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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 11698 | . . 3 ⊢ i ≠ 0 | |
| 2 | 1 | neii 2942 | . 2 ⊢ ¬ i = 0 |
| 3 | 0lt1 11785 | . . . . 5 ⊢ 0 < 1 | |
| 4 | 0re 11263 | . . . . . 6 ⊢ 0 ∈ ℝ | |
| 5 | 1re 11261 | . . . . . 6 ⊢ 1 ∈ ℝ | |
| 6 | 4, 5 | ltnsymi 11380 | . . . . 5 ⊢ (0 < 1 → ¬ 1 < 0) |
| 7 | 3, 6 | ax-mp 5 | . . . 4 ⊢ ¬ 1 < 0 |
| 8 | ixi 11892 | . . . . . . 7 ⊢ (i · i) = -1 | |
| 9 | 5 | renegcli 11570 | . . . . . . 7 ⊢ -1 ∈ ℝ |
| 10 | 8, 9 | eqeltri 2837 | . . . . . 6 ⊢ (i · i) ∈ ℝ |
| 11 | 4, 10, 5 | ltadd1i 11817 | . . . . 5 ⊢ (0 < (i · i) ↔ (0 + 1) < ((i · i) + 1)) |
| 12 | ax-1cn 11213 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
| 13 | 12 | addlidi 11449 | . . . . . 6 ⊢ (0 + 1) = 1 |
| 14 | ax-i2m1 11223 | . . . . . 6 ⊢ ((i · i) + 1) = 0 | |
| 15 | 13, 14 | breq12i 5152 | . . . . 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 11783 | . . . . 5 ⊢ ((i ∈ ℝ ∧ i ≠ 0) → 0 < (i · i)) | |
| 19 | 18 | ex 412 | . . . 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 197 | 1 ⊢ ¬ i ∈ ℝ |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 ici 11157 + caddc 11158 · cmul 11160 < clt 11295 -cneg 11493 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 |
| This theorem is referenced by: rimul 12257 nthruc 16288 areacirclem4 37718 ine1 42349 itrere 42353 sqrtnegnre 47319 requad01 47608 |
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