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Theorem nmobndi 30755

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 11209	. . . 4 ⊢ ((𝑁‘𝑇) ∈ ℝ → (𝑁‘𝑇) ≤ (𝑁‘𝑇))
2		breq2 5093	. . . . 5 ⊢ (𝑟 = (𝑁‘𝑇) → ((𝑁‘𝑇) ≤ 𝑟 ↔ (𝑁‘𝑇) ≤ (𝑁‘𝑇)))
3	2	rspcev 3572	. . . 4 ⊢ (((𝑁‘𝑇) ∈ ℝ ∧ (𝑁‘𝑇) ≤ (𝑁‘𝑇)) → ∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟)
4	1, 3	mpdan 687	. . 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 30746	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ^*)
11	5, 6, 10	mp3an12 1453	. . . . . 6 ⊢ (𝑇:𝑋⟶𝑌 → (𝑁‘𝑇) ∈ ℝ^*)
12	11	adantr 480	. . . . 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 30750	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → -∞ < (𝑁‘𝑇))
15	5, 6, 14	mp3an12 1453	. . . . . 6 ⊢ (𝑇:𝑋⟶𝑌 → -∞ < (𝑁‘𝑇))
16	15	adantr 480	. . . . 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 13068	. . . . 5 ⊢ ((((𝑁‘𝑇) ∈ ℝ^* ∧ 𝑟 ∈ ℝ) ∧ (-∞ < (𝑁‘𝑇) ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ∈ ℝ)
19	12, 13, 16, 17, 18	syl22anc 838	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ∈ ℝ)
20	19	rexlimdvaa 3134	. . 3 ⊢ (𝑇:𝑋⟶𝑌 → (∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟 → (𝑁‘𝑇) ∈ ℝ))
21	4, 20	impbid2 226	. 2 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟))
22		rexr 11158	. . . 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 30752	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ^*) → ((𝑁‘𝑇) ≤ 𝑟 ↔ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
26	22, 25	sylan2 593	. . 3 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ) → ((𝑁‘𝑇) ≤ 𝑟 ↔ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
27	26	rexbidva 3154	. 2 ⊢ (𝑇:𝑋⟶𝑌 → (∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟 ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
28	21, 27	bitrd 279	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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 class class class wbr 5089 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ℝcr 11005 1c1 11007 -∞cmnf 11144 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 NrmCVeccnv 30564 BaseSetcba 30566 norm_CVcnmcv 30570 normOp_OLD cnmoo 30721

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-z 12469 df-uz 12733 df-rp 12891 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-grpo 30473 df-gid 30474 df-ginv 30475 df-ablo 30525 df-vc 30539 df-nv 30572 df-va 30575 df-ba 30576 df-sm 30577 df-0v 30578 df-nmcv 30580 df-nmoo 30725

This theorem is referenced by: nmounbi 30756 nmobndseqi 30759 nmobndseqiALT 30760 htthlem 30897

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