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Mirrors > Home > HSE Home > Th. List > hlim2 | Structured version Visualization version GIF version |
Description: The limit of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlim2 | ⊢ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝐹 ⇝𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5070 | . . . . 5 ⊢ (𝑤 = 𝐴 → (𝐹 ⇝𝑣 𝑤 ↔ 𝐹 ⇝𝑣 𝐴)) | |
2 | oveq2 7164 | . . . . . . . . 9 ⊢ (𝑤 = 𝐴 → ((𝐹‘𝑧) −ℎ 𝑤) = ((𝐹‘𝑧) −ℎ 𝐴)) | |
3 | 2 | fveq2d 6674 | . . . . . . . 8 ⊢ (𝑤 = 𝐴 → (normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) = (normℎ‘((𝐹‘𝑧) −ℎ 𝐴))) |
4 | 3 | breq1d 5076 | . . . . . . 7 ⊢ (𝑤 = 𝐴 → ((normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ (normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
5 | 4 | rexralbidv 3301 | . . . . . 6 ⊢ (𝑤 = 𝐴 → (∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
6 | 5 | ralbidv 3197 | . . . . 5 ⊢ (𝑤 = 𝐴 → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
7 | 1, 6 | bibi12d 348 | . . . 4 ⊢ (𝑤 = 𝐴 → ((𝐹 ⇝𝑣 𝑤 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥) ↔ (𝐹 ⇝𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥))) |
8 | 7 | imbi2d 343 | . . 3 ⊢ (𝑤 = 𝐴 → ((𝐹:ℕ⟶ ℋ → (𝐹 ⇝𝑣 𝑤 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥)) ↔ (𝐹:ℕ⟶ ℋ → (𝐹 ⇝𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)))) |
9 | vex 3497 | . . . . . 6 ⊢ 𝑤 ∈ V | |
10 | 9 | hlimi 28965 | . . . . 5 ⊢ (𝐹 ⇝𝑣 𝑤 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥)) |
11 | 10 | baib 538 | . . . 4 ⊢ ((𝐹:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) → (𝐹 ⇝𝑣 𝑤 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥)) |
12 | 11 | expcom 416 | . . 3 ⊢ (𝑤 ∈ ℋ → (𝐹:ℕ⟶ ℋ → (𝐹 ⇝𝑣 𝑤 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥))) |
13 | 8, 12 | vtoclga 3574 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐹:ℕ⟶ ℋ → (𝐹 ⇝𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥))) |
14 | 13 | impcom 410 | 1 ⊢ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝐹 ⇝𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 class class class wbr 5066 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 < clt 10675 ℕcn 11638 ℤ≥cuz 12244 ℝ+crp 12390 ℋchba 28696 normℎcno 28700 −ℎ cmv 28702 ⇝𝑣 chli 28704 |
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 ax-cnex 10593 ax-1cn 10595 ax-addcl 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 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-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 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-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-nn 11639 df-hlim 28749 |
This theorem is referenced by: (None) |
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