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Mirrors > Home > ILE Home > Th. List > caucvgrelemrec | GIF version |
Description: Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.) |
Ref | Expression |
---|---|
caucvgrelemrec | ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (℩𝑟 ∈ ℝ (𝐴 · 𝑟) = 1) = (1 / 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rerecclap 8636 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ) | |
2 | simpr 109 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑟 ∈ ℝ) → 𝑟 ∈ ℝ) | |
3 | 2 | recnd 7937 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑟 ∈ ℝ) → 𝑟 ∈ ℂ) |
4 | simpll 524 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑟 ∈ ℝ) → 𝐴 ∈ ℝ) | |
5 | 4 | recnd 7937 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑟 ∈ ℝ) → 𝐴 ∈ ℂ) |
6 | simplr 525 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑟 ∈ ℝ) → 𝐴 # 0) | |
7 | ax-1cn 7856 | . . . . 5 ⊢ 1 ∈ ℂ | |
8 | divmulap 8581 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝑟 ∈ ℂ ∧ (𝐴 ∈ ℂ ∧ 𝐴 # 0)) → ((1 / 𝐴) = 𝑟 ↔ (𝐴 · 𝑟) = 1)) | |
9 | 7, 8 | mp3an1 1319 | . . . 4 ⊢ ((𝑟 ∈ ℂ ∧ (𝐴 ∈ ℂ ∧ 𝐴 # 0)) → ((1 / 𝐴) = 𝑟 ↔ (𝐴 · 𝑟) = 1)) |
10 | 3, 5, 6, 9 | syl12anc 1231 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑟 ∈ ℝ) → ((1 / 𝐴) = 𝑟 ↔ (𝐴 · 𝑟) = 1)) |
11 | eqcom 2172 | . . 3 ⊢ ((1 / 𝐴) = 𝑟 ↔ 𝑟 = (1 / 𝐴)) | |
12 | 10, 11 | bitr3di 194 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑟 ∈ ℝ) → ((𝐴 · 𝑟) = 1 ↔ 𝑟 = (1 / 𝐴))) |
13 | 1, 12 | riota5 5832 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (℩𝑟 ∈ ℝ (𝐴 · 𝑟) = 1) = (1 / 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 class class class wbr 3987 ℩crio 5806 (class class class)co 5851 ℂcc 7761 ℝcr 7762 0cc0 7763 1c1 7764 · cmul 7768 # cap 8489 / cdiv 8578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-mulrcl 7862 ax-addcom 7863 ax-mulcom 7864 ax-addass 7865 ax-mulass 7866 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-1rid 7870 ax-0id 7871 ax-rnegex 7872 ax-precex 7873 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-apti 7878 ax-pre-ltadd 7879 ax-pre-mulgt0 7880 ax-pre-mulext 7881 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-reap 8483 df-ap 8490 df-div 8579 |
This theorem is referenced by: caucvgrelemcau 10933 |
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