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Mirrors > Home > MPE Home > Th. List > Mathboxes > resopunitintvd | Structured version Visualization version GIF version |
Description: Restrict continuous function on open unit interval. (Contributed by metakunt, 12-May-2024.) |
Ref | Expression |
---|---|
resopunitintvd.1 | ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐴) ∈ (ℂ–cn→ℂ)) |
Ref | Expression |
---|---|
resopunitintvd | ⊢ (𝜑 → (𝑥 ∈ (0(,)1) ↦ 𝐴) ∈ ((0(,)1)–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioosscn 12826 | . . 3 ⊢ (0(,)1) ⊆ ℂ | |
2 | resmpt 5870 | . . 3 ⊢ ((0(,)1) ⊆ ℂ → ((𝑥 ∈ ℂ ↦ 𝐴) ↾ (0(,)1)) = (𝑥 ∈ (0(,)1) ↦ 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝑥 ∈ ℂ ↦ 𝐴) ↾ (0(,)1)) = (𝑥 ∈ (0(,)1) ↦ 𝐴) |
4 | resopunitintvd.1 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐴) ∈ (ℂ–cn→ℂ)) | |
5 | rescncf 23583 | . . . 4 ⊢ ((0(,)1) ⊆ ℂ → ((𝑥 ∈ ℂ ↦ 𝐴) ∈ (ℂ–cn→ℂ) → ((𝑥 ∈ ℂ ↦ 𝐴) ↾ (0(,)1)) ∈ ((0(,)1)–cn→ℂ))) | |
6 | 1, 5 | ax-mp 5 | . . 3 ⊢ ((𝑥 ∈ ℂ ↦ 𝐴) ∈ (ℂ–cn→ℂ) → ((𝑥 ∈ ℂ ↦ 𝐴) ↾ (0(,)1)) ∈ ((0(,)1)–cn→ℂ)) |
7 | 4, 6 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ 𝐴) ↾ (0(,)1)) ∈ ((0(,)1)–cn→ℂ)) |
8 | 3, 7 | eqeltrrid 2856 | 1 ⊢ (𝜑 → (𝑥 ∈ (0(,)1) ↦ 𝐴) ∈ ((0(,)1)–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 ⊆ wss 3854 ↦ cmpt 5105 ↾ cres 5519 (class class class)co 7143 ℂcc 10558 0cc0 10560 1c1 10561 (,)cioo 12764 –cn→ccncf 23562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-pre-lttri 10634 ax-pre-lttrn 10635 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-po 5436 df-so 5437 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-1st 7686 df-2nd 7687 df-er 8292 df-map 8411 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-ioo 12768 df-cncf 23564 |
This theorem is referenced by: lcmineqlem10 39590 lcmineqlem12 39592 |
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