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| Mirrors > Home > HSE Home > Th. List > hoadd32i | Structured version Visualization version GIF version | ||
| Description: Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| hods.1 | ⊢ 𝑅: ℋ⟶ ℋ | 
| hods.2 | ⊢ 𝑆: ℋ⟶ ℋ | 
| hods.3 | ⊢ 𝑇: ℋ⟶ ℋ | 
| Ref | Expression | 
|---|---|
| hoadd32i | ⊢ ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hods.2 | . . . 4 ⊢ 𝑆: ℋ⟶ ℋ | |
| 2 | hods.3 | . . . 4 ⊢ 𝑇: ℋ⟶ ℋ | |
| 3 | 1, 2 | hoaddcomi 31792 | . . 3 ⊢ (𝑆 +op 𝑇) = (𝑇 +op 𝑆) | 
| 4 | 3 | oveq2i 7443 | . 2 ⊢ (𝑅 +op (𝑆 +op 𝑇)) = (𝑅 +op (𝑇 +op 𝑆)) | 
| 5 | hods.1 | . . 3 ⊢ 𝑅: ℋ⟶ ℋ | |
| 6 | 5, 1, 2 | hoaddassi 31796 | . 2 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇)) | 
| 7 | 5, 2, 1 | hoaddassi 31796 | . 2 ⊢ ((𝑅 +op 𝑇) +op 𝑆) = (𝑅 +op (𝑇 +op 𝑆)) | 
| 8 | 4, 6, 7 | 3eqtr4i 2774 | 1 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ⟶wf 6556 (class class class)co 7432 ℋchba 30939 +op chos 30958 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-hilex 31019 ax-hfvadd 31020 ax-hvcom 31021 ax-hvass 31022 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-hosum 31750 | 
| This theorem is referenced by: hosubeq0i 31846 | 
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