![]() |
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 10874 | . . 3 ⊢ i ≠ 0 | |
2 | 1 | neii 2963 | . 2 ⊢ ¬ i = 0 |
3 | 0lt1 10961 | . . . . 5 ⊢ 0 < 1 | |
4 | 0re 10439 | . . . . . 6 ⊢ 0 ∈ ℝ | |
5 | 1re 10437 | . . . . . 6 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | ltnsymi 10557 | . . . . 5 ⊢ (0 < 1 → ¬ 1 < 0) |
7 | 3, 6 | ax-mp 5 | . . . 4 ⊢ ¬ 1 < 0 |
8 | ixi 11068 | . . . . . . 7 ⊢ (i · i) = -1 | |
9 | 5 | renegcli 10746 | . . . . . . 7 ⊢ -1 ∈ ℝ |
10 | 8, 9 | eqeltri 2856 | . . . . . 6 ⊢ (i · i) ∈ ℝ |
11 | 4, 10, 5 | ltadd1i 10993 | . . . . 5 ⊢ (0 < (i · i) ↔ (0 + 1) < ((i · i) + 1)) |
12 | ax-1cn 10391 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
13 | 12 | addid2i 10626 | . . . . . 6 ⊢ (0 + 1) = 1 |
14 | ax-i2m1 10401 | . . . . . 6 ⊢ ((i · i) + 1) = 0 | |
15 | 13, 14 | breq12i 4934 | . . . . 5 ⊢ ((0 + 1) < ((i · i) + 1) ↔ 1 < 0) |
16 | 11, 15 | bitri 267 | . . . 4 ⊢ (0 < (i · i) ↔ 1 < 0) |
17 | 7, 16 | mtbir 315 | . . 3 ⊢ ¬ 0 < (i · i) |
18 | msqgt0 10959 | . . . . 5 ⊢ ((i ∈ ℝ ∧ i ≠ 0) → 0 < (i · i)) | |
19 | 18 | ex 405 | . . . 4 ⊢ (i ∈ ℝ → (i ≠ 0 → 0 < (i · i))) |
20 | 19 | necon1bd 2979 | . . 3 ⊢ (i ∈ ℝ → (¬ 0 < (i · i) → i = 0)) |
21 | 17, 20 | mpi 20 | . 2 ⊢ (i ∈ ℝ → i = 0) |
22 | 2, 21 | mto 189 | 1 ⊢ ¬ i ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 class class class wbr 4925 (class class class)co 6974 ℝcr 10332 0cc0 10333 1c1 10334 ici 10335 + caddc 10336 · cmul 10338 < clt 10472 -cneg 10669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-po 5322 df-so 5323 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 |
This theorem is referenced by: rimul 11428 nthruc 15463 areacirclem4 34455 sqrtnegnre 42938 requad01 43179 |
Copyright terms: Public domain | W3C validator |