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Mirrors > Home > HSE Home > Th. List > hodcl | Structured version Visualization version GIF version |
Description: Closure of the difference of two Hilbert space operators. (Contributed by NM, 15-Nov-2002.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hodcl | ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝐴) ∈ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hodval 31489 | . . 3 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝐴) = ((𝑆‘𝐴) −ℎ (𝑇‘𝐴))) | |
2 | ffvelcdm 7074 | . . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝑆‘𝐴) ∈ ℋ) | |
3 | 2 | 3adant2 1128 | . . . 4 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝑆‘𝐴) ∈ ℋ) |
4 | ffvelcdm 7074 | . . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝑇‘𝐴) ∈ ℋ) | |
5 | 4 | 3adant1 1127 | . . . 4 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝑇‘𝐴) ∈ ℋ) |
6 | hvsubcl 30764 | . . . 4 ⊢ (((𝑆‘𝐴) ∈ ℋ ∧ (𝑇‘𝐴) ∈ ℋ) → ((𝑆‘𝐴) −ℎ (𝑇‘𝐴)) ∈ ℋ) | |
7 | 3, 5, 6 | syl2anc 583 | . . 3 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆‘𝐴) −ℎ (𝑇‘𝐴)) ∈ ℋ) |
8 | 1, 7 | eqeltrd 2825 | . 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝐴) ∈ ℋ) |
9 | 8 | 3expa 1115 | 1 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝐴) ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 ℋchba 30666 −ℎ cmv 30672 −op chod 30687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-hilex 30746 ax-hfvadd 30747 ax-hfvmul 30752 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-ltxr 11252 df-sub 11445 df-neg 11446 df-hvsub 30718 df-hodif 31479 |
This theorem is referenced by: hodcli 31510 |
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