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Theorem lnopcnbd 31845

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 31838	. . . 4 ⊢ ((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp) → (norm_op‘𝑇) ∈ ℝ)
2	1	ex 412	. . 3 ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp → (norm_op‘𝑇) ∈ ℝ))
3		elbdop2 31680	. . . 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 31679	. . . 4 ⊢ (𝑇 ∈ BndLinOp → (norm_op‘𝑇) ∈ ℝ)
7		nmbdoplb 31834	. . . . 5 ⊢ ((𝑇 ∈ BndLinOp ∧ 𝑦 ∈ ℋ) → (norm_ℎ‘(𝑇‘𝑦)) ≤ ((norm_op‘𝑇) · (norm_ℎ‘𝑦)))
8	7	ralrimiva 3143	. . . 4 ⊢ (𝑇 ∈ BndLinOp → ∀𝑦 ∈ ℋ (norm_ℎ‘(𝑇‘𝑦)) ≤ ((norm_op‘𝑇) · (norm_ℎ‘𝑦)))
9		oveq1 7427	. . . . . . 7 ⊢ (𝑥 = (norm_op‘𝑇) → (𝑥 · (norm_ℎ‘𝑦)) = ((norm_op‘𝑇) · (norm_ℎ‘𝑦)))
10	9	breq2d 5160	. . . . . 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 3174	. . . . 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 3609	. . . 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 31844	. . 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 1534 ∈ wcel 2099 ∀wral 3058 ∃wrex 3067 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 ℝcr 11137 · cmul 11143 ≤ cle 11279 ℋchba 30728 norm_ℎcno 30732 norm_opcnop 30754 ContOpccop 30755 LinOpclo 30756 BndLinOpcbo 30757

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 ax-cc 10458 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 ax-hilex 30808 ax-hfvadd 30809 ax-hvcom 30810 ax-hvass 30811 ax-hv0cl 30812 ax-hvaddid 30813 ax-hfvmul 30814 ax-hvmulid 30815 ax-hvmulass 30816 ax-hvdistr1 30817 ax-hvdistr2 30818 ax-hvmul0 30819 ax-hfi 30888 ax-his1 30891 ax-his2 30892 ax-his3 30893 ax-his4 30894 ax-hcompl 31011

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-oadd 8490 df-omul 8491 df-er 8724 df-map 8846 df-pm 8847 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-acn 9965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-rlim 15465 df-sum 15665 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19023 df-cntz 19267 df-cmn 19736 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-fbas 21275 df-fg 21276 df-cnfld 21279 df-top 22795 df-topon 22812 df-topsp 22834 df-bases 22848 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-cn 23130 df-cnp 23131 df-lm 23132 df-haus 23218 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24225 df-ms 24226 df-tms 24227 df-cfil 25182 df-cau 25183 df-cmet 25184 df-grpo 30302 df-gid 30303 df-ginv 30304 df-gdiv 30305 df-ablo 30354 df-vc 30368 df-nv 30401 df-va 30404 df-ba 30405 df-sm 30406 df-0v 30407 df-vs 30408 df-nmcv 30409 df-ims 30410 df-dip 30510 df-ssp 30531 df-ph 30622 df-cbn 30672 df-hnorm 30777 df-hba 30778 df-hvsub 30780 df-hlim 30781 df-hcau 30782 df-sh 31016 df-ch 31030 df-oc 31061 df-ch0 31062 df-shs 31117 df-pjh 31204 df-h0op 31557 df-nmop 31648 df-cnop 31649 df-lnop 31650 df-bdop 31651 df-unop 31652 df-hmop 31653

This theorem is referenced by: lncnopbd 31846 lnopcnre 31848 hmopidmchi 31960

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