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Mirrors > Home > ILE Home > Th. List > inelr | GIF version |
Description: The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.) |
Ref | Expression |
---|---|
inelr | ⊢ ¬ i ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ine0 8283 | . . 3 ⊢ i ≠ 0 | |
2 | 1 | neii 2336 | . 2 ⊢ ¬ i = 0 |
3 | 0lt1 8016 | . . . . . 6 ⊢ 0 < 1 | |
4 | 0re 7890 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
5 | 1re 7889 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | ltnsymi 7989 | . . . . . 6 ⊢ (0 < 1 → ¬ 1 < 0) |
7 | 3, 6 | ax-mp 5 | . . . . 5 ⊢ ¬ 1 < 0 |
8 | ixi 8472 | . . . . . . . 8 ⊢ (i · i) = -1 | |
9 | 5 | renegcli 8151 | . . . . . . . 8 ⊢ -1 ∈ ℝ |
10 | 8, 9 | eqeltri 2237 | . . . . . . 7 ⊢ (i · i) ∈ ℝ |
11 | 4, 10, 5 | ltadd1i 8391 | . . . . . 6 ⊢ (0 < (i · i) ↔ (0 + 1) < ((i · i) + 1)) |
12 | ax-1cn 7837 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
13 | 12 | addid2i 8032 | . . . . . . 7 ⊢ (0 + 1) = 1 |
14 | ax-i2m1 7849 | . . . . . . 7 ⊢ ((i · i) + 1) = 0 | |
15 | 13, 14 | breq12i 3985 | . . . . . 6 ⊢ ((0 + 1) < ((i · i) + 1) ↔ 1 < 0) |
16 | 11, 15 | bitri 183 | . . . . 5 ⊢ (0 < (i · i) ↔ 1 < 0) |
17 | 7, 16 | mtbir 661 | . . . 4 ⊢ ¬ 0 < (i · i) |
18 | mullt0 8369 | . . . . . 6 ⊢ (((i ∈ ℝ ∧ i < 0) ∧ (i ∈ ℝ ∧ i < 0)) → 0 < (i · i)) | |
19 | 18 | anidms 395 | . . . . 5 ⊢ ((i ∈ ℝ ∧ i < 0) → 0 < (i · i)) |
20 | 19 | ex 114 | . . . 4 ⊢ (i ∈ ℝ → (i < 0 → 0 < (i · i))) |
21 | 17, 20 | mtoi 654 | . . 3 ⊢ (i ∈ ℝ → ¬ i < 0) |
22 | mulgt0 7964 | . . . . . 6 ⊢ (((i ∈ ℝ ∧ 0 < i) ∧ (i ∈ ℝ ∧ 0 < i)) → 0 < (i · i)) | |
23 | 22 | anidms 395 | . . . . 5 ⊢ ((i ∈ ℝ ∧ 0 < i) → 0 < (i · i)) |
24 | 23 | ex 114 | . . . 4 ⊢ (i ∈ ℝ → (0 < i → 0 < (i · i))) |
25 | 17, 24 | mtoi 654 | . . 3 ⊢ (i ∈ ℝ → ¬ 0 < i) |
26 | lttri3 7969 | . . . 4 ⊢ ((i ∈ ℝ ∧ 0 ∈ ℝ) → (i = 0 ↔ (¬ i < 0 ∧ ¬ 0 < i))) | |
27 | 4, 26 | mpan2 422 | . . 3 ⊢ (i ∈ ℝ → (i = 0 ↔ (¬ i < 0 ∧ ¬ 0 < i))) |
28 | 21, 25, 27 | mpbir2and 933 | . 2 ⊢ (i ∈ ℝ → i = 0) |
29 | 2, 28 | mto 652 | 1 ⊢ ¬ i ∈ ℝ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 (class class class)co 5836 ℝcr 7743 0cc0 7744 1c1 7745 ici 7746 + caddc 7747 · cmul 7749 < clt 7924 -cneg 8061 |
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-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 |
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-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-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-ltxr 7929 df-sub 8062 df-neg 8063 |
This theorem is referenced by: rimul 8474 |
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