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Mirrors > Home > MPE Home > Th. List > Mathboxes > prelrrx2 | Structured version Visualization version GIF version |
Description: An unordered pair of ordered pairs with first components 1 and 2 and real numbers as second components is a point in a real Euclidean space of dimension 2. (Contributed by AV, 4-Feb-2023.) |
Ref | Expression |
---|---|
prelrrx2.i | ⊢ 𝐼 = {1, 2} |
prelrrx2.b | ⊢ 𝑃 = (ℝ ↑m 𝐼) |
Ref | Expression |
---|---|
prelrrx2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉} ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ex 10626 | . . . . . . . 8 ⊢ 1 ∈ V | |
2 | 2ex 11702 | . . . . . . . 8 ⊢ 2 ∈ V | |
3 | 1, 2 | pm3.2i 474 | . . . . . . 7 ⊢ (1 ∈ V ∧ 2 ∈ V) |
4 | 3 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1 ∈ V ∧ 2 ∈ V)) |
5 | id 22 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) | |
6 | 1ne2 11833 | . . . . . . 7 ⊢ 1 ≠ 2 | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 1 ≠ 2) |
8 | 4, 5, 7 | 3jca 1125 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((1 ∈ V ∧ 2 ∈ V) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 1 ≠ 2)) |
9 | fprg 6894 | . . . . 5 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 1 ≠ 2) → {〈1, 𝐴〉, 〈2, 𝐵〉}:{1, 2}⟶{𝐴, 𝐵}) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉}:{1, 2}⟶{𝐴, 𝐵}) |
11 | prssi 4714 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝐴, 𝐵} ⊆ ℝ) | |
12 | 10, 11 | fssd 6502 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉}:{1, 2}⟶ℝ) |
13 | reex 10617 | . . . . 5 ⊢ ℝ ∈ V | |
14 | prex 5298 | . . . . 5 ⊢ {1, 2} ∈ V | |
15 | 13, 14 | pm3.2i 474 | . . . 4 ⊢ (ℝ ∈ V ∧ {1, 2} ∈ V) |
16 | elmapg 8402 | . . . 4 ⊢ ((ℝ ∈ V ∧ {1, 2} ∈ V) → ({〈1, 𝐴〉, 〈2, 𝐵〉} ∈ (ℝ ↑m {1, 2}) ↔ {〈1, 𝐴〉, 〈2, 𝐵〉}:{1, 2}⟶ℝ)) | |
17 | 15, 16 | ax-mp 5 | . . 3 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉} ∈ (ℝ ↑m {1, 2}) ↔ {〈1, 𝐴〉, 〈2, 𝐵〉}:{1, 2}⟶ℝ) |
18 | 12, 17 | sylibr 237 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉} ∈ (ℝ ↑m {1, 2})) |
19 | prelrrx2.b | . . . 4 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
20 | prelrrx2.i | . . . . 5 ⊢ 𝐼 = {1, 2} | |
21 | 20 | oveq2i 7146 | . . . 4 ⊢ (ℝ ↑m 𝐼) = (ℝ ↑m {1, 2}) |
22 | 19, 21 | eqtri 2821 | . . 3 ⊢ 𝑃 = (ℝ ↑m {1, 2}) |
23 | 22 | eleq2i 2881 | . 2 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉} ∈ 𝑃 ↔ {〈1, 𝐴〉, 〈2, 𝐵〉} ∈ (ℝ ↑m {1, 2})) |
24 | 18, 23 | sylibr 237 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉} ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 {cpr 4527 〈cop 4531 ⟶wf 6320 (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-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 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-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 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-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 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-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-2 11688 |
This theorem is referenced by: prelrrx2b 45128 rrx2xpref1o 45132 rrx2plordisom 45137 line2ylem 45165 line2xlem 45167 itscnhlinecirc02p 45199 inlinecirc02plem 45200 |
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