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Mirrors > Home > HSE Home > Th. List > hoddii | Structured version Visualization version GIF version |
Description: Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 29718 does not require linearity.) (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hoddi.1 | ⊢ 𝑅 ∈ LinOp |
hoddi.2 | ⊢ 𝑆: ℋ⟶ ℋ |
hoddi.3 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
hoddii | ⊢ (𝑅 ∘ (𝑆 −op 𝑇)) = ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoddi.2 | . . . . . . 7 ⊢ 𝑆: ℋ⟶ ℋ | |
2 | 1 | ffvelrni 6863 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
3 | hoddi.3 | . . . . . . 7 ⊢ 𝑇: ℋ⟶ ℋ | |
4 | 3 | ffvelrni 6863 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
5 | hoddi.1 | . . . . . . 7 ⊢ 𝑅 ∈ LinOp | |
6 | 5 | lnopsubi 29912 | . . . . . 6 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → (𝑅‘((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) = ((𝑅‘(𝑆‘𝑥)) −ℎ (𝑅‘(𝑇‘𝑥)))) |
7 | 2, 4, 6 | syl2anc 587 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑅‘((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) = ((𝑅‘(𝑆‘𝑥)) −ℎ (𝑅‘(𝑇‘𝑥)))) |
8 | hodval 29680 | . . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) | |
9 | 1, 3, 8 | mp3an12 1452 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
10 | 9 | fveq2d 6681 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑅‘((𝑆 −op 𝑇)‘𝑥)) = (𝑅‘((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))) |
11 | 5 | lnopfi 29907 | . . . . . . 7 ⊢ 𝑅: ℋ⟶ ℋ |
12 | 11, 1 | hocoi 29702 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑆)‘𝑥) = (𝑅‘(𝑆‘𝑥))) |
13 | 11, 3 | hocoi 29702 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑇)‘𝑥) = (𝑅‘(𝑇‘𝑥))) |
14 | 12, 13 | oveq12d 7191 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑆)‘𝑥) −ℎ ((𝑅 ∘ 𝑇)‘𝑥)) = ((𝑅‘(𝑆‘𝑥)) −ℎ (𝑅‘(𝑇‘𝑥)))) |
15 | 7, 10, 14 | 3eqtr4d 2784 | . . . 4 ⊢ (𝑥 ∈ ℋ → (𝑅‘((𝑆 −op 𝑇)‘𝑥)) = (((𝑅 ∘ 𝑆)‘𝑥) −ℎ ((𝑅 ∘ 𝑇)‘𝑥))) |
16 | 1, 3 | hosubcli 29707 | . . . . 5 ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ |
17 | 11, 16 | hocoi 29702 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ (𝑆 −op 𝑇))‘𝑥) = (𝑅‘((𝑆 −op 𝑇)‘𝑥))) |
18 | 11, 1 | hocofi 29704 | . . . . 5 ⊢ (𝑅 ∘ 𝑆): ℋ⟶ ℋ |
19 | 11, 3 | hocofi 29704 | . . . . 5 ⊢ (𝑅 ∘ 𝑇): ℋ⟶ ℋ |
20 | hodval 29680 | . . . . 5 ⊢ (((𝑅 ∘ 𝑆): ℋ⟶ ℋ ∧ (𝑅 ∘ 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆)‘𝑥) −ℎ ((𝑅 ∘ 𝑇)‘𝑥))) | |
21 | 18, 19, 20 | mp3an12 1452 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆)‘𝑥) −ℎ ((𝑅 ∘ 𝑇)‘𝑥))) |
22 | 15, 17, 21 | 3eqtr4d 2784 | . . 3 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ (𝑆 −op 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥)) |
23 | 22 | rgen 3064 | . 2 ⊢ ∀𝑥 ∈ ℋ ((𝑅 ∘ (𝑆 −op 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥) |
24 | 11, 16 | hocofi 29704 | . . 3 ⊢ (𝑅 ∘ (𝑆 −op 𝑇)): ℋ⟶ ℋ |
25 | 18, 19 | hosubcli 29707 | . . 3 ⊢ ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇)): ℋ⟶ ℋ |
26 | 24, 25 | hoeqi 29699 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑅 ∘ (𝑆 −op 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥) ↔ (𝑅 ∘ (𝑆 −op 𝑇)) = ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))) |
27 | 23, 26 | mpbi 233 | 1 ⊢ (𝑅 ∘ (𝑆 −op 𝑇)) = ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 ∀wral 3054 ∘ ccom 5530 ⟶wf 6336 ‘cfv 6340 (class class class)co 7173 ℋchba 28857 −ℎ cmv 28863 −op chod 28878 LinOpclo 28885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-hilex 28937 ax-hfvadd 28938 ax-hvass 28940 ax-hv0cl 28941 ax-hvaddid 28942 ax-hfvmul 28943 ax-hvmulid 28944 ax-hvdistr2 28947 ax-hvmul0 28948 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-po 5443 df-so 5444 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-er 8323 df-map 8442 df-en 8559 df-dom 8560 df-sdom 8561 df-pnf 10758 df-mnf 10759 df-ltxr 10761 df-sub 10953 df-neg 10954 df-hvsub 28909 df-hodif 29670 df-lnop 29779 |
This theorem is referenced by: hoddi 29928 unierri 30042 |
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