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Mirrors > Home > MPE Home > Th. List > inelr | Structured version Visualization version 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 10503 | . . 3 ⊢ i ≠ 0 | |
2 | 1 | neii 2825 | . 2 ⊢ ¬ i = 0 |
3 | 0lt1 10588 | . . . . 5 ⊢ 0 < 1 | |
4 | 0re 10078 | . . . . . 6 ⊢ 0 ∈ ℝ | |
5 | 1re 10077 | . . . . . 6 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | ltnsymi 10194 | . . . . 5 ⊢ (0 < 1 → ¬ 1 < 0) |
7 | 3, 6 | ax-mp 5 | . . . 4 ⊢ ¬ 1 < 0 |
8 | ixi 10694 | . . . . . . 7 ⊢ (i · i) = -1 | |
9 | 5 | renegcli 10380 | . . . . . . 7 ⊢ -1 ∈ ℝ |
10 | 8, 9 | eqeltri 2726 | . . . . . 6 ⊢ (i · i) ∈ ℝ |
11 | 4, 10, 5 | ltadd1i 10620 | . . . . 5 ⊢ (0 < (i · i) ↔ (0 + 1) < ((i · i) + 1)) |
12 | ax-1cn 10032 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
13 | 12 | addid2i 10262 | . . . . . 6 ⊢ (0 + 1) = 1 |
14 | ax-i2m1 10042 | . . . . . 6 ⊢ ((i · i) + 1) = 0 | |
15 | 13, 14 | breq12i 4694 | . . . . 5 ⊢ ((0 + 1) < ((i · i) + 1) ↔ 1 < 0) |
16 | 11, 15 | bitri 264 | . . . 4 ⊢ (0 < (i · i) ↔ 1 < 0) |
17 | 7, 16 | mtbir 312 | . . 3 ⊢ ¬ 0 < (i · i) |
18 | msqgt0 10586 | . . . . 5 ⊢ ((i ∈ ℝ ∧ i ≠ 0) → 0 < (i · i)) | |
19 | 18 | ex 449 | . . . 4 ⊢ (i ∈ ℝ → (i ≠ 0 → 0 < (i · i))) |
20 | 19 | necon1bd 2841 | . . 3 ⊢ (i ∈ ℝ → (¬ 0 < (i · i) → i = 0)) |
21 | 17, 20 | mpi 20 | . 2 ⊢ (i ∈ ℝ → i = 0) |
22 | 2, 21 | mto 188 | 1 ⊢ ¬ i ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 class class class wbr 4685 (class class class)co 6690 ℝcr 9973 0cc0 9974 1c1 9975 ici 9976 + caddc 9977 · cmul 9979 < clt 10112 -cneg 10305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 |
This theorem is referenced by: rimul 11049 nthruc 15025 areacirclem4 33633 |
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