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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 28023 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 6266 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
3 | hoddi.3 | . . . . . . 7 ⊢ 𝑇: ℋ⟶ ℋ | |
4 | 3 | ffvelrni 6266 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
5 | hoddi.1 | . . . . . . 7 ⊢ 𝑅 ∈ LinOp | |
6 | 5 | lnopsubi 28217 | . . . . . 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 691 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑅‘((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) = ((𝑅‘(𝑆‘𝑥)) −ℎ (𝑅‘(𝑇‘𝑥)))) |
8 | hodval 27985 | . . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) | |
9 | 1, 3, 8 | mp3an12 1406 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
10 | 9 | fveq2d 6107 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑅‘((𝑆 −op 𝑇)‘𝑥)) = (𝑅‘((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))) |
11 | 5 | lnopfi 28212 | . . . . . . 7 ⊢ 𝑅: ℋ⟶ ℋ |
12 | 11, 1 | hocoi 28007 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑆)‘𝑥) = (𝑅‘(𝑆‘𝑥))) |
13 | 11, 3 | hocoi 28007 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑇)‘𝑥) = (𝑅‘(𝑇‘𝑥))) |
14 | 12, 13 | oveq12d 6567 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑆)‘𝑥) −ℎ ((𝑅 ∘ 𝑇)‘𝑥)) = ((𝑅‘(𝑆‘𝑥)) −ℎ (𝑅‘(𝑇‘𝑥)))) |
15 | 7, 10, 14 | 3eqtr4d 2654 | . . . 4 ⊢ (𝑥 ∈ ℋ → (𝑅‘((𝑆 −op 𝑇)‘𝑥)) = (((𝑅 ∘ 𝑆)‘𝑥) −ℎ ((𝑅 ∘ 𝑇)‘𝑥))) |
16 | 1, 3 | hosubcli 28012 | . . . . 5 ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ |
17 | 11, 16 | hocoi 28007 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ (𝑆 −op 𝑇))‘𝑥) = (𝑅‘((𝑆 −op 𝑇)‘𝑥))) |
18 | 11, 1 | hocofi 28009 | . . . . 5 ⊢ (𝑅 ∘ 𝑆): ℋ⟶ ℋ |
19 | 11, 3 | hocofi 28009 | . . . . 5 ⊢ (𝑅 ∘ 𝑇): ℋ⟶ ℋ |
20 | hodval 27985 | . . . . 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 1406 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆)‘𝑥) −ℎ ((𝑅 ∘ 𝑇)‘𝑥))) |
22 | 15, 17, 21 | 3eqtr4d 2654 | . . 3 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ (𝑆 −op 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥)) |
23 | 22 | rgen 2906 | . 2 ⊢ ∀𝑥 ∈ ℋ ((𝑅 ∘ (𝑆 −op 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥) |
24 | 11, 16 | hocofi 28009 | . . 3 ⊢ (𝑅 ∘ (𝑆 −op 𝑇)): ℋ⟶ ℋ |
25 | 18, 19 | hosubcli 28012 | . . 3 ⊢ ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇)): ℋ⟶ ℋ |
26 | 24, 25 | hoeqi 28004 | . 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 219 | 1 ⊢ (𝑅 ∘ (𝑆 −op 𝑇)) = ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℋchil 27160 −ℎ cmv 27166 −op chod 27181 LinOpclo 27188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-hilex 27240 ax-hfvadd 27241 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvdistr2 27250 ax-hvmul0 27251 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 df-hvsub 27212 df-hodif 27975 df-lnop 28084 |
This theorem is referenced by: hoddi 28233 unierri 28347 |
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