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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 10634 | . . . . . . . 8 ⊢ 1 ∈ V | |
2 | 2ex 11712 | . . . . . . . 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 11843 | . . . . . . 7 ⊢ 1 ≠ 2 | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 1 ≠ 2) |
8 | 4, 5, 7 | 3jca 1123 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((1 ∈ V ∧ 2 ∈ V) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 1 ≠ 2)) |
9 | fprg 6914 | . . . . 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 4751 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝐴, 𝐵} ⊆ ℝ) | |
12 | 10, 11 | fssd 6525 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {〈1, 𝐴〉, 〈2, 𝐵〉}:{1, 2}⟶ℝ) |
13 | reex 10625 | . . . . 5 ⊢ ℝ ∈ V | |
14 | prex 5330 | . . . . 5 ⊢ {1, 2} ∈ V | |
15 | 13, 14 | pm3.2i 473 | . . . 4 ⊢ (ℝ ∈ V ∧ {1, 2} ∈ V) |
16 | elmapg 8416 | . . . 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 7164 | . . . 4 ⊢ (ℝ ↑m 𝐼) = (ℝ ↑m {1, 2}) |
22 | 19, 21 | eqtri 2843 | . . 3 ⊢ 𝑃 = (ℝ ↑m {1, 2}) |
23 | 22 | eleq2i 2903 | . 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 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 Vcvv 3493 {cpr 4566 〈cop 4570 ⟶wf 6348 (class class class)co 7153 ↑m cmap 8403 ℝcr 10533 1c1 10535 2c2 11690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-mpt 5144 df-id 5457 df-po 5471 df-so 5472 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-er 8286 df-map 8405 df-en 8507 df-dom 8508 df-sdom 8509 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-2 11698 |
This theorem is referenced by: prelrrx2b 44775 rrx2xpref1o 44779 rrx2plordisom 44784 line2ylem 44812 line2xlem 44814 itscnhlinecirc02p 44846 inlinecirc02plem 44847 |
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