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Mirrors > Home > HSE Home > Th. List > bdophsi | Structured version Visualization version GIF version |
Description: The sum of two bounded linear operators is a bounded linear operator. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoptri.1 | ⊢ 𝑆 ∈ BndLinOp |
nmoptri.2 | ⊢ 𝑇 ∈ BndLinOp |
Ref | Expression |
---|---|
bdophsi | ⊢ (𝑆 +op 𝑇) ∈ BndLinOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoptri.1 | . . . 4 ⊢ 𝑆 ∈ BndLinOp | |
2 | bdopln 29632 | . . . 4 ⊢ (𝑆 ∈ BndLinOp → 𝑆 ∈ LinOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 𝑆 ∈ LinOp |
4 | nmoptri.2 | . . . 4 ⊢ 𝑇 ∈ BndLinOp | |
5 | bdopln 29632 | . . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ 𝑇 ∈ LinOp |
7 | 3, 6 | lnophsi 29772 | . 2 ⊢ (𝑆 +op 𝑇) ∈ LinOp |
8 | bdopf 29633 | . . . . . 6 ⊢ (𝑆 ∈ BndLinOp → 𝑆: ℋ⟶ ℋ) | |
9 | 1, 8 | ax-mp 5 | . . . . 5 ⊢ 𝑆: ℋ⟶ ℋ |
10 | bdopf 29633 | . . . . . 6 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ) | |
11 | 4, 10 | ax-mp 5 | . . . . 5 ⊢ 𝑇: ℋ⟶ ℋ |
12 | 9, 11 | hoaddcli 29539 | . . . 4 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
13 | nmopxr 29637 | . . . 4 ⊢ ((𝑆 +op 𝑇): ℋ⟶ ℋ → (normop‘(𝑆 +op 𝑇)) ∈ ℝ*) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (normop‘(𝑆 +op 𝑇)) ∈ ℝ* |
15 | nmopre 29641 | . . . . 5 ⊢ (𝑆 ∈ BndLinOp → (normop‘𝑆) ∈ ℝ) | |
16 | 1, 15 | ax-mp 5 | . . . 4 ⊢ (normop‘𝑆) ∈ ℝ |
17 | nmopre 29641 | . . . . 5 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ) | |
18 | 4, 17 | ax-mp 5 | . . . 4 ⊢ (normop‘𝑇) ∈ ℝ |
19 | 16, 18 | readdcli 10650 | . . 3 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ |
20 | nmopgtmnf 29639 | . . . 4 ⊢ ((𝑆 +op 𝑇): ℋ⟶ ℋ → -∞ < (normop‘(𝑆 +op 𝑇))) | |
21 | 12, 20 | ax-mp 5 | . . 3 ⊢ -∞ < (normop‘(𝑆 +op 𝑇)) |
22 | 1, 4 | nmoptrii 29865 | . . 3 ⊢ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)) |
23 | xrre 12556 | . . 3 ⊢ ((((normop‘(𝑆 +op 𝑇)) ∈ ℝ* ∧ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ) ∧ (-∞ < (normop‘(𝑆 +op 𝑇)) ∧ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)))) → (normop‘(𝑆 +op 𝑇)) ∈ ℝ) | |
24 | 14, 19, 21, 22, 23 | mp4an 691 | . 2 ⊢ (normop‘(𝑆 +op 𝑇)) ∈ ℝ |
25 | elbdop2 29642 | . 2 ⊢ ((𝑆 +op 𝑇) ∈ BndLinOp ↔ ((𝑆 +op 𝑇) ∈ LinOp ∧ (normop‘(𝑆 +op 𝑇)) ∈ ℝ)) | |
26 | 7, 24, 25 | mpbir2an 709 | 1 ⊢ (𝑆 +op 𝑇) ∈ BndLinOp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 class class class wbr 5058 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 + caddc 10534 -∞cmnf 10667 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 ℋchba 28690 +op chos 28709 normopcnop 28716 LinOpclo 28718 BndLinOpcbo 28719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-hilex 28770 ax-hfvadd 28771 ax-hvcom 28772 ax-hvass 28773 ax-hv0cl 28774 ax-hvaddid 28775 ax-hfvmul 28776 ax-hvmulid 28777 ax-hvmulass 28778 ax-hvdistr1 28779 ax-hvdistr2 28780 ax-hvmul0 28781 ax-hfi 28850 ax-his1 28853 ax-his2 28854 ax-his3 28855 ax-his4 28856 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-grpo 28264 df-gid 28265 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-nmcv 28371 df-hnorm 28739 df-hba 28740 df-hvsub 28742 df-hosum 29501 df-nmop 29610 df-lnop 29612 df-bdop 29613 |
This theorem is referenced by: bdophdi 29868 nmoptri2i 29870 unierri 29875 |
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