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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 10359 | . . . . . . . 8 ⊢ 1 ∈ V | |
2 | 2ex 11435 | . . . . . . . 8 ⊢ 2 ∈ V | |
3 | 1, 2 | pm3.2i 464 | . . . . . . 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 11573 | . . . . . . 7 ⊢ 1 ≠ 2 | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 1 ≠ 2) |
8 | 4, 5, 7 | 3jca 1162 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((1 ∈ V ∧ 2 ∈ V) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 1 ≠ 2)) |
9 | fprg 6678 | . . . . 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 4572 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝐴, 𝐵} ⊆ ℝ) | |
12 | 10, 11 | fssd 6296 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉}:{1, 2}⟶ℝ) |
13 | reex 10350 | . . . . 5 ⊢ ℝ ∈ V | |
14 | prex 5132 | . . . . 5 ⊢ {1, 2} ∈ V | |
15 | 13, 14 | pm3.2i 464 | . . . 4 ⊢ (ℝ ∈ V ∧ {1, 2} ∈ V) |
16 | elmapg 8140 | . . . 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 226 | . 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 6921 | . . . 4 ⊢ (ℝ ↑𝑚 𝐼) = (ℝ ↑𝑚 {1, 2}) |
22 | 19, 21 | eqtri 2849 | . . 3 ⊢ 𝑃 = (ℝ ↑𝑚 {1, 2}) |
23 | 22 | eleq2i 2898 | . 2 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉} ∈ 𝑃 ↔ {〈1, 𝐴〉, 〈2, 𝐵〉} ∈ (ℝ ↑𝑚 {1, 2})) |
24 | 18, 23 | sylibr 226 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉} ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 Vcvv 3414 {cpr 4401 〈cop 4405 ⟶wf 6123 (class class class)co 6910 ↑𝑚 cmap 8127 ℝcr 10258 1c1 10260 2c2 11413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-2 11421 |
This theorem is referenced by: rrx2xpref1o 42276 rrx2plordisom 42281 line2ylem 43313 line2xlem 43315 |
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