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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 11673 | . . 3 ⊢ i ≠ 0 | |
2 | 1 | neii 2938 | . 2 ⊢ ¬ i = 0 |
3 | 0lt1 11760 | . . . . 5 ⊢ 0 < 1 | |
4 | 0re 11240 | . . . . . 6 ⊢ 0 ∈ ℝ | |
5 | 1re 11238 | . . . . . 6 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | ltnsymi 11357 | . . . . 5 ⊢ (0 < 1 → ¬ 1 < 0) |
7 | 3, 6 | ax-mp 5 | . . . 4 ⊢ ¬ 1 < 0 |
8 | ixi 11867 | . . . . . . 7 ⊢ (i · i) = -1 | |
9 | 5 | renegcli 11545 | . . . . . . 7 ⊢ -1 ∈ ℝ |
10 | 8, 9 | eqeltri 2825 | . . . . . 6 ⊢ (i · i) ∈ ℝ |
11 | 4, 10, 5 | ltadd1i 11792 | . . . . 5 ⊢ (0 < (i · i) ↔ (0 + 1) < ((i · i) + 1)) |
12 | ax-1cn 11190 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
13 | 12 | addlidi 11426 | . . . . . 6 ⊢ (0 + 1) = 1 |
14 | ax-i2m1 11200 | . . . . . 6 ⊢ ((i · i) + 1) = 0 | |
15 | 13, 14 | breq12i 5151 | . . . . 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 11758 | . . . . 5 ⊢ ((i ∈ ℝ ∧ i ≠ 0) → 0 < (i · i)) | |
19 | 18 | ex 412 | . . . 4 ⊢ (i ∈ ℝ → (i ≠ 0 → 0 < (i · i))) |
20 | 19 | necon1bd 2954 | . . 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 1534 ∈ wcel 2099 ≠ wne 2936 class class class wbr 5142 (class class class)co 7414 ℝcr 11131 0cc0 11132 1c1 11133 ici 11134 + caddc 11135 · cmul 11137 < clt 11272 -cneg 11469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 |
This theorem is referenced by: rimul 12227 nthruc 16222 areacirclem4 37178 ine1 41869 itrere 41873 sqrtnegnre 46681 requad01 46955 |
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