nmosetre - Metamath Proof Explorer

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

Theorem nmosetre 28541

Description: The set in the supremum of the operator norm definition df-nmoo 28522 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nmosetre.2	⊢ 𝑌 = (BaseSet‘𝑊)
nmosetre.4	⊢ 𝑁 = (norm_CV‘𝑊)

**Assertion**
Ref	Expression
nmosetre	⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧)))} ⊆ ℝ)

Distinct variable groups: 𝑥,𝑧,𝑇 𝑥,𝑊,𝑧 𝑥,𝑋,𝑧 𝑥,𝑌,𝑧 Allowed substitution hints: 𝑀(𝑥,𝑧) 𝑁(𝑥,𝑧)

**Proof of Theorem nmosetre**
Step	Hyp	Ref	Expression
1		ffvelrn 6849	. . . . . . . 8 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑧 ∈ 𝑋) → (𝑇‘𝑧) ∈ 𝑌)
2		nmosetre.2	. . . . . . . . 9 ⊢ 𝑌 = (BaseSet‘𝑊)
3		nmosetre.4	. . . . . . . . 9 ⊢ 𝑁 = (norm_CV‘𝑊)
4	2, 3	nvcl 28438	. . . . . . . 8 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝑧) ∈ 𝑌) → (𝑁‘(𝑇‘𝑧)) ∈ ℝ)
5	1, 4	sylan2 594	. . . . . . 7 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇:𝑋⟶𝑌 ∧ 𝑧 ∈ 𝑋)) → (𝑁‘(𝑇‘𝑧)) ∈ ℝ)
6	5	anassrs 470	. . . . . 6 ⊢ (((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ 𝑧 ∈ 𝑋) → (𝑁‘(𝑇‘𝑧)) ∈ ℝ)
7		eleq1 2900	. . . . . 6 ⊢ (𝑥 = (𝑁‘(𝑇‘𝑧)) → (𝑥 ∈ ℝ ↔ (𝑁‘(𝑇‘𝑧)) ∈ ℝ))
8	6, 7	syl5ibr 248	. . . . 5 ⊢ (𝑥 = (𝑁‘(𝑇‘𝑧)) → (((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ 𝑧 ∈ 𝑋) → 𝑥 ∈ ℝ))
9	8	impcom 410	. . . 4 ⊢ ((((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 = (𝑁‘(𝑇‘𝑧))) → 𝑥 ∈ ℝ)
10	9	adantrl 714	. . 3 ⊢ ((((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ 𝑧 ∈ 𝑋) ∧ ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧)))) → 𝑥 ∈ ℝ)
11	10	rexlimdva2 3287	. 2 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (∃𝑧 ∈ 𝑋 ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧))) → 𝑥 ∈ ℝ))
12	11	abssdv 4045	1 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧)))} ⊆ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2799 ∃wrex 3139 ⊆ wss 3936 class class class wbr 5066 ⟶wf 6351 ‘cfv 6355 ℝcr 10536 1c1 10538 ≤ cle 10676 NrmCVeccnv 28361 BaseSetcba 28363 norm_CVcnmcv 28367

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-1st 7689 df-2nd 7690 df-vc 28336 df-nv 28369 df-va 28372 df-ba 28373 df-sm 28374 df-0v 28375 df-nmcv 28377

This theorem is referenced by: nmoxr 28543 nmooge0 28544 nmorepnf 28545 nmoolb 28548 nmoubi 28549 nmlno0lem 28570 nmopsetretHIL 29641

W3C validator