Theorem hoadd12i 31539

Description: Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hods.1	⊢ 𝑅: ℋ⟶ ℋ
hods.2	⊢ 𝑆: ℋ⟶ ℋ
hods.3	⊢ 𝑇: ℋ⟶ ℋ

Assertion

Ref	Expression
hoadd12i	⊢ (𝑅 +op (𝑆 +op 𝑇)) = (𝑆 +op (𝑅 +op 𝑇))

Proof of Theorem hoadd12i

Step	Hyp	Ref	Expression
1		hods.1	. . . 4 ⊢ 𝑅: ℋ⟶ ℋ
2		hods.2	. . . 4 ⊢ 𝑆: ℋ⟶ ℋ
3	1, 2	hoaddcomi 31534	. . 3 ⊢ (𝑅 +op 𝑆) = (𝑆 +op 𝑅)
4	3	oveq1i 7415	. 2 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = ((𝑆 +op 𝑅) +op 𝑇)
5		hods.3	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
6	1, 2, 5	hoaddassi 31538	. 2 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))
7	2, 1, 5	hoaddassi 31538	. 2 ⊢ ((𝑆 +op 𝑅) +op 𝑇) = (𝑆 +op (𝑅 +op 𝑇))
8	4, 6, 7	3eqtr3i 2762	1 ⊢ (𝑅 +op (𝑆 +op 𝑇)) = (𝑆 +op (𝑅 +op 𝑇))

Syntax hints:   = wceq 1533  ⟶wf 6533  (class class class)co 7405   ℋchba 30681   +_op chos 30700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-hilex 30761  ax-hfvadd 30762  ax-hvcom 30763  ax-hvass 30764

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8824  df-hosum 31492

This theorem is referenced by:  ho0subi  31557