nmblore - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > nmblore		Structured version Visualization version GIF version

Theorem nmblore 30857

Description: The norm of a bounded operator is a real number. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nmblore.1	⊢ 𝑋 = (BaseSet‘𝑈)
nmblore.2	⊢ 𝑌 = (BaseSet‘𝑊)
nmblore.3	⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
nmblore.5	⊢ 𝐵 = (𝑈 BLnOp 𝑊)

**Assertion**
Ref	Expression
nmblore	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) ∈ ℝ)

**Proof of Theorem nmblore**
Step	Hyp	Ref	Expression
1		nmblore.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
2		nmblore.2	. . . 4 ⊢ 𝑌 = (BaseSet‘𝑊)
3		nmblore.5	. . . 4 ⊢ 𝐵 = (𝑈 BLnOp 𝑊)
4	1, 2, 3	blof 30856	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇:𝑋⟶𝑌)
5		nmblore.3	. . . 4 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
6	1, 2, 5	nmogtmnf 30841	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → -∞ < (𝑁‘𝑇))
7	4, 6	syld3an3 1412	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → -∞ < (𝑁‘𝑇))
8		eqid 2737	. . . . 5 ⊢ (𝑈 LnOp 𝑊) = (𝑈 LnOp 𝑊)
9	5, 8, 3	isblo 30853	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ (𝑈 LnOp 𝑊) ∧ (𝑁‘𝑇) < +∞)))
10	9	simplbda 499	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) < +∞)
11	10	3impa 1110	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) < +∞)
12	1, 2, 5	nmoxr 30837	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ^*)
13	4, 12	syld3an3 1412	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) ∈ ℝ^*)
14		xrrebnd 13120	. . 3 ⊢ ((𝑁‘𝑇) ∈ ℝ^* → ((𝑁‘𝑇) ∈ ℝ ↔ (-∞ < (𝑁‘𝑇) ∧ (𝑁‘𝑇) < +∞)))
15	13, 14	syl 17	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → ((𝑁‘𝑇) ∈ ℝ ↔ (-∞ < (𝑁‘𝑇) ∧ (𝑁‘𝑇) < +∞)))
16	7, 11, 15	mpbir2and 714	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 class class class wbr 5086 ⟶wf 6495 ‘cfv 6499 (class class class)co 7367 ℝcr 11037 +∞cpnf 11176 -∞cmnf 11177 ℝ^*cxr 11178 < clt 11179 NrmCVeccnv 30655 BaseSetcba 30657 LnOp clno 30811 normOp_OLD cnmoo 30812 BLnOp cblo 30813

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

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 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-rp 12943 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-grpo 30564 df-gid 30565 df-ginv 30566 df-ablo 30616 df-vc 30630 df-nv 30663 df-va 30666 df-ba 30667 df-sm 30668 df-0v 30669 df-nmcv 30671 df-lno 30815 df-nmoo 30816 df-blo 30817

This theorem is referenced by: nmblolbii 30870 isblo3i 30872 blocni 30876 htthlem 30988

W3C validator