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Mirrors > Home > ILE Home > Th. List > recdivap | GIF version |
Description: The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.) |
Ref | Expression |
---|---|
recdivap | ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1div1e1 8591 | . . . 4 ⊢ (1 / 1) = 1 | |
2 | 1 | oveq1i 5846 | . . 3 ⊢ ((1 / 1) / (𝐴 / 𝐵)) = (1 / (𝐴 / 𝐵)) |
3 | ax-1cn 7837 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | 1ap0 8479 | . . . . 5 ⊢ 1 # 0 | |
5 | 3, 4 | pm3.2i 270 | . . . 4 ⊢ (1 ∈ ℂ ∧ 1 # 0) |
6 | divdivdivap 8600 | . . . 4 ⊢ (((1 ∈ ℂ ∧ (1 ∈ ℂ ∧ 1 # 0)) ∧ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0))) → ((1 / 1) / (𝐴 / 𝐵)) = ((1 · 𝐵) / (1 · 𝐴))) | |
7 | 3, 5, 6 | mpanl12 433 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 1) / (𝐴 / 𝐵)) = ((1 · 𝐵) / (1 · 𝐴))) |
8 | 2, 7 | eqtr3id 2211 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (1 / (𝐴 / 𝐵)) = ((1 · 𝐵) / (1 · 𝐴))) |
9 | mulid2 7888 | . . . 4 ⊢ (𝐵 ∈ ℂ → (1 · 𝐵) = 𝐵) | |
10 | mulid2 7888 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
11 | 9, 10 | oveqan12rd 5856 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 · 𝐵) / (1 · 𝐴)) = (𝐵 / 𝐴)) |
12 | 11 | ad2ant2r 501 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 · 𝐵) / (1 · 𝐴)) = (𝐵 / 𝐴)) |
13 | 8, 12 | eqtrd 2197 | 1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 (class class class)co 5836 ℂcc 7742 0cc0 7744 1c1 7745 · cmul 7749 # cap 8470 / cdiv 8559 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 |
This theorem is referenced by: divcanap6 8606 recdivapd 8694 |
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