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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 12293 | . . . . 5 ⊢ 2 ∈ V | |
4 | 3 | prid2 4762 | . . . 4 ⊢ 2 ∈ {1, 2} |
5 | rrx2px.i | . . . 4 ⊢ 𝐼 = {1, 2} | |
6 | 4, 5 | eleqtrri 2826 | . . 3 ⊢ 2 ∈ 𝐼 |
7 | 6 | a1i 11 | . 2 ⊢ (𝑋 ∈ 𝑃 → 2 ∈ 𝐼) |
8 | 1, 2, 7 | mapfvd 8875 | 1 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {cpr 4625 ‘cfv 6537 (class class class)co 7405 ↑m cmap 8822 ℝcr 11111 1c1 11113 2c2 12271 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-1cn 11170 ax-addcl 11172 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-map 8824 df-2 12279 |
This theorem is referenced by: rrx2pnedifcoorneor 47682 rrx2pnedifcoorneorr 47683 ehl2eudisval0 47691 ehl2eudis0lt 47692 rrx2vlinest 47707 rrx2linest 47708 rrx2linest2 47710 2sphere 47715 2sphere0 47716 line2 47718 line2x 47720 line2y 47721 itsclc0 47737 itsclc0b 47738 itsclinecirc0 47739 itsclinecirc0b 47740 itsclinecirc0in 47741 itscnhlinecirc02plem3 47750 itscnhlinecirc02p 47751 inlinecirc02plem 47752 inlinecirc02p 47753 |
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