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Mirrors > Home > MPE Home > Th. List > elee | Structured version Visualization version GIF version |
Description: Membership in a Euclidean space. We define Euclidean space here using Cartesian coordinates over 𝑁 space. We later abstract away from this using Tarski's geometry axioms, so this exact definition is unimportant. (Contributed by Scott Fenton, 3-Jun-2013.) |
Ref | Expression |
---|---|
elee | ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6557 | . . . . 5 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
2 | 1 | oveq2d 6565 | . . . 4 ⊢ (𝑛 = 𝑁 → (ℝ ↑𝑚 (1...𝑛)) = (ℝ ↑𝑚 (1...𝑁))) |
3 | df-ee 25571 | . . . 4 ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑𝑚 (1...𝑛))) | |
4 | ovex 6577 | . . . 4 ⊢ (ℝ ↑𝑚 (1...𝑁)) ∈ V | |
5 | 2, 3, 4 | fvmpt 6191 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (ℝ ↑𝑚 (1...𝑁))) |
6 | 5 | eleq2d 2673 | . 2 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (ℝ ↑𝑚 (1...𝑁)))) |
7 | reex 9906 | . . 3 ⊢ ℝ ∈ V | |
8 | ovex 6577 | . . 3 ⊢ (1...𝑁) ∈ V | |
9 | 7, 8 | elmap 7772 | . 2 ⊢ (𝐴 ∈ (ℝ ↑𝑚 (1...𝑁)) ↔ 𝐴:(1...𝑁)⟶ℝ) |
10 | 6, 9 | syl6bb 275 | 1 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ℝcr 9814 1c1 9816 ℕcn 10897 ...cfz 12197 𝔼cee 25568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-ee 25571 |
This theorem is referenced by: mptelee 25575 eleei 25577 axlowdimlem5 25626 axlowdimlem7 25628 axlowdimlem10 25631 axlowdimlem14 25635 axlowdim1 25639 |
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