nmfnrepnf - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > nmfnrepnf		Structured version Visualization version GIF version

Theorem nmfnrepnf 31683

Description: The norm of a Hilbert space functional is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

**Assertion**
Ref	Expression
nmfnrepnf	⊢ (𝑇: ℋ⟶ℂ → ((norm_fn‘𝑇) ∈ ℝ ↔ (norm_fn‘𝑇) ≠ +∞))

Proof of Theorem nmfnrepnf Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmfnsetre 31680	. . 3 ⊢ (𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm_ℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ)
2		nmfnsetn0 31681	. . . 4 ⊢ (abs‘(𝑇‘0_ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm_ℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}
3	2	ne0ii 4333	. . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm_ℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ≠ ∅
4		supxrre2 13336	. . 3 ⊢ (({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm_ℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ ∧ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm_ℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ≠ ∅) → (sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm_ℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ^, < ) ∈ ℝ ↔ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm_ℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ^, < ) ≠ +∞))
5	1, 3, 4	sylancl 585	. 2 ⊢ (𝑇: ℋ⟶ℂ → (sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm_ℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ^, < ) ∈ ℝ ↔ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm_ℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ^, < ) ≠ +∞))
6		nmfnval 31679	. . 3 ⊢ (𝑇: ℋ⟶ℂ → (norm_fn‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm_ℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ^*, < ))
7	6	eleq1d 2814	. 2 ⊢ (𝑇: ℋ⟶ℂ → ((norm_fn‘𝑇) ∈ ℝ ↔ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm_ℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ^*, < ) ∈ ℝ))
8	6	neeq1d 2996	. 2 ⊢ (𝑇: ℋ⟶ℂ → ((norm_fn‘𝑇) ≠ +∞ ↔ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm_ℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ^*, < ) ≠ +∞))
9	5, 7, 8	3bitr4d 311	1 ⊢ (𝑇: ℋ⟶ℂ → ((norm_fn‘𝑇) ∈ ℝ ↔ (norm_fn‘𝑇) ≠ +∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {cab 2705 ≠ wne 2936 ∃wrex 3066 ⊆ wss 3945 ∅c0 4318 class class class wbr 5142 ⟶wf 6538 ‘cfv 6542 supcsup 9457 ℂcc 11130 ℝcr 11131 1c1 11133 +∞cpnf 11269 ℝ^*cxr 11271 < clt 11272 ≤ cle 11273 abscabs 15207 ℋchba 30722 norm_ℎcno 30726 0_ℎc0v 30727 norm_fncnmf 30754

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-hilex 30802 ax-hv0cl 30806 ax-hvmul0 30813 ax-hfi 30882 ax-his3 30887

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-hnorm 30771 df-nmfn 31648

This theorem is referenced by: (None)

W3C validator

	
		OSZAR »