hoadd12i - Hilbert Space Explorer

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Theorem hoadd12i 31926

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 31921	. . 3 ⊢ (𝑅 +_op 𝑆) = (𝑆 +_op 𝑅)
4	3	oveq1i 7402	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = ((𝑆 +_op 𝑅) +_op 𝑇)
5		hods.3	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
6	1, 2, 5	hoaddassi 31925	. 2 ⊢ ((𝑅 +_op 𝑆) +_op 𝑇) = (𝑅 +_op (𝑆 +_op 𝑇))
7	2, 1, 5	hoaddassi 31925	. 2 ⊢ ((𝑆 +_op 𝑅) +_op 𝑇) = (𝑆 +_op (𝑅 +_op 𝑇))
8	4, 6, 7	3eqtr3i 2792	1 ⊢ (𝑅 +_op (𝑆 +_op 𝑇)) = (𝑆 +_op (𝑅 +_op 𝑇))

Colors of variables: wff setvar class

Syntax hints: = wceq 1559 ⟶wf 6513 (class class class)co 7392 ℋchba 31068 +_op chos 31087

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-hilex 31148 ax-hfvadd 31149 ax-hvcom 31150 ax-hvass 31151

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-ov 7395 df-oprab 7396 df-mpo 7397 df-map 8805 df-hosum 31879

This theorem is referenced by: ho0subi 31944

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