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Theorem nmobndi 27910

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 10296	. . . 4 ⊢ ((𝑁‘𝑇) ∈ ℝ → (𝑁‘𝑇) ≤ (𝑁‘𝑇))
2		breq2 4796	. . . . 5 ⊢ (𝑟 = (𝑁‘𝑇) → ((𝑁‘𝑇) ≤ 𝑟 ↔ (𝑁‘𝑇) ≤ (𝑁‘𝑇)))
3	2	rspcev 3437	. . . 4 ⊢ (((𝑁‘𝑇) ∈ ℝ ∧ (𝑁‘𝑇) ≤ (𝑁‘𝑇)) → ∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟)
4	1, 3	mpdan 705	. . 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 27901	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ^*)
11	5, 6, 10	mp3an12 1551	. . . . . 6 ⊢ (𝑇:𝑋⟶𝑌 → (𝑁‘𝑇) ∈ ℝ^*)
12	11	adantr 472	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ∈ ℝ^*)
13		simprl 811	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → 𝑟 ∈ ℝ)
14	7, 8, 9	nmogtmnf 27905	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → -∞ < (𝑁‘𝑇))
15	5, 6, 14	mp3an12 1551	. . . . . 6 ⊢ (𝑇:𝑋⟶𝑌 → -∞ < (𝑁‘𝑇))
16	15	adantr 472	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → -∞ < (𝑁‘𝑇))
17		simprr 813	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ≤ 𝑟)
18		xrre 12164	. . . . 5 ⊢ ((((𝑁‘𝑇) ∈ ℝ^* ∧ 𝑟 ∈ ℝ) ∧ (-∞ < (𝑁‘𝑇) ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ∈ ℝ)
19	12, 13, 16, 17, 18	syl22anc 1464	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ∈ ℝ)
20	19	rexlimdvaa 3158	. . 3 ⊢ (𝑇:𝑋⟶𝑌 → (∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟 → (𝑁‘𝑇) ∈ ℝ))
21	4, 20	impbid2 216	. 2 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟))
22		rexr 10248	. . . 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 27907	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ^*) → ((𝑁‘𝑇) ≤ 𝑟 ↔ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
26	22, 25	sylan2 492	. . 3 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ) → ((𝑁‘𝑇) ≤ 𝑟 ↔ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
27	26	rexbidva 3175	. 2 ⊢ (𝑇:𝑋⟶𝑌 → (∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟 ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
28	21, 27	bitrd 268	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 196 ∧ wa 383 = wceq 1620 ∈ wcel 2127 ∀wral 3038 ∃wrex 3039 class class class wbr 4792 ⟶wf 6033 ‘cfv 6037 (class class class)co 6801 ℝcr 10098 1c1 10100 -∞cmnf 10235 ℝ^*cxr 10236 < clt 10237 ≤ cle 10238 NrmCVeccnv 27719 BaseSetcba 27721 norm_CVcnmcv 27725 normOp_OLD cnmoo 27876

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 ax-pre-sup 10177

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-om 7219 df-1st 7321 df-2nd 7322 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-er 7899 df-map 8013 df-en 8110 df-dom 8111 df-sdom 8112 df-sup 8501 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-div 10848 df-nn 11184 df-2 11242 df-3 11243 df-n0 11456 df-z 11541 df-uz 11851 df-rp 11997 df-seq 12967 df-exp 13026 df-cj 14009 df-re 14010 df-im 14011 df-sqrt 14145 df-abs 14146 df-grpo 27627 df-gid 27628 df-ginv 27629 df-ablo 27679 df-vc 27694 df-nv 27727 df-va 27730 df-ba 27731 df-sm 27732 df-0v 27733 df-nmcv 27735 df-nmoo 27880

This theorem is referenced by: nmounbi 27911 nmobndseqi 27914 nmobndseqiALT 27915 htthlem 28054

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