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Mirrors > Home > ILE Home > Th. List > recrecap | GIF version |
Description: A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.) |
Ref | Expression |
---|---|
recrecap | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / (1 / 𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recidap2 8591 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((1 / 𝐴) · 𝐴) = 1) | |
2 | 1cnd 7923 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → 1 ∈ ℂ) | |
3 | simpl 108 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → 𝐴 ∈ ℂ) | |
4 | recclap 8583 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℂ) | |
5 | recap0 8589 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) # 0) | |
6 | divmulap 8579 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ ((1 / 𝐴) ∈ ℂ ∧ (1 / 𝐴) # 0)) → ((1 / (1 / 𝐴)) = 𝐴 ↔ ((1 / 𝐴) · 𝐴) = 1)) | |
7 | 2, 3, 4, 5, 6 | syl112anc 1237 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((1 / (1 / 𝐴)) = 𝐴 ↔ ((1 / 𝐴) · 𝐴) = 1)) |
8 | 1, 7 | mpbird 166 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / (1 / 𝐴)) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 class class class wbr 3987 (class class class)co 5850 ℂcc 7759 0cc0 7761 1c1 7762 · cmul 7766 # cap 8487 / cdiv 8576 |
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 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 |
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 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 |
This theorem is referenced by: recrecapi 8648 recrecapd 8689 |
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