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Mirrors > Home > HSE Home > Th. List > hocsubdiri | Structured version Visualization version GIF version |
Description: Distributive law for Hilbert space operator difference. (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 |
---|---|
hocsubdiri | ⊢ ((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 | hosubcli 29552 | . . . . 5 ⊢ (𝑅 −op 𝑆): ℋ⟶ ℋ |
4 | hods.3 | . . . . 5 ⊢ 𝑇: ℋ⟶ ℋ | |
5 | 3, 4 | hocoi 29547 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 −op 𝑆) ∘ 𝑇)‘𝑥) = ((𝑅 −op 𝑆)‘(𝑇‘𝑥))) |
6 | 1, 4 | hocofi 29549 | . . . . . 6 ⊢ (𝑅 ∘ 𝑇): ℋ⟶ ℋ |
7 | 2, 4 | hocofi 29549 | . . . . . 6 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ |
8 | hodval 29525 | . . . . . 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 1448 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇) −op (𝑆 ∘ 𝑇))‘𝑥) = (((𝑅 ∘ 𝑇)‘𝑥) −ℎ ((𝑆 ∘ 𝑇)‘𝑥))) |
10 | 4 | ffvelrni 6827 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
11 | hodval 29525 | . . . . . . . 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 1448 | . . . . . . 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 29547 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑇)‘𝑥) = (𝑅‘(𝑇‘𝑥))) |
15 | 2, 4 | hocoi 29547 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑥) = (𝑆‘(𝑇‘𝑥))) |
16 | 14, 15 | oveq12d 7153 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇)‘𝑥) −ℎ ((𝑆 ∘ 𝑇)‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) −ℎ (𝑆‘(𝑇‘𝑥)))) |
17 | 13, 16 | eqtr4d 2836 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 −op 𝑆)‘(𝑇‘𝑥)) = (((𝑅 ∘ 𝑇)‘𝑥) −ℎ ((𝑆 ∘ 𝑇)‘𝑥))) |
18 | 9, 17 | eqtr4d 2836 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇) −op (𝑆 ∘ 𝑇))‘𝑥) = ((𝑅 −op 𝑆)‘(𝑇‘𝑥))) |
19 | 5, 18 | eqtr4d 2836 | . . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 −op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) −op (𝑆 ∘ 𝑇))‘𝑥)) |
20 | 19 | rgen 3116 | . 2 ⊢ ∀𝑥 ∈ ℋ (((𝑅 −op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) −op (𝑆 ∘ 𝑇))‘𝑥) |
21 | 3, 4 | hocofi 29549 | . . 3 ⊢ ((𝑅 −op 𝑆) ∘ 𝑇): ℋ⟶ ℋ |
22 | 6, 7 | hosubcli 29552 | . . 3 ⊢ ((𝑅 ∘ 𝑇) −op (𝑆 ∘ 𝑇)): ℋ⟶ ℋ |
23 | 21, 22 | hoeqi 29544 | . 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 233 | 1 ⊢ ((𝑅 −op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) −op (𝑆 ∘ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℋchba 28702 −ℎ cmv 28708 −op chod 28723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-hilex 28782 ax-hfvadd 28783 ax-hfvmul 28788 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-neg 10862 df-hvsub 28754 df-hodif 29515 |
This theorem is referenced by: hocsubdir 29568 unierri 29887 pjclem3 29980 |
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