 836	. . 3 |- ((A e. CC /\ (B e. H~ /\ C e. H~)) -> (T e. LinOp -> (T` ((A .h B) +h C)) = ((A .h (T` B)) +h (T` C))))
24	23	impcom 351	. 2 |- ((T e. LinOp /\ (A e. CC /\ (B e. H~ /\ C e. H~))) -> (T` ((A .h B) +h C)) = ((A .h (T` B)) +h (T` C)))
25	24	anassrs 443	1 |- (((T e. LinOp /\ A e. CC) /\ (B e. H~ /\ C e. H~)) -> (T` ((A .h B) +h C)) = ((A .h (T` B)) +h (T` C)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  -->wf 3184  ` cfv 3188  (class class class)co 3969  CCcc 5244  H~chil 8783   +h cva 8784   .h csm 8785  LinOpclo 8811

This theorem is referenced by:  lnop0t 9885  lnopmult 9886  lnopl 9887

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-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-pow 2748  ax-pr 2785  ax-un 2872  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  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-opr 3971  df-lnop 9762

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