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Theorem nmobndi 28662

Description: Two ways to express that an operator is bounded. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nmoubi.1	⊢ 𝑋 = (BaseSet‘𝑈)
nmoubi.y	⊢ 𝑌 = (BaseSet‘𝑊)
nmoubi.l	⊢ 𝐿 = (norm_CV‘𝑈)
nmoubi.m	⊢ 𝑀 = (norm_CV‘𝑊)
nmoubi.3	⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
nmoubi.u	⊢ 𝑈 ∈ NrmCVec
nmoubi.w	⊢ 𝑊 ∈ NrmCVec

**Assertion**
Ref	Expression
nmobndi	⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))

Distinct variable groups: 𝑦,𝑟,𝐿 𝑦,𝑈 𝑦,𝑊 𝑌,𝑟,𝑦 𝑀,𝑟,𝑦 𝑇,𝑟,𝑦 𝑋,𝑟,𝑦 𝑁,𝑟,𝑦 Allowed substitution hints: 𝑈(𝑟) 𝑊(𝑟)

**Proof of Theorem nmobndi**
Step	Hyp	Ref	Expression
1		leid 10779	. . . 4 ⊢ ((𝑁‘𝑇) ∈ ℝ → (𝑁‘𝑇) ≤ (𝑁‘𝑇))
2		breq2 5039	. . . . 5 ⊢ (𝑟 = (𝑁‘𝑇) → ((𝑁‘𝑇) ≤ 𝑟 ↔ (𝑁‘𝑇) ≤ (𝑁‘𝑇)))
3	2	rspcev 3543	. . . 4 ⊢ (((𝑁‘𝑇) ∈ ℝ ∧ (𝑁‘𝑇) ≤ (𝑁‘𝑇)) → ∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟)
4	1, 3	mpdan 686	. . 3 ⊢ ((𝑁‘𝑇) ∈ ℝ → ∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟)
5		nmoubi.u	. . . . . . 7 ⊢ 𝑈 ∈ NrmCVec
6		nmoubi.w	. . . . . . 7 ⊢ 𝑊 ∈ NrmCVec
7		nmoubi.1	. . . . . . . 8 ⊢ 𝑋 = (BaseSet‘𝑈)
8		nmoubi.y	. . . . . . . 8 ⊢ 𝑌 = (BaseSet‘𝑊)
9		nmoubi.3	. . . . . . . 8 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
10	7, 8, 9	nmoxr 28653	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ^*)
11	5, 6, 10	mp3an12 1448	. . . . . 6 ⊢ (𝑇:𝑋⟶𝑌 → (𝑁‘𝑇) ∈ ℝ^*)
12	11	adantr 484	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ∈ ℝ^*)
13		simprl 770	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → 𝑟 ∈ ℝ)
14	7, 8, 9	nmogtmnf 28657	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → -∞ < (𝑁‘𝑇))
15	5, 6, 14	mp3an12 1448	. . . . . 6 ⊢ (𝑇:𝑋⟶𝑌 → -∞ < (𝑁‘𝑇))
16	15	adantr 484	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → -∞ < (𝑁‘𝑇))
17		simprr 772	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ≤ 𝑟)
18		xrre 12608	. . . . 5 ⊢ ((((𝑁‘𝑇) ∈ ℝ^* ∧ 𝑟 ∈ ℝ) ∧ (-∞ < (𝑁‘𝑇) ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ∈ ℝ)
19	12, 13, 16, 17, 18	syl22anc 837	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ∈ ℝ)
20	19	rexlimdvaa 3209	. . 3 ⊢ (𝑇:𝑋⟶𝑌 → (∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟 → (𝑁‘𝑇) ∈ ℝ))
21	4, 20	impbid2 229	. 2 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟))
22		rexr 10730	. . . 4 ⊢ (𝑟 ∈ ℝ → 𝑟 ∈ ℝ^*)
23		nmoubi.l	. . . . 5 ⊢ 𝐿 = (norm_CV‘𝑈)
24		nmoubi.m	. . . . 5 ⊢ 𝑀 = (norm_CV‘𝑊)
25	7, 8, 23, 24, 9, 5, 6	nmoubi 28659	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ^*) → ((𝑁‘𝑇) ≤ 𝑟 ↔ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
26	22, 25	sylan2 595	. . 3 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ) → ((𝑁‘𝑇) ≤ 𝑟 ↔ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
27	26	rexbidva 3220	. 2 ⊢ (𝑇:𝑋⟶𝑌 → (∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟 ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
28	21, 27	bitrd 282	1 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3070 ∃wrex 3071 class class class wbr 5035 ⟶wf 6335 ‘cfv 6339 (class class class)co 7155 ℝcr 10579 1c1 10581 -∞cmnf 10716 ℝ^*cxr 10717 < clt 10718 ≤ cle 10719 NrmCVeccnv 28471 BaseSetcba 28473 norm_CVcnmcv 28477 normOp_OLD cnmoo 28628

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-cnex 10636 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656 ax-pre-mulgt0 10657 ax-pre-sup 10658

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7585 df-1st 7698 df-2nd 7699 df-wrecs 7962 df-recs 8023 df-rdg 8061 df-er 8304 df-map 8423 df-en 8533 df-dom 8534 df-sdom 8535 df-sup 8944 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 df-div 11341 df-nn 11680 df-2 11742 df-3 11743 df-n0 11940 df-z 12026 df-uz 12288 df-rp 12436 df-seq 13424 df-exp 13485 df-cj 14511 df-re 14512 df-im 14513 df-sqrt 14647 df-abs 14648 df-grpo 28380 df-gid 28381 df-ginv 28382 df-ablo 28432 df-vc 28446 df-nv 28479 df-va 28482 df-ba 28483 df-sm 28484 df-0v 28485 df-nmcv 28487 df-nmoo 28632

This theorem is referenced by: nmounbi 28663 nmobndseqi 28666 nmobndseqiALT 28667 htthlem 28804

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