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Theorem lnopcnbd 31783

Description: A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
lnopcnbd	⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp))

Proof of Theorem lnopcnbd Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmcopex 31776	. . . 4 ⊢ ((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp) → (norm_op‘𝑇) ∈ ℝ)
2	1	ex 412	. . 3 ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp → (norm_op‘𝑇) ∈ ℝ))
3		elbdop2 31618	. . . 4 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (norm_op‘𝑇) ∈ ℝ))
4	3	baibr 536	. . 3 ⊢ (𝑇 ∈ LinOp → ((norm_op‘𝑇) ∈ ℝ ↔ 𝑇 ∈ BndLinOp))
5	2, 4	sylibd 238	. 2 ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp → 𝑇 ∈ BndLinOp))
6		nmopre 31617	. . . 4 ⊢ (𝑇 ∈ BndLinOp → (norm_op‘𝑇) ∈ ℝ)
7		nmbdoplb 31772	. . . . 5 ⊢ ((𝑇 ∈ BndLinOp ∧ 𝑦 ∈ ℋ) → (norm_ℎ‘(𝑇‘𝑦)) ≤ ((norm_op‘𝑇) · (norm_ℎ‘𝑦)))
8	7	ralrimiva 3138	. . . 4 ⊢ (𝑇 ∈ BndLinOp → ∀𝑦 ∈ ℋ (norm_ℎ‘(𝑇‘𝑦)) ≤ ((norm_op‘𝑇) · (norm_ℎ‘𝑦)))
9		oveq1 7409	. . . . . . 7 ⊢ (𝑥 = (norm_op‘𝑇) → (𝑥 · (norm_ℎ‘𝑦)) = ((norm_op‘𝑇) · (norm_ℎ‘𝑦)))
10	9	breq2d 5151	. . . . . 6 ⊢ (𝑥 = (norm_op‘𝑇) → ((norm_ℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (norm_ℎ‘𝑦)) ↔ (norm_ℎ‘(𝑇‘𝑦)) ≤ ((norm_op‘𝑇) · (norm_ℎ‘𝑦))))
11	10	ralbidv 3169	. . . . 5 ⊢ (𝑥 = (norm_op‘𝑇) → (∀𝑦 ∈ ℋ (norm_ℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (norm_ℎ‘𝑦)) ↔ ∀𝑦 ∈ ℋ (norm_ℎ‘(𝑇‘𝑦)) ≤ ((norm_op‘𝑇) · (norm_ℎ‘𝑦))))
12	11	rspcev 3604	. . . 4 ⊢ (((norm_op‘𝑇) ∈ ℝ ∧ ∀𝑦 ∈ ℋ (norm_ℎ‘(𝑇‘𝑦)) ≤ ((norm_op‘𝑇) · (norm_ℎ‘𝑦))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (norm_ℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (norm_ℎ‘𝑦)))
13	6, 8, 12	syl2anc 583	. . 3 ⊢ (𝑇 ∈ BndLinOp → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (norm_ℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (norm_ℎ‘𝑦)))
14		lnopcon 31782	. . 3 ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (norm_ℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (norm_ℎ‘𝑦))))
15	13, 14	imbitrrid 245	. 2 ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ BndLinOp → 𝑇 ∈ ContOp))
16	5, 15	impbid 211	1 ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 class class class wbr 5139 ‘cfv 6534 (class class class)co 7402 ℝcr 11106 · cmul 11112 ≤ cle 11248 ℋchba 30666 norm_ℎcno 30670 norm_opcnop 30692 ContOpccop 30693 LinOpclo 30694 BndLinOpcbo 30695

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-inf2 9633 ax-cc 10427 ax-cnex 11163 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-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 ax-hilex 30746 ax-hfvadd 30747 ax-hvcom 30748 ax-hvass 30749 ax-hv0cl 30750 ax-hvaddid 30751 ax-hfvmul 30752 ax-hvmulid 30753 ax-hvmulass 30754 ax-hvdistr1 30755 ax-hvdistr2 30756 ax-hvmul0 30757 ax-hfi 30826 ax-his1 30829 ax-his2 30830 ax-his3 30831 ax-his4 30832 ax-hcompl 30949

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-rmo 3368 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-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 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-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 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-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-acn 9934 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ioo 13329 df-ico 13331 df-icc 13332 df-fz 13486 df-fzo 13629 df-fl 13758 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-rlim 15435 df-sum 15635 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-mulg 18992 df-cntz 19229 df-cmn 19698 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-fbas 21231 df-fg 21232 df-cnfld 21235 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-cld 22867 df-ntr 22868 df-cls 22869 df-nei 22946 df-cn 23075 df-cnp 23076 df-lm 23077 df-haus 23163 df-tx 23410 df-hmeo 23603 df-fil 23694 df-fm 23786 df-flim 23787 df-flf 23788 df-xms 24170 df-ms 24171 df-tms 24172 df-cfil 25127 df-cau 25128 df-cmet 25129 df-grpo 30240 df-gid 30241 df-ginv 30242 df-gdiv 30243 df-ablo 30292 df-vc 30306 df-nv 30339 df-va 30342 df-ba 30343 df-sm 30344 df-0v 30345 df-vs 30346 df-nmcv 30347 df-ims 30348 df-dip 30448 df-ssp 30469 df-ph 30560 df-cbn 30610 df-hnorm 30715 df-hba 30716 df-hvsub 30718 df-hlim 30719 df-hcau 30720 df-sh 30954 df-ch 30968 df-oc 30999 df-ch0 31000 df-shs 31055 df-pjh 31142 df-h0op 31495 df-nmop 31586 df-cnop 31587 df-lnop 31588 df-bdop 31589 df-unop 31590 df-hmop 31591

This theorem is referenced by: lncnopbd 31784 lnopcnre 31786 hmopidmchi 31898

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