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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 10637 | . . . . . . . 8 ⊢ 1 ∈ V | |
2 | 2ex 11715 | . . . . . . . 8 ⊢ 2 ∈ V | |
3 | 1, 2 | pm3.2i 473 | . . . . . . 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 11846 | . . . . . . 7 ⊢ 1 ≠ 2 | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 1 ≠ 2) |
8 | 4, 5, 7 | 3jca 1124 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((1 ∈ V ∧ 2 ∈ V) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 1 ≠ 2)) |
9 | fprg 6917 | . . . . 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 4754 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝐴, 𝐵} ⊆ ℝ) | |
12 | 10, 11 | fssd 6528 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉}:{1, 2}⟶ℝ) |
13 | reex 10628 | . . . . 5 ⊢ ℝ ∈ V | |
14 | prex 5333 | . . . . 5 ⊢ {1, 2} ∈ V | |
15 | 13, 14 | pm3.2i 473 | . . . 4 ⊢ (ℝ ∈ V ∧ {1, 2} ∈ V) |
16 | elmapg 8419 | . . . 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 236 | . 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 7167 | . . . 4 ⊢ (ℝ ↑m 𝐼) = (ℝ ↑m {1, 2}) |
22 | 19, 21 | eqtri 2844 | . . 3 ⊢ 𝑃 = (ℝ ↑m {1, 2}) |
23 | 22 | eleq2i 2904 | . 2 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉} ∈ 𝑃 ↔ {〈1, 𝐴〉, 〈2, 𝐵〉} ∈ (ℝ ↑m {1, 2})) |
24 | 18, 23 | sylibr 236 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉} ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 {cpr 4569 〈cop 4573 ⟶wf 6351 (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-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 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-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 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-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-2 11701 |
This theorem is referenced by: prelrrx2b 44721 rrx2xpref1o 44725 rrx2plordisom 44730 line2ylem 44758 line2xlem 44760 itscnhlinecirc02p 44792 inlinecirc02plem 44793 |
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