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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 11247 | . . . . 5 ⊢ 1 ∈ V | |
4 | 3 | prid1 4768 | . . . 4 ⊢ 1 ∈ {1, 2} |
5 | rrx2px.i | . . . 4 ⊢ 𝐼 = {1, 2} | |
6 | 4, 5 | eleqtrri 2824 | . . 3 ⊢ 1 ∈ 𝐼 |
7 | 6 | a1i 11 | . 2 ⊢ (𝑋 ∈ 𝑃 → 1 ∈ 𝐼) |
8 | 1, 2, 7 | mapfvd 8898 | 1 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {cpr 4632 ‘cfv 6549 (class class class)co 7419 ↑m cmap 8845 ℝcr 11144 1c1 11146 2c2 12305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-1cn 11203 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-map 8847 |
This theorem is referenced by: rrx2pnedifcoorneor 47980 rrx2plord2 47986 ehl2eudisval0 47989 ehl2eudis0lt 47990 rrx2vlinest 48005 rrx2linest 48006 rrx2linest2 48008 2sphere 48013 2sphere0 48014 line2 48016 line2x 48018 line2y 48019 itsclc0 48035 itsclc0b 48036 itsclinecirc0 48037 itsclinecirc0b 48038 itsclinecirc0in 48039 itscnhlinecirc02plem3 48048 itscnhlinecirc02p 48049 inlinecirc02plem 48050 inlinecirc02p 48051 |
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