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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 29539 | . . . . 5 ⊢ (𝑅 +op 𝑆): ℋ⟶ ℋ |
4 | hods.3 | . . . . 5 ⊢ 𝑇: ℋ⟶ ℋ | |
5 | 3, 4 | hocoi 29535 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇‘𝑥))) |
6 | 1, 4 | hocofi 29537 | . . . . . 6 ⊢ (𝑅 ∘ 𝑇): ℋ⟶ ℋ |
7 | 2, 4 | hocofi 29537 | . . . . . 6 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ |
8 | hosval 29511 | . . . . . 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 1447 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) = (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥))) |
10 | 4 | ffvelrni 6844 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
11 | hosval 29511 | . . . . . . . 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 1447 | . . . . . . 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 29535 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑇)‘𝑥) = (𝑅‘(𝑇‘𝑥))) |
15 | 2, 4 | hocoi 29535 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑥) = (𝑆‘(𝑇‘𝑥))) |
16 | 14, 15 | oveq12d 7168 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) +ℎ (𝑆‘(𝑇‘𝑥)))) |
17 | 13, 16 | eqtr4d 2859 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇‘𝑥)) = (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥))) |
18 | 9, 17 | eqtr4d 2859 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇‘𝑥))) |
19 | 5, 18 | eqtr4d 2859 | . . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥)) |
20 | 19 | rgen 3148 | . 2 ⊢ ∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) |
21 | 3, 4 | hocofi 29537 | . . 3 ⊢ ((𝑅 +op 𝑆) ∘ 𝑇): ℋ⟶ ℋ |
22 | 6, 7 | hoaddcli 29539 | . . 3 ⊢ ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇)): ℋ⟶ ℋ |
23 | 21, 22 | hoeqi 29532 | . 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 232 | 1 ⊢ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∘ ccom 5553 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℋchba 28690 +ℎ cva 28691 +op chos 28709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-hilex 28770 ax-hfvadd 28771 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-hosum 29501 |
This theorem is referenced by: (None) |
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