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| Mirrors > Home > HSE Home > Th. List > hocadddiri | Structured version Visualization version GIF version | ||
| Description: Distributive law for Hilbert space operator sum. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| hods.1 | ⊢ 𝑅: ℋ⟶ ℋ | 
| hods.2 | ⊢ 𝑆: ℋ⟶ ℋ | 
| hods.3 | ⊢ 𝑇: ℋ⟶ ℋ | 
| Ref | Expression | 
|---|---|
| hocadddiri | ⊢ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hods.1 | . . . . . 6 ⊢ 𝑅: ℋ⟶ ℋ | |
| 2 | hods.2 | . . . . . 6 ⊢ 𝑆: ℋ⟶ ℋ | |
| 3 | 1, 2 | hoaddcli 31787 | . . . . 5 ⊢ (𝑅 +op 𝑆): ℋ⟶ ℋ | 
| 4 | hods.3 | . . . . 5 ⊢ 𝑇: ℋ⟶ ℋ | |
| 5 | 3, 4 | hocoi 31783 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇‘𝑥))) | 
| 6 | 1, 4 | hocofi 31785 | . . . . . 6 ⊢ (𝑅 ∘ 𝑇): ℋ⟶ ℋ | 
| 7 | 2, 4 | hocofi 31785 | . . . . . 6 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ | 
| 8 | hosval 31759 | . . . . . 6 ⊢ (((𝑅 ∘ 𝑇): ℋ⟶ ℋ ∧ (𝑆 ∘ 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) = (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥))) | |
| 9 | 6, 7, 8 | mp3an12 1453 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) = (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥))) | 
| 10 | 4 | ffvelcdmi 7103 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) | 
| 11 | hosval 31759 | . . . . . . . 8 ⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑅 +op 𝑆)‘(𝑇‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) +ℎ (𝑆‘(𝑇‘𝑥)))) | |
| 12 | 1, 2, 11 | mp3an12 1453 | . . . . . . 7 ⊢ ((𝑇‘𝑥) ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) +ℎ (𝑆‘(𝑇‘𝑥)))) | 
| 13 | 10, 12 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) +ℎ (𝑆‘(𝑇‘𝑥)))) | 
| 14 | 1, 4 | hocoi 31783 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑇)‘𝑥) = (𝑅‘(𝑇‘𝑥))) | 
| 15 | 2, 4 | hocoi 31783 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑥) = (𝑆‘(𝑇‘𝑥))) | 
| 16 | 14, 15 | oveq12d 7449 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) +ℎ (𝑆‘(𝑇‘𝑥)))) | 
| 17 | 13, 16 | eqtr4d 2780 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇‘𝑥)) = (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥))) | 
| 18 | 9, 17 | eqtr4d 2780 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇‘𝑥))) | 
| 19 | 5, 18 | eqtr4d 2780 | . . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥)) | 
| 20 | 19 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) | 
| 21 | 3, 4 | hocofi 31785 | . . 3 ⊢ ((𝑅 +op 𝑆) ∘ 𝑇): ℋ⟶ ℋ | 
| 22 | 6, 7 | hoaddcli 31787 | . . 3 ⊢ ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇)): ℋ⟶ ℋ | 
| 23 | 21, 22 | hoeqi 31780 | . 2 ⊢ (∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) ↔ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))) | 
| 24 | 20, 23 | mpbi 230 | 1 ⊢ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℋchba 30938 +ℎ cva 30939 +op chos 30957 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-hilex 31018 ax-hfvadd 31019 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-hosum 31749 | 
| This theorem is referenced by: (None) | 
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