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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 13410 | . . 3 ⊢ (0(,)1) ⊆ ℂ | |
2 | resmpt 6035 | . . 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 24804 | . . . 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 2833 | 1 ⊢ (𝜑 → (𝑥 ∈ (0(,)1) ↦ 𝐴) ∈ ((0(,)1)–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3944 ↦ cmpt 5225 ↾ cres 5674 (class class class)co 7414 ℂcc 11128 0cc0 11130 1c1 11131 (,)cioo 13348 –cn→ccncf 24783 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-pre-lttri 11204 ax-pre-lttrn 11205 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 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-id 5570 df-po 5584 df-so 5585 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-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-ioo 13352 df-cncf 24785 |
This theorem is referenced by: lcmineqlem10 41446 lcmineqlem12 41448 |
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