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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrx2pxel | Structured version Visualization version GIF version |
Description: The x-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 |
---|---|
rrx2pxel | ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrx2px.b | . 2 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
2 | id 22 | . 2 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝑃) | |
3 | 1ex 10626 | . . . . 5 ⊢ 1 ∈ V | |
4 | 3 | prid1 4658 | . . . 4 ⊢ 1 ∈ {1, 2} |
5 | rrx2px.i | . . . 4 ⊢ 𝐼 = {1, 2} | |
6 | 4, 5 | eleqtrri 2889 | . . 3 ⊢ 1 ∈ 𝐼 |
7 | 6 | a1i 11 | . 2 ⊢ (𝑋 ∈ 𝑃 → 1 ∈ 𝐼) |
8 | 1, 2, 7 | mapfvd 8426 | 1 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {cpr 4527 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 ℝcr 10525 1c1 10527 2c2 11680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-1cn 10584 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 |
This theorem is referenced by: rrx2pnedifcoorneor 45130 rrx2plord2 45136 ehl2eudisval0 45139 ehl2eudis0lt 45140 rrx2vlinest 45155 rrx2linest 45156 rrx2linest2 45158 2sphere 45163 2sphere0 45164 line2 45166 line2x 45168 line2y 45169 itsclc0 45185 itsclc0b 45186 itsclinecirc0 45187 itsclinecirc0b 45188 itsclinecirc0in 45189 itscnhlinecirc02plem3 45198 itscnhlinecirc02p 45199 inlinecirc02plem 45200 inlinecirc02p 45201 |
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