![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11598 | . . 3 ⊢ i ≠ 0 | |
2 | 1 | neii 2942 | . 2 ⊢ ¬ i = 0 |
3 | 0lt1 11685 | . . . . 5 ⊢ 0 < 1 | |
4 | 0re 11165 | . . . . . 6 ⊢ 0 ∈ ℝ | |
5 | 1re 11163 | . . . . . 6 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | ltnsymi 11282 | . . . . 5 ⊢ (0 < 1 → ¬ 1 < 0) |
7 | 3, 6 | ax-mp 5 | . . . 4 ⊢ ¬ 1 < 0 |
8 | ixi 11792 | . . . . . . 7 ⊢ (i · i) = -1 | |
9 | 5 | renegcli 11470 | . . . . . . 7 ⊢ -1 ∈ ℝ |
10 | 8, 9 | eqeltri 2830 | . . . . . 6 ⊢ (i · i) ∈ ℝ |
11 | 4, 10, 5 | ltadd1i 11717 | . . . . 5 ⊢ (0 < (i · i) ↔ (0 + 1) < ((i · i) + 1)) |
12 | ax-1cn 11117 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
13 | 12 | addlidi 11351 | . . . . . 6 ⊢ (0 + 1) = 1 |
14 | ax-i2m1 11127 | . . . . . 6 ⊢ ((i · i) + 1) = 0 | |
15 | 13, 14 | breq12i 5118 | . . . . 5 ⊢ ((0 + 1) < ((i · i) + 1) ↔ 1 < 0) |
16 | 11, 15 | bitri 275 | . . . 4 ⊢ (0 < (i · i) ↔ 1 < 0) |
17 | 7, 16 | mtbir 323 | . . 3 ⊢ ¬ 0 < (i · i) |
18 | msqgt0 11683 | . . . . 5 ⊢ ((i ∈ ℝ ∧ i ≠ 0) → 0 < (i · i)) | |
19 | 18 | ex 414 | . . . 4 ⊢ (i ∈ ℝ → (i ≠ 0 → 0 < (i · i))) |
20 | 19 | necon1bd 2958 | . . 3 ⊢ (i ∈ ℝ → (¬ 0 < (i · i) → i = 0)) |
21 | 17, 20 | mpi 20 | . 2 ⊢ (i ∈ ℝ → i = 0) |
22 | 2, 21 | mto 196 | 1 ⊢ ¬ i ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 class class class wbr 5109 (class class class)co 7361 ℝcr 11058 0cc0 11059 1c1 11060 ici 11061 + caddc 11062 · cmul 11064 < clt 11197 -cneg 11394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 |
This theorem is referenced by: rimul 12152 nthruc 16142 areacirclem4 36219 sqrtnegnre 45629 requad01 45903 |
Copyright terms: Public domain | W3C validator |