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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 11258 | . . . . . . . 8 ⊢ 1 ∈ V | |
| 2 | 2ex 12344 | . . . . . . . 8 ⊢ 2 ∈ V | |
| 3 | 1, 2 | pm3.2i 470 | . . . . . . 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 12475 | . . . . . . 7 ⊢ 1 ≠ 2 | |
| 7 | 6 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 1 ≠ 2) | 
| 8 | 4, 5, 7 | 3jca 1128 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((1 ∈ V ∧ 2 ∈ V) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 1 ≠ 2)) | 
| 9 | fprg 7174 | . . . . 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 4820 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝐴, 𝐵} ⊆ ℝ) | |
| 12 | 10, 11 | fssd 6752 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉}:{1, 2}⟶ℝ) | 
| 13 | reex 11247 | . . . . 5 ⊢ ℝ ∈ V | |
| 14 | prex 5436 | . . . . 5 ⊢ {1, 2} ∈ V | |
| 15 | 13, 14 | pm3.2i 470 | . . . 4 ⊢ (ℝ ∈ V ∧ {1, 2} ∈ V) | 
| 16 | elmapg 8880 | . . . 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 234 | . 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 7443 | . . . 4 ⊢ (ℝ ↑m 𝐼) = (ℝ ↑m {1, 2}) | 
| 22 | 19, 21 | eqtri 2764 | . . 3 ⊢ 𝑃 = (ℝ ↑m {1, 2}) | 
| 23 | 22 | eleq2i 2832 | . 2 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉} ∈ 𝑃 ↔ {〈1, 𝐴〉, 〈2, 𝐵〉} ∈ (ℝ ↑m {1, 2})) | 
| 24 | 18, 23 | sylibr 234 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉} ∈ 𝑃) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 {cpr 4627 〈cop 4631 ⟶wf 6556 (class class class)co 7432 ↑m cmap 8867 ℝcr 11155 1c1 11157 2c2 12322 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-2 12330 | 
| This theorem is referenced by: prelrrx2b 48640 rrx2xpref1o 48644 rrx2plordisom 48649 line2ylem 48677 line2xlem 48679 itscnhlinecirc02p 48711 inlinecirc02plem 48712 | 
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