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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrx2pyel | Structured version Visualization version GIF version |
Description: The y-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.) |
Ref | Expression |
---|---|
rrx2px.i | ⊢ 𝐼 = {1, 2} |
rrx2px.b | ⊢ 𝑃 = (ℝ ↑m 𝐼) |
Ref | Expression |
---|---|
rrx2pyel | ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrx2px.b | . 2 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
2 | id 22 | . 2 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝑃) | |
3 | 2ex 12050 | . . . . 5 ⊢ 2 ∈ V | |
4 | 3 | prid2 4699 | . . . 4 ⊢ 2 ∈ {1, 2} |
5 | rrx2px.i | . . . 4 ⊢ 𝐼 = {1, 2} | |
6 | 4, 5 | eleqtrri 2838 | . . 3 ⊢ 2 ∈ 𝐼 |
7 | 6 | a1i 11 | . 2 ⊢ (𝑋 ∈ 𝑃 → 2 ∈ 𝐼) |
8 | 1, 2, 7 | mapfvd 8667 | 1 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 {cpr 4563 ‘cfv 6433 (class class class)co 7275 ↑m cmap 8615 ℝcr 10870 1c1 10872 2c2 12028 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-1cn 10929 ax-addcl 10931 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-map 8617 df-2 12036 |
This theorem is referenced by: rrx2pnedifcoorneor 46062 rrx2pnedifcoorneorr 46063 ehl2eudisval0 46071 ehl2eudis0lt 46072 rrx2vlinest 46087 rrx2linest 46088 rrx2linest2 46090 2sphere 46095 2sphere0 46096 line2 46098 line2x 46100 line2y 46101 itsclc0 46117 itsclc0b 46118 itsclinecirc0 46119 itsclinecirc0b 46120 itsclinecirc0in 46121 itscnhlinecirc02plem3 46130 itscnhlinecirc02p 46131 inlinecirc02plem 46132 inlinecirc02p 46133 |
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