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Theorem nmobndi 30854

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 11233	. . . 4 ⊢ ((𝑁‘𝑇) ∈ ℝ → (𝑁‘𝑇) ≤ (𝑁‘𝑇))
2		breq2 5103	. . . . 5 ⊢ (𝑟 = (𝑁‘𝑇) → ((𝑁‘𝑇) ≤ 𝑟 ↔ (𝑁‘𝑇) ≤ (𝑁‘𝑇)))
3	2	rspcev 3577	. . . 4 ⊢ (((𝑁‘𝑇) ∈ ℝ ∧ (𝑁‘𝑇) ≤ (𝑁‘𝑇)) → ∃𝑟 ∈ ℝ (𝑁‘𝑇) ≤ 𝑟)
4	1, 3	mpdan 688	. . 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 30845	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ^*)
11	5, 6, 10	mp3an12 1454	. . . . . 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 771	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → 𝑟 ∈ ℝ)
14	7, 8, 9	nmogtmnf 30849	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → -∞ < (𝑁‘𝑇))
15	5, 6, 14	mp3an12 1454	. . . . . 6 ⊢ (𝑇:𝑋⟶𝑌 → -∞ < (𝑁‘𝑇))
16	15	adantr 480	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → -∞ < (𝑁‘𝑇))
17		simprr 773	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ≤ 𝑟)
18		xrre 13088	. . . . 5 ⊢ ((((𝑁‘𝑇) ∈ ℝ^* ∧ 𝑟 ∈ ℝ) ∧ (-∞ < (𝑁‘𝑇) ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ∈ ℝ)
19	12, 13, 16, 17, 18	syl22anc 839	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑟 ∈ ℝ ∧ (𝑁‘𝑇) ≤ 𝑟)) → (𝑁‘𝑇) ∈ ℝ)
20	19	rexlimdvaa 3139	. . 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 11182	. . . 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 30851	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ^*) → ((𝑁‘𝑇) ≤ 𝑟 ↔ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
26	22, 25	sylan2 594	. . 3 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ) → ((𝑁‘𝑇) ≤ 𝑟 ↔ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
27	26	rexbidva 3159	. 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 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3061 class class class wbr 5099 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 ℝcr 11029 1c1 11031 -∞cmnf 11168 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 NrmCVeccnv 30663 BaseSetcba 30665 norm_CVcnmcv 30669 normOp_OLD cnmoo 30820

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-sup 9349 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-n0 12406 df-z 12493 df-uz 12756 df-rp 12910 df-seq 13929 df-exp 13989 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-grpo 30572 df-gid 30573 df-ginv 30574 df-ablo 30624 df-vc 30638 df-nv 30671 df-va 30674 df-ba 30675 df-sm 30676 df-0v 30677 df-nmcv 30679 df-nmoo 30824

This theorem is referenced by: nmounbi 30855 nmobndseqi 30858 nmobndseqiALT 30859 htthlem 30996

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