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| Description: Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 31799 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 | ffvelcdmi 7103 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) | 
| 3 | hoddi.3 | . . . . . . 7 ⊢ 𝑇: ℋ⟶ ℋ | |
| 4 | 3 | ffvelcdmi 7103 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) | 
| 5 | hoddi.1 | . . . . . . 7 ⊢ 𝑅 ∈ LinOp | |
| 6 | 5 | lnopsubi 31993 | . . . . . 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 31761 | . . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) | |
| 9 | 1, 3, 8 | mp3an12 1453 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) | 
| 10 | 9 | fveq2d 6910 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑅‘((𝑆 −op 𝑇)‘𝑥)) = (𝑅‘((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))) | 
| 11 | 5 | lnopfi 31988 | . . . . . . 7 ⊢ 𝑅: ℋ⟶ ℋ | 
| 12 | 11, 1 | hocoi 31783 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑆)‘𝑥) = (𝑅‘(𝑆‘𝑥))) | 
| 13 | 11, 3 | hocoi 31783 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑇)‘𝑥) = (𝑅‘(𝑇‘𝑥))) | 
| 14 | 12, 13 | oveq12d 7449 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑆)‘𝑥) −ℎ ((𝑅 ∘ 𝑇)‘𝑥)) = ((𝑅‘(𝑆‘𝑥)) −ℎ (𝑅‘(𝑇‘𝑥)))) | 
| 15 | 7, 10, 14 | 3eqtr4d 2787 | . . . 4 ⊢ (𝑥 ∈ ℋ → (𝑅‘((𝑆 −op 𝑇)‘𝑥)) = (((𝑅 ∘ 𝑆)‘𝑥) −ℎ ((𝑅 ∘ 𝑇)‘𝑥))) | 
| 16 | 1, 3 | hosubcli 31788 | . . . . 5 ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ | 
| 17 | 11, 16 | hocoi 31783 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ (𝑆 −op 𝑇))‘𝑥) = (𝑅‘((𝑆 −op 𝑇)‘𝑥))) | 
| 18 | 11, 1 | hocofi 31785 | . . . . 5 ⊢ (𝑅 ∘ 𝑆): ℋ⟶ ℋ | 
| 19 | 11, 3 | hocofi 31785 | . . . . 5 ⊢ (𝑅 ∘ 𝑇): ℋ⟶ ℋ | 
| 20 | hodval 31761 | . . . . 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 1453 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆)‘𝑥) −ℎ ((𝑅 ∘ 𝑇)‘𝑥))) | 
| 22 | 15, 17, 21 | 3eqtr4d 2787 | . . 3 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ (𝑆 −op 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥)) | 
| 23 | 22 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ ℋ ((𝑅 ∘ (𝑆 −op 𝑇))‘𝑥) = (((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))‘𝑥) | 
| 24 | 11, 16 | hocofi 31785 | . . 3 ⊢ (𝑅 ∘ (𝑆 −op 𝑇)): ℋ⟶ ℋ | 
| 25 | 18, 19 | hosubcli 31788 | . . 3 ⊢ ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇)): ℋ⟶ ℋ | 
| 26 | 24, 25 | hoeqi 31780 | . 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 230 | 1 ⊢ (𝑅 ∘ (𝑆 −op 𝑇)) = ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℋchba 30938 −ℎ cmv 30944 −op chod 30959 LinOpclo 30966 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-hilex 31018 ax-hfvadd 31019 ax-hvass 31021 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvdistr2 31028 ax-hvmul0 31029 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 df-hvsub 30990 df-hodif 31751 df-lnop 31860 | 
| This theorem is referenced by: hoddi 32009 unierri 32123 | 
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