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Theorem nmobndi 30980

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 11281	. . . 4 ⊢ ((𝑁‘𝑇) ∈ ℝ → (𝑁‘𝑇) ≤ (𝑁‘𝑇))
2		breq2 5106	. . . . 5 ⊢ (𝑟 = (𝑁‘𝑇) → ((𝑁‘𝑇) ≤ 𝑟 ↔ (𝑁‘𝑇) ≤ (𝑁‘𝑇)))
3	2	rspcev 3583	. . . 4 ⊢ (((𝑁‘𝑇) ∈ ℝ ∧ (𝑁‘𝑇) ≤ (𝑁‘𝑇)) → ∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟)
4	1, 3	mpdan 697	. . 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 30971	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ^*)
11	5, 6, 10	mp3an12 1474	. . . . . 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 780	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → 𝑟 ∈ ℝ)
14	7, 8, 9	nmogtmnf 30975	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → -∞ < (𝑁‘𝑇))
15	5, 6, 14	mp3an12 1474	. . . . . 6 ⊢ (𝑇:𝑋⟶𝑌 → -∞ < (𝑁‘𝑇))
16	15	adantr 484	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → -∞ < (𝑁‘𝑇))
17		simprr 782	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ≤ 𝑟)
18		xrre 13174	. . . . 5 ⊢ ((((𝑁‘𝑇) ∈ ℝ^* ∧ 𝑟 ∈ ℝ) ∧ (-∞ < (𝑁‘𝑇) ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ∈ ℝ)
19	12, 13, 16, 17, 18	syl22anc 849	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ∈ ℝ)
20	19	rexlimdvaa 3166	. . 3 ⊢ (𝑇:𝑋⟶𝑌 → (∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟 → (𝑁‘𝑇) ∈ ℝ))
21	4, 20	impbid2 228	. 2 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟))
22		rexr 11230	. . . 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 30977	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ^*) → ((𝑁‘𝑇) ≤ 𝑟 ↔ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
26	22, 25	sylan2 602	. . 3 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ) → ((𝑁‘𝑇) ≤ 𝑟 ↔ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
27	26	rexbidva 3186	. 2 ⊢ (𝑇:𝑋⟶𝑌 → (∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟 ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
28	21, 27	bitrd 281	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 208 ∧ wa 399 = wceq 1562 ∈ wcel 2144 ∀wral 3078 ∃wrex 3088 class class class wbr 5102 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 ℝcr 11074 1c1 11076 -∞cmnf 11216 ℝ^*cxr 11217 < clt 11218 ≤ cle 11219 NrmCVeccnv 30789 BaseSetcba 30791 norm_CVcnmcv 30795 normOp_OLD cnmoo 30946

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-er 8680 df-map 8812 df-en 8930 df-dom 8931 df-sdom 8932 df-sup 9390 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-n0 12484 df-z 12571 df-uz 12842 df-rp 12996 df-seq 14017 df-exp 14077 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-grpo 30698 df-gid 30699 df-ginv 30700 df-ablo 30750 df-vc 30764 df-nv 30797 df-va 30800 df-ba 30801 df-sm 30802 df-0v 30803 df-nmcv 30805 df-nmoo 30950

This theorem is referenced by: nmounbi 30981 nmobndseqi 30984 nmobndseqiALT 30985 htthlem 31122

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