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Mirrors > Home > ILE Home > Th. List > rerecclap | GIF version |
Description: Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.) |
Ref | Expression |
---|---|
rerecclap | ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7987 | . . . . . 6 ⊢ 0 ∈ ℝ | |
2 | apreap 8574 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 # 0 ↔ 𝐴 #ℝ 0)) | |
3 | 1, 2 | mpan2 425 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 # 0 ↔ 𝐴 #ℝ 0)) |
4 | 3 | pm5.32i 454 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) ↔ (𝐴 ∈ ℝ ∧ 𝐴 #ℝ 0)) |
5 | recexre 8565 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 #ℝ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) | |
6 | 4, 5 | sylbi 121 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) |
7 | eqcom 2191 | . . . . 5 ⊢ (𝑥 = (1 / 𝐴) ↔ (1 / 𝐴) = 𝑥) | |
8 | 1cnd 8003 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ) | |
9 | simpr 110 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
10 | 9 | recnd 8016 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ) |
11 | simpll 527 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℝ) | |
12 | 11 | recnd 8016 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℂ) |
13 | simplr 528 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑥 ∈ ℝ) → 𝐴 # 0) | |
14 | divmulap 8662 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝐴 ∈ ℂ ∧ 𝐴 # 0)) → ((1 / 𝐴) = 𝑥 ↔ (𝐴 · 𝑥) = 1)) | |
15 | 8, 10, 12, 13, 14 | syl112anc 1253 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑥 ∈ ℝ) → ((1 / 𝐴) = 𝑥 ↔ (𝐴 · 𝑥) = 1)) |
16 | 7, 15 | bitrid 192 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 # 0) ∧ 𝑥 ∈ ℝ) → (𝑥 = (1 / 𝐴) ↔ (𝐴 · 𝑥) = 1)) |
17 | 16 | rexbidva 2487 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (∃𝑥 ∈ ℝ 𝑥 = (1 / 𝐴) ↔ ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)) |
18 | 6, 17 | mpbird 167 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → ∃𝑥 ∈ ℝ 𝑥 = (1 / 𝐴)) |
19 | risset 2518 | . 2 ⊢ ((1 / 𝐴) ∈ ℝ ↔ ∃𝑥 ∈ ℝ 𝑥 = (1 / 𝐴)) | |
20 | 18, 19 | sylibr 134 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ∃wrex 2469 class class class wbr 4018 (class class class)co 5896 ℂcc 7839 ℝcr 7840 0cc0 7841 1c1 7842 · cmul 7846 #ℝ creap 8561 # cap 8568 / cdiv 8659 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 ax-pre-mulext 7959 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 df-div 8660 |
This theorem is referenced by: redivclap 8718 rerecclapzi 8763 rerecclapd 8821 rerecapb 8830 ltdiv2 8874 recnz 9376 reexpclzap 10571 redivap 10915 imdivap 10922 caucvgrelemrec 11020 trirec0 15251 |
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