Theorem hoddi 9914

Description: Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdir 9706 does not require linearity.)

Hypotheses

Ref	Expression
hoddi.1	|- R e. LinOp
hoddi.2	|- S:H~-->H~
hoddi.3	|- T:H~-->H~

Assertion

Ref	Expression
hoddi	|- (R o. (S -op T)) = ((R o. S) -op (R o. T))

Proof of Theorem hoddi

Step	Hyp	Ref	Expression
1		hoddi.1	. . . . . . 7 |- R e. LinOp
2	1	lnopsub 9898	. . . . . 6 |- (((S` x) e. H~ /\ (T` x) e. H~) -> (R` ((S` x) -h (T` x))) = ((R` (S` x)) -h (R` (T` x))))
3		hoddi.2	. . . . . . 7 |- S:H~-->H~
4	3	ffvelrni 3815	. . . . . 6 |- (x e. H~ -> (S` x) e. H~)
5		hoddi.3	. . . . . . 7 |- T:H~-->H~
6	5	ffvelrni 3815	. . . . . 6 |- (x e. H~ -> (T` x) e. H~)
7	2, 4, 6	sylanc 471	. . . . 5 |- (x e. H~ -> (R` ((S` x) -h (T` x))) = ((R` (S` x)) -h (R` (T` x))))
8		hodvaltOLD 9520	. . . . . . 7 |- (((S:H~-->H~ /\ T:H~-->H~) /\ x e. H~) -> ((S -op T)` x) = ((S` x) -h (T` x)))
9	3, 5, 8	mpanl12 708	. . . . . 6 |- (x e. H~ -> ((S -op T)` x) = ((S` x) -h (T` x)))
10	9	fveq2d 3728	. . . . 5 |- (x e. H~ -> (R` ((S -op T)` x)) = (R` ((S` x) -h (T` x))))
11	1	lnopf 9893	. . . . . . 7 |- R:H~-->H~
12	11, 3	hoco 9690	. . . . . 6 |- (x e. H~ -> ((R o. S)` x) = (R` (S` x)))
13	11, 5	hoco 9690	. . . . . 6 |- (x e. H~ -> ((R o. T)` x) = (R` (T` x)))
14	12, 13	opreq12d 3978	. . . . 5 |- (x e. H~ -> (((R o. S)` x) -h ((R o. T)` x)) = ((R` (S` x)) -h (R` (T` x))))
15	7, 10, 14	3eqtr4d 1517	. . . 4 |- (x e. H~ -> (R` ((S -op T)` x)) = (((R o. S)` x) -h ((R o. T)` x)))
16	3, 5	hosubcl 9695	. . . . 5 |- (S -op T):H~-->H~
17	11, 16	hoco 9690	. . . 4 |- (x e. H~ -> ((R o. (S -op T))` x) = (R` ((S -op T)` x)))
18	11, 3	hocof 9692	. . . . 5 |- (R o. S):H~-->H~
19	11, 5	hocof 9692	. . . . 5 |- (R o. T):H~-->H~
20		hodvaltOLD 9520	. . . . 5 |- ((((R o. S):H~-->H~ /\ (R o. T):H~-->H~) /\ x e. H~) -> (((R o. S) -op (R o. T))` x) = (((R o. S)` x) -h ((R o. T)` x)))
21	18, 19, 20	mpanl12 708	. . . 4 |- (x e. H~ -> (((R o. S) -op (R o. T))` x) = (((R o. S)` x) -h ((R o. T)` x)))
22	15, 17, 21	3eqtr4d 1517	. . 3 |- (x e. H~ -> ((R o. (S -op T))` x) = (((R o. S) -op (R o. T))` x))
23	22	rgen 1698	. 2 |- A.x e. H~ ((R o. (S -op T))` x) = (((R o. S) -op (R o. T))` x)
24	11, 16	hocof 9692	. . 3 |- (R o. (S -op T)):H~-->H~
25	18, 19	hosubcl 9695	. . 3 |- ((R o. S) -op (R o. T)):H~-->H~
26	24, 25	hoeq 9687	. 2 |- (A.x e. H~ ((R o. (S -op T))` x) = (((R o. S) -op (R o. T))` x) <-> (R o. (S -op T)) = ((R o. S) -op (R o. T)))
27	23, 26	mpbi 189	1 |- (R o. (S -op T)) = ((R o. S) -op (R o. T))

Syntax hints:   = wceq 956   e. wcel 958  A.wral 1645   o. ccom 3174  -->wf 3178  ` cfv 3182  (class class class)co 3963  H~chil 8788   -h cmv 8792   -op chod 8809  LinOpclo 8816

This theorem is referenced by:  hoddit 9915  unierr 10037

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625  ax-hilex 8869  ax-hfvadd 8870  ax-hvass 8872  ax-hv0cl 8873  ax-hvaddid 8874  ax-hfvmul 8875  ax-hvmulid 8876  ax-hvdistr2 8879  ax-hvmul0 8880

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-map 4324  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-sub 5356  df-neg 5358  df-hvsub 8840  df-hodif 9508  df-lnop 9767

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