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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 11715 | . . . . 5 ⊢ 2 ∈ V | |
4 | 3 | prid2 4699 | . . . 4 ⊢ 2 ∈ {1, 2} |
5 | rrx2px.i | . . . 4 ⊢ 𝐼 = {1, 2} | |
6 | 4, 5 | eleqtrri 2912 | . . 3 ⊢ 2 ∈ 𝐼 |
7 | 6 | a1i 11 | . 2 ⊢ (𝑋 ∈ 𝑃 → 2 ∈ 𝐼) |
8 | 1, 2, 7 | mapfvd 8443 | 1 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {cpr 4569 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 ℝcr 10536 1c1 10538 2c2 11693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-1cn 10595 ax-addcl 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408 df-2 11701 |
This theorem is referenced by: rrx2pnedifcoorneor 44723 rrx2pnedifcoorneorr 44724 ehl2eudisval0 44732 ehl2eudis0lt 44733 rrx2vlinest 44748 rrx2linest 44749 rrx2linest2 44751 2sphere 44756 2sphere0 44757 line2 44759 line2x 44761 line2y 44762 itsclc0 44778 itsclc0b 44779 itsclinecirc0 44780 itsclinecirc0b 44781 itsclinecirc0in 44782 itscnhlinecirc02plem3 44791 itscnhlinecirc02p 44792 inlinecirc02plem 44793 inlinecirc02p 44794 |
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