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Mirrors > Home > ILE Home > Th. List > recexgt0 | GIF version |
Description: Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Ref | Expression |
---|---|
recexgt0 | ⊢ ((A ∈ ℝ ∧ 0 < A) → ∃x ∈ ℝ (0 < x ∧ (A · x) = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-precex 6793 | . 2 ⊢ ((A ∈ ℝ ∧ 0 <ℝ A) → ∃x ∈ ℝ (0 <ℝ x ∧ (A · x) = 1)) | |
2 | 0re 6825 | . . . 4 ⊢ 0 ∈ ℝ | |
3 | ltxrlt 6882 | . . . 4 ⊢ ((0 ∈ ℝ ∧ A ∈ ℝ) → (0 < A ↔ 0 <ℝ A)) | |
4 | 2, 3 | mpan 400 | . . 3 ⊢ (A ∈ ℝ → (0 < A ↔ 0 <ℝ A)) |
5 | 4 | pm5.32i 427 | . 2 ⊢ ((A ∈ ℝ ∧ 0 < A) ↔ (A ∈ ℝ ∧ 0 <ℝ A)) |
6 | ltxrlt 6882 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ x ∈ ℝ) → (0 < x ↔ 0 <ℝ x)) | |
7 | 2, 6 | mpan 400 | . . . 4 ⊢ (x ∈ ℝ → (0 < x ↔ 0 <ℝ x)) |
8 | 7 | anbi1d 438 | . . 3 ⊢ (x ∈ ℝ → ((0 < x ∧ (A · x) = 1) ↔ (0 <ℝ x ∧ (A · x) = 1))) |
9 | 8 | rexbiia 2333 | . 2 ⊢ (∃x ∈ ℝ (0 < x ∧ (A · x) = 1) ↔ ∃x ∈ ℝ (0 <ℝ x ∧ (A · x) = 1)) |
10 | 1, 5, 9 | 3imtr4i 190 | 1 ⊢ ((A ∈ ℝ ∧ 0 < A) → ∃x ∈ ℝ (0 < x ∧ (A · x) = 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ∃wrex 2301 class class class wbr 3755 (class class class)co 5455 ℝcr 6710 0cc0 6711 1c1 6712 <ℝ cltrr 6715 · cmul 6716 < clt 6857 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-1re 6777 ax-addrcl 6780 ax-rnegex 6792 ax-precex 6793 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-xp 4294 df-pnf 6859 df-mnf 6860 df-ltxr 6862 |
This theorem is referenced by: ltmul1 7376 |
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