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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 7440 | . . . . 5 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
| 2 | 1 | oveq2d 7448 | . . . 4 ⊢ (𝑛 = 𝑁 → (ℝ ↑m (1...𝑛)) = (ℝ ↑m (1...𝑁))) | 
| 3 | df-ee 28907 | . . . 4 ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛))) | |
| 4 | ovex 7465 | . . . 4 ⊢ (ℝ ↑m (1...𝑁)) ∈ V | |
| 5 | 2, 3, 4 | fvmpt 7015 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (ℝ ↑m (1...𝑁))) | 
| 6 | 5 | eleq2d 2826 | . 2 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (ℝ ↑m (1...𝑁)))) | 
| 7 | reex 11247 | . . 3 ⊢ ℝ ∈ V | |
| 8 | ovex 7465 | . . 3 ⊢ (1...𝑁) ∈ V | |
| 9 | 7, 8 | elmap 8912 | . 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 206 = wceq 1539 ∈ wcel 2107 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 ℝcr 11155 1c1 11157 ℕcn 12267 ...cfz 13548 𝔼cee 28904 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-ee 28907 | 
| This theorem is referenced by: mptelee 28911 eleei 28913 axlowdimlem5 28962 axlowdimlem7 28964 axlowdimlem10 28967 axlowdimlem14 28971 axlowdim1 28975 elntg2 29001 | 
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