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Mirrors > Home > HSE Home > Th. List > bdophdi | Structured version Visualization version GIF version |
Description: The difference between two bounded linear operators is bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoptri.1 | ⊢ 𝑆 ∈ BndLinOp |
nmoptri.2 | ⊢ 𝑇 ∈ BndLinOp |
Ref | Expression |
---|---|
bdophdi | ⊢ (𝑆 −op 𝑇) ∈ BndLinOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoptri.1 | . . . 4 ⊢ 𝑆 ∈ BndLinOp | |
2 | bdopf 31093 | . . . 4 ⊢ (𝑆 ∈ BndLinOp → 𝑆: ℋ⟶ ℋ) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 𝑆: ℋ⟶ ℋ |
4 | nmoptri.2 | . . . 4 ⊢ 𝑇 ∈ BndLinOp | |
5 | bdopf 31093 | . . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ 𝑇: ℋ⟶ ℋ |
7 | 3, 6 | honegsubi 31027 | . 2 ⊢ (𝑆 +op (-1 ·op 𝑇)) = (𝑆 −op 𝑇) |
8 | neg1cn 12322 | . . . 4 ⊢ -1 ∈ ℂ | |
9 | 4 | bdophmi 31263 | . . . 4 ⊢ (-1 ∈ ℂ → (-1 ·op 𝑇) ∈ BndLinOp) |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ (-1 ·op 𝑇) ∈ BndLinOp |
11 | 1, 10 | bdophsi 31327 | . 2 ⊢ (𝑆 +op (-1 ·op 𝑇)) ∈ BndLinOp |
12 | 7, 11 | eqeltrri 2831 | 1 ⊢ (𝑆 −op 𝑇) ∈ BndLinOp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ⟶wf 6536 (class class class)co 7404 ℂcc 11104 1c1 11107 -cneg 11441 ℋchba 30150 +op chos 30169 ·op chot 30170 −op chod 30171 BndLinOpcbo 30179 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-hilex 30230 ax-hfvadd 30231 ax-hvcom 30232 ax-hvass 30233 ax-hv0cl 30234 ax-hvaddid 30235 ax-hfvmul 30236 ax-hvmulid 30237 ax-hvmulass 30238 ax-hvdistr1 30239 ax-hvdistr2 30240 ax-hvmul0 30241 ax-hfi 30310 ax-his1 30313 ax-his2 30314 ax-his3 30315 ax-his4 30316 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-grpo 29724 df-gid 29725 df-ablo 29776 df-vc 29790 df-nv 29823 df-va 29826 df-ba 29827 df-sm 29828 df-0v 29829 df-nmcv 29831 df-hnorm 30199 df-hba 30200 df-hvsub 30202 df-hosum 30961 df-homul 30962 df-hodif 30963 df-nmop 31070 df-lnop 31072 df-bdop 31073 |
This theorem is referenced by: unierri 31335 |
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