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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 28596 | . . 3 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 −op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))) | |
4 | 1, 2, 3 | mp2an 708 | . 2 ⊢ (𝑆 −op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
5 | 1 | ffvelrni 6358 | . . 3 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
6 | 2 | ffvelrni 6358 | . . 3 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
7 | hvsubcl 27874 | . . 3 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) ∈ ℋ) | |
8 | 5, 6, 7 | syl2anc 693 | . 2 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) ∈ ℋ) |
9 | 4, 8 | fmpti 6383 | 1 ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℋchil 27776 −ℎ cmv 27782 −op chod 27797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-hilex 27856 ax-hfvadd 27857 ax-hfvmul 27862 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-hvsub 27828 df-hodif 28591 |
This theorem is referenced by: hosubfni 28630 hosubcl 28632 hodsi 28634 hocsubdiri 28639 hodseqi 28653 ho0subi 28654 honegsubi 28655 hoaddsubi 28680 hosd1i 28681 honpncani 28686 hoddii 28848 unierri 28963 pjddii 29015 |
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