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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 7414 | . . . . 5 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
2 | 1 | oveq2d 7422 | . . . 4 ⊢ (𝑛 = 𝑁 → (ℝ ↑m (1...𝑛)) = (ℝ ↑m (1...𝑁))) |
3 | df-ee 28139 | . . . 4 ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛))) | |
4 | ovex 7439 | . . . 4 ⊢ (ℝ ↑m (1...𝑁)) ∈ V | |
5 | 2, 3, 4 | fvmpt 6996 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (ℝ ↑m (1...𝑁))) |
6 | 5 | eleq2d 2820 | . 2 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (ℝ ↑m (1...𝑁)))) |
7 | reex 11198 | . . 3 ⊢ ℝ ∈ V | |
8 | ovex 7439 | . . 3 ⊢ (1...𝑁) ∈ V | |
9 | 7, 8 | elmap 8862 | . 2 ⊢ (𝐴 ∈ (ℝ ↑m (1...𝑁)) ↔ 𝐴:(1...𝑁)⟶ℝ) |
10 | 6, 9 | bitrdi 287 | 1 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ↑m cmap 8817 ℝcr 11106 1c1 11108 ℕcn 12209 ...cfz 13481 𝔼cee 28136 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-map 8819 df-ee 28139 |
This theorem is referenced by: mptelee 28143 eleei 28145 axlowdimlem5 28194 axlowdimlem7 28196 axlowdimlem10 28199 axlowdimlem14 28203 axlowdim1 28207 elntg2 28233 |
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