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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 30142 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 6960 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
3 | hoddi.3 | . . . . . . 7 ⊢ 𝑇: ℋ⟶ ℋ | |
4 | 3 | ffvelrni 6960 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
5 | hoddi.1 | . . . . . . 7 ⊢ 𝑅 ∈ LinOp | |
6 | 5 | lnopsubi 30336 | . . . . . 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 584 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑅‘((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) = ((𝑅‘(𝑆‘𝑥)) −ℎ (𝑅‘(𝑇‘𝑥)))) |
8 | hodval 30104 | . . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) | |
9 | 1, 3, 8 | mp3an12 1450 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
10 | 9 | fveq2d 6778 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑅‘((𝑆 −op 𝑇)‘𝑥)) = (𝑅‘((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))) |
11 | 5 | lnopfi 30331 | . . . . . . 7 ⊢ 𝑅: ℋ⟶ ℋ |
12 | 11, 1 | hocoi 30126 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑆)‘𝑥) = (𝑅‘(𝑆‘𝑥))) |
13 | 11, 3 | hocoi 30126 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑇)‘𝑥) = (𝑅‘(𝑇‘𝑥))) |
14 | 12, 13 | oveq12d 7293 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑆)‘𝑥) −ℎ ((𝑅 ∘ 𝑇)‘𝑥)) = ((𝑅‘(𝑆‘𝑥)) −ℎ (𝑅‘(𝑇‘𝑥)))) |
15 | 7, 10, 14 | 3eqtr4d 2788 | . . . 4 ⊢ (𝑥 ∈ ℋ → (𝑅‘((𝑆 −op 𝑇)‘𝑥)) = (((𝑅 ∘ 𝑆)‘𝑥) −ℎ ((𝑅 ∘ 𝑇)‘𝑥))) |
16 | 1, 3 | hosubcli 30131 | . . . . 5 ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ |
17 | 11, 16 | hocoi 30126 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ (𝑆 −op 𝑇))‘𝑥) = (𝑅‘((𝑆 −op 𝑇)‘𝑥))) |
18 | 11, 1 | hocofi 30128 | . . . . 5 ⊢ (𝑅 ∘ 𝑆): ℋ⟶ ℋ |
19 | 11, 3 | hocofi 30128 | . . . . 5 ⊢ (𝑅 ∘ 𝑇): ℋ⟶ ℋ |
20 | hodval 30104 | . . . . 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 1450 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆)‘𝑥) −ℎ ((𝑅 ∘ 𝑇)‘𝑥))) |
22 | 15, 17, 21 | 3eqtr4d 2788 | . . 3 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ (𝑆 −op 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥)) |
23 | 22 | rgen 3074 | . 2 ⊢ ∀𝑥 ∈ ℋ ((𝑅 ∘ (𝑆 −op 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥) |
24 | 11, 16 | hocofi 30128 | . . 3 ⊢ (𝑅 ∘ (𝑆 −op 𝑇)): ℋ⟶ ℋ |
25 | 18, 19 | hosubcli 30131 | . . 3 ⊢ ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇)): ℋ⟶ ℋ |
26 | 24, 25 | hoeqi 30123 | . 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 229 | 1 ⊢ (𝑅 ∘ (𝑆 −op 𝑇)) = ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∘ ccom 5593 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℋchba 29281 −ℎ cmv 29287 −op chod 29302 LinOpclo 29309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-hilex 29361 ax-hfvadd 29362 ax-hvass 29364 ax-hv0cl 29365 ax-hvaddid 29366 ax-hfvmul 29367 ax-hvmulid 29368 ax-hvdistr2 29371 ax-hvmul0 29372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-ltxr 11014 df-sub 11207 df-neg 11208 df-hvsub 29333 df-hodif 30094 df-lnop 30203 |
This theorem is referenced by: hoddi 30352 unierri 30466 |
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