Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfrp2 | Structured version Visualization version GIF version |
Description: Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.) |
Ref | Expression |
---|---|
dfrp2 | ⊢ ℝ+ = (0(,)+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltpnf 12502 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
2 | 1 | adantr 483 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝑥 < +∞) |
3 | 2 | pm4.71i 562 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝑥 < +∞)) |
4 | df-3an 1085 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < +∞) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝑥 < +∞)) | |
5 | 3, 4 | bitr4i 280 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < +∞)) |
6 | elrp 12378 | . . 3 ⊢ (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) | |
7 | 0xr 10674 | . . . 4 ⊢ 0 ∈ ℝ* | |
8 | pnfxr 10681 | . . . 4 ⊢ +∞ ∈ ℝ* | |
9 | elioo2 12766 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑥 ∈ (0(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < +∞))) | |
10 | 7, 8, 9 | mp2an 690 | . . 3 ⊢ (𝑥 ∈ (0(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < +∞)) |
11 | 5, 6, 10 | 3bitr4i 305 | . 2 ⊢ (𝑥 ∈ ℝ+ ↔ 𝑥 ∈ (0(,)+∞)) |
12 | 11 | eqriv 2818 | 1 ⊢ ℝ+ = (0(,)+∞) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5052 (class class class)co 7142 ℝcr 10522 0cc0 10523 +∞cpnf 10658 ℝ*cxr 10660 < clt 10661 ℝ+crp 12376 (,)cioo 12725 |
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 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-addrcl 10584 ax-rnegex 10594 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 |
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-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-po 5460 df-so 5461 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-1st 7675 df-2nd 7676 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-rp 12377 df-ioo 12729 |
This theorem is referenced by: omssubadd 31565 |
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