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Mirrors > Home > ILE Home > Th. List > inelr | 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 8069 | . . 3 ⊢ i ≠ 0 | |
2 | 1 | neii 2282 | . 2 ⊢ ¬ i = 0 |
3 | 0lt1 7806 | . . . . . 6 ⊢ 0 < 1 | |
4 | 0re 7684 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
5 | 1re 7683 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | ltnsymi 7780 | . . . . . 6 ⊢ (0 < 1 → ¬ 1 < 0) |
7 | 3, 6 | ax-mp 7 | . . . . 5 ⊢ ¬ 1 < 0 |
8 | ixi 8257 | . . . . . . . 8 ⊢ (i · i) = -1 | |
9 | 5 | renegcli 7941 | . . . . . . . 8 ⊢ -1 ∈ ℝ |
10 | 8, 9 | eqeltri 2185 | . . . . . . 7 ⊢ (i · i) ∈ ℝ |
11 | 4, 10, 5 | ltadd1i 8177 | . . . . . 6 ⊢ (0 < (i · i) ↔ (0 + 1) < ((i · i) + 1)) |
12 | ax-1cn 7632 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
13 | 12 | addid2i 7822 | . . . . . . 7 ⊢ (0 + 1) = 1 |
14 | ax-i2m1 7644 | . . . . . . 7 ⊢ ((i · i) + 1) = 0 | |
15 | 13, 14 | breq12i 3902 | . . . . . 6 ⊢ ((0 + 1) < ((i · i) + 1) ↔ 1 < 0) |
16 | 11, 15 | bitri 183 | . . . . 5 ⊢ (0 < (i · i) ↔ 1 < 0) |
17 | 7, 16 | mtbir 643 | . . . 4 ⊢ ¬ 0 < (i · i) |
18 | mullt0 8155 | . . . . . 6 ⊢ (((i ∈ ℝ ∧ i < 0) ∧ (i ∈ ℝ ∧ i < 0)) → 0 < (i · i)) | |
19 | 18 | anidms 392 | . . . . 5 ⊢ ((i ∈ ℝ ∧ i < 0) → 0 < (i · i)) |
20 | 19 | ex 114 | . . . 4 ⊢ (i ∈ ℝ → (i < 0 → 0 < (i · i))) |
21 | 17, 20 | mtoi 636 | . . 3 ⊢ (i ∈ ℝ → ¬ i < 0) |
22 | mulgt0 7756 | . . . . . 6 ⊢ (((i ∈ ℝ ∧ 0 < i) ∧ (i ∈ ℝ ∧ 0 < i)) → 0 < (i · i)) | |
23 | 22 | anidms 392 | . . . . 5 ⊢ ((i ∈ ℝ ∧ 0 < i) → 0 < (i · i)) |
24 | 23 | ex 114 | . . . 4 ⊢ (i ∈ ℝ → (0 < i → 0 < (i · i))) |
25 | 17, 24 | mtoi 636 | . . 3 ⊢ (i ∈ ℝ → ¬ 0 < i) |
26 | lttri3 7761 | . . . 4 ⊢ ((i ∈ ℝ ∧ 0 ∈ ℝ) → (i = 0 ↔ (¬ i < 0 ∧ ¬ 0 < i))) | |
27 | 4, 26 | mpan2 419 | . . 3 ⊢ (i ∈ ℝ → (i = 0 ↔ (¬ i < 0 ∧ ¬ 0 < i))) |
28 | 21, 25, 27 | mpbir2and 909 | . 2 ⊢ (i ∈ ℝ → i = 0) |
29 | 2, 28 | mto 634 | 1 ⊢ ¬ i ∈ ℝ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 = wceq 1312 ∈ wcel 1461 class class class wbr 3893 (class class class)co 5726 ℝcr 7540 0cc0 7541 1c1 7542 ici 7543 + caddc 7544 · cmul 7546 < clt 7718 -cneg 7851 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-mulrcl 7638 ax-addcom 7639 ax-mulcom 7640 ax-addass 7641 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-0id 7647 ax-rnegex 7648 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-lttrn 7653 ax-pre-apti 7654 ax-pre-ltadd 7655 ax-pre-mulgt0 7656 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-iota 5044 df-fun 5081 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-pnf 7720 df-mnf 7721 df-ltxr 7723 df-sub 7852 df-neg 7853 |
This theorem is referenced by: rimul 8259 |
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