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Mirrors > Home > HSE Home > Th. List > hosubcli | Structured version Visualization version GIF version |
Description: Mapping of difference of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hoeq.1 | ⊢ 𝑆: ℋ⟶ ℋ |
hoeq.2 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
hosubcli | ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoeq.1 | . . 3 ⊢ 𝑆: ℋ⟶ ℋ | |
2 | hoeq.2 | . . 3 ⊢ 𝑇: ℋ⟶ ℋ | |
3 | hodmval 31760 | . . 3 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 −op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))) | |
4 | 1, 2, 3 | mp2an 691 | . 2 ⊢ (𝑆 −op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
5 | 1 | ffvelcdmi 7115 | . . 3 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
6 | 2 | ffvelcdmi 7115 | . . 3 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
7 | hvsubcl 31040 | . . 3 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) ∈ ℋ) | |
8 | 5, 6, 7 | syl2anc 583 | . 2 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) ∈ ℋ) |
9 | 4, 8 | fmpti 7144 | 1 ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2103 ↦ cmpt 5252 ⟶wf 6568 ‘cfv 6572 (class class class)co 7445 ℋchba 30942 −ℎ cmv 30948 −op chod 30963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-hilex 31022 ax-hfvadd 31023 ax-hfvmul 31028 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-po 5611 df-so 5612 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-ltxr 11325 df-sub 11518 df-neg 11519 df-hvsub 30994 df-hodif 31755 |
This theorem is referenced by: hosubfni 31794 hosubcl 31796 hodsi 31798 hocsubdiri 31803 hodseqi 31817 ho0subi 31818 honegsubi 31819 hoaddsubi 31844 hosd1i 31845 honpncani 31850 hoddii 32012 unierri 32127 pjddii 32179 |
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