Theorem lnoplt 9829

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 3965	. . . . . . . . 9 |- (x = A -> (x .h y) = (A .h y))
2	1	opreq1d 3972	. . . . . . . 8 |- (x = A -> ((x .h y) +h z) = ((A .h y) +h z))
3	2	fveq2d 3725	. . . . . . 7 |- (x = A -> (T` ((x .h y) +h z)) = (T` ((A .h y) +h z)))
4		opreq1 3965	. . . . . . . 8 |- (x = A -> (x .h (T` y)) = (A .h (T` y)))
5	4	opreq1d 3972	. . . . . . 7 |- (x = A -> ((x .h (T` y)) +h (T` z)) = ((A .h (T` y)) +h (T` z)))
6	3, 5	eqeq12d 1488	. . . . . 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 3966	. . . . . . . . 9 |- (y = B -> (A .h y) = (A .h B))
8	7	opreq1d 3972	. . . . . . . 8 |- (y = B -> ((A .h y) +h z) = ((A .h B) +h z))
9	8	fveq2d 3725	. . . . . . 7 |- (y = B -> (T` ((A .h y) +h z)) = (T` ((A .h B) +h z)))
10		fveq2 3721	. . . . . . . . 9 |- (y = B -> (T` y) = (T` B))
11	10	opreq2d 3973	. . . . . . . 8 |- (y = B -> (A .h (T` y)) = (A .h (T` B)))
12	11	opreq1d 3972	. . . . . . 7 |- (y = B -> ((A .h (T` y)) +h (T` z)) = ((A .h (T` B)) +h (T` z)))
13	9, 12	eqeq12d 1488	. . . . . 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 3966	. . . . . . . 8 |- (z = C -> ((A .h B) +h z) = ((A .h B) +h C))
15	14	fveq2d 3725	. . . . . . 7 |- (z = C -> (T` ((A .h B) +h z)) = (T` ((A .h B) +h C)))
16		fveq2 3721	. . . . . . . 8 |- (z = C -> (T` z) = (T` C))
17	16	opreq2d 3973	. . . . . . 7 |- (z = C -> ((A .h (T` B)) +h (T` z)) = ((A .h (T` B)) +h (T` C)))
18	15, 17	eqeq12d 1488	. . . . . 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 1880	. . . . 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 9775	. . . . . 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 833	. . 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 441	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 774   = wceq 955   e. wcel 957  A.wral 1644  -->wf 3175  ` cfv 3179  (class class class)co 3960  CCcc 5219  H~chil 8772   +h cva 8773   .h csm 8774  LinOpclo 8800

This theorem is referenced by:  lnop0t 9881  lnopmult 9882  lnopl 9883

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-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 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-hilex 8853

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fv 3195  df-opr 3962  df-lnop 9758

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