Theorem hoaddcom 9638

Description: Commutativity of sum of Hilbert space operators.

Hypotheses

Ref	Expression
hoeq.1	|- S:H~-->H~
hoeq.2	|- T:H~-->H~

Assertion

Ref	Expression
hoaddcom	|- (S +op T) = (T +op S)

Proof of Theorem hoaddcom

Step	Hyp	Ref	Expression
1		ax-hvcom 8810	. . . . 5 |- (((S` x) e. H~ /\ (T` x) e. H~) -> ((S` x) +h (T` x)) = ((T` x) +h (S` x)))
2		hoeq.1	. . . . . 6 |- S:H~-->H~
3	2	ffvelrni 3806	. . . . 5 |- (x e. H~ -> (S` x) e. H~)
4		hoeq.2	. . . . . 6 |- T:H~-->H~
5	4	ffvelrni 3806	. . . . 5 |- (x e. H~ -> (T` x) e. H~)
6	1, 3, 5	sylanc 471	. . . 4 |- (x e. H~ -> ((S` x) +h (T` x)) = ((T` x) +h (S` x)))
7		hosvaltOLD 9457	. . . . 5 |- (((S:H~-->H~ /\ T:H~-->H~) /\ x e. H~) -> ((S +op T)` x) = ((S` x) +h (T` x)))
8	2, 4, 7	mpanl12 707	. . . 4 |- (x e. H~ -> ((S +op T)` x) = ((S` x) +h (T` x)))
9		hosvaltOLD 9457	. . . . 5 |- (((T:H~-->H~ /\ S:H~-->H~) /\ x e. H~) -> ((T +op S)` x) = ((T` x) +h (S` x)))
10	4, 2, 9	mpanl12 707	. . . 4 |- (x e. H~ -> ((T +op S)` x) = ((T` x) +h (S` x)))
11	6, 8, 10	3eqtr4d 1514	. . 3 |- (x e. H~ -> ((S +op T)` x) = ((T +op S)` x))
12	11	rgen 1695	. 2 |- A.x e. H~ ((S +op T)` x) = ((T +op S)` x)
13	2, 4	hoaddcl 9634	. . 3 |- (S +op T):H~-->H~
14	4, 2	hoaddcl 9634	. . 3 |- (T +op S):H~-->H~
15	13, 14	hoeq 9627	. 2 |- (A.x e. H~ ((S +op T)` x) = ((T +op S)` x) <-> (S +op T) = (T +op S))
16	12, 15	mpbi 189	1 |- (S +op T) = (T +op S)

Syntax hints:   = wceq 954   e. wcel 956  A.wral 1642  -->wf 3173  ` cfv 3177  (class class class)co 3954  H~chil 8727   +h cva 8728   +op chos 8746

This theorem is referenced by:  hoaddcomt 9640  hoadd12 9643  hoadd23 9644  hoaddsub 9687  hosd1 9688  hosubeq0 9692

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-hilex 8808  ax-hfvadd 8809  ax-hvcom 8810

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-opr 3956  df-oprab 3957  df-map 4314  df-hosum 9446

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