Theorem hods 9696

Description: Relationship between Hilbert space operator difference and sum.

Hypotheses

Ref	Expression
hods.1	|- R:H~-->H~
hods.2	|- S:H~-->H~
hods.3	|- T:H~-->H~

Assertion

Ref	Expression
hods	|- ((R -op S) = T <-> (S +op T) = R)

Proof of Theorem hods

Step	Hyp	Ref	Expression
1		hvsubaddt 8939	. . . . 5 |- (((R` x) e. H~ /\ (S` x) e. H~ /\ (T` x) e. H~) -> (((R` x) -h (S` x)) = (T` x) <-> ((S` x) +h (T` x)) = (R` x)))
2		hods.1	. . . . . 6 |- R:H~-->H~
3	2	ffvelrni 3821	. . . . 5 |- (x e. H~ -> (R` x) e. H~)
4		hods.2	. . . . . 6 |- S:H~-->H~
5	4	ffvelrni 3821	. . . . 5 |- (x e. H~ -> (S` x) e. H~)
6		hods.3	. . . . . 6 |- T:H~-->H~
7	6	ffvelrni 3821	. . . . 5 |- (x e. H~ -> (T` x) e. H~)
8	1, 3, 5, 7	syl3anc 860	. . . 4 |- (x e. H~ -> (((R` x) -h (S` x)) = (T` x) <-> ((S` x) +h (T` x)) = (R` x)))
9		hodvaltOLD 9515	. . . . . 6 |- (((R:H~-->H~ /\ S:H~-->H~) /\ x e. H~) -> ((R -op S)` x) = ((R` x) -h (S` x)))
10	2, 4, 9	mpanl12 710	. . . . 5 |- (x e. H~ -> ((R -op S)` x) = ((R` x) -h (S` x)))
11	10	eqeq1d 1486	. . . 4 |- (x e. H~ -> (((R -op S)` x) = (T` x) <-> ((R` x) -h (S` x)) = (T` x)))
12		hosvaltOLD 9512	. . . . . 6 |- (((S:H~-->H~ /\ T:H~-->H~) /\ x e. H~) -> ((S +op T)` x) = ((S` x) +h (T` x)))
13	4, 6, 12	mpanl12 710	. . . . 5 |- (x e. H~ -> ((S +op T)` x) = ((S` x) +h (T` x)))
14	13	eqeq1d 1486	. . . 4 |- (x e. H~ -> (((S +op T)` x) = (R` x) <-> ((S` x) +h (T` x)) = (R` x)))
15	8, 11, 14	3bitr4d 552	. . 3 |- (x e. H~ -> (((R -op S)` x) = (T` x) <-> ((S +op T)` x) = (R` x)))
16	15	ralbiia 1676	. 2 |- (A.x e. H~ ((R -op S)` x) = (T` x) <-> A.x e. H~ ((S +op T)` x) = (R` x))
17	2, 4	hosubcl 9690	. . 3 |- (R -op S):H~-->H~
18	17, 6	hoeq 9682	. 2 |- (A.x e. H~ ((R -op S)` x) = (T` x) <-> (R -op S) = T)
19	4, 6	hoaddcl 9689	. . 3 |- (S +op T):H~-->H~
20	19, 2	hoeq 9682	. 2 |- (A.x e. H~ ((S +op T)` x) = (R` x) <-> (S +op T) = R)
21	16, 18, 20	3bitr3 181	1 |- ((R -op S) = T <-> (S +op T) = R)

Syntax hints:   <-> wb 146   = wceq 958   e. wcel 960  A.wral 1648  -->wf 3184  ` cfv 3188  (class class class)co 3969  H~chil 8783   +h cva 8784   -h cmv 8787   +op chos 8802   -op chod 8804

This theorem is referenced by:  hodid 9708  hodseq 9715  ho0sub 9716  hosd1 9743  pjoc 10103

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-9 967  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-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634  ax-hilex 8864  ax-hfvadd 8865  ax-hvcom 8866  ax-hvass 8867  ax-hv0cl 8868  ax-hvaddid 8869  ax-hfvmul 8870  ax-hvmulid 8871  ax-hvdistr2 8874  ax-hvmul0 8875

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  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-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  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-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-map 4330  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-sub 5368  df-neg 5370  df-hvsub 8835  df-hosum 9501  df-hodif 9503

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