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Mirrors > Home > MPE Home > Th. List > ehlval | Structured version Visualization version GIF version |
Description: Value of the Euclidean space of dimension 𝑁. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
ehlval.e | ⊢ 𝐸 = (𝔼hil‘𝑁) |
Ref | Expression |
---|---|
ehlval | ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ehlval.e | . 2 ⊢ 𝐸 = (𝔼hil‘𝑁) | |
2 | oveq2 7153 | . . . 4 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
3 | 2 | fveq2d 6667 | . . 3 ⊢ (𝑛 = 𝑁 → (ℝ^‘(1...𝑛)) = (ℝ^‘(1...𝑁))) |
4 | df-ehl 23916 | . . 3 ⊢ 𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛))) | |
5 | fvex 6676 | . . 3 ⊢ (ℝ^‘(1...𝑁)) ∈ V | |
6 | 3, 4, 5 | fvmpt 6761 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝔼hil‘𝑁) = (ℝ^‘(1...𝑁))) |
7 | 1, 6 | syl5eq 2865 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 1c1 10526 ℕ0cn0 11885 ...cfz 12880 ℝ^crrx 23913 𝔼hilcehl 23914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-ehl 23916 |
This theorem is referenced by: ehlbase 23945 ehl0 23947 ehleudis 23948 ehleudisval 23949 eenglngeehlnm 44654 2sphere 44664 itscnhlinecirc02plem3 44699 inlinecirc02p 44702 |
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