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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 11725 | . . 3 ⊢ i ≠ 0 | |
2 | 1 | neii 2948 | . 2 ⊢ ¬ i = 0 |
3 | 0lt1 11812 | . . . . 5 ⊢ 0 < 1 | |
4 | 0re 11292 | . . . . . 6 ⊢ 0 ∈ ℝ | |
5 | 1re 11290 | . . . . . 6 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | ltnsymi 11409 | . . . . 5 ⊢ (0 < 1 → ¬ 1 < 0) |
7 | 3, 6 | ax-mp 5 | . . . 4 ⊢ ¬ 1 < 0 |
8 | ixi 11919 | . . . . . . 7 ⊢ (i · i) = -1 | |
9 | 5 | renegcli 11597 | . . . . . . 7 ⊢ -1 ∈ ℝ |
10 | 8, 9 | eqeltri 2840 | . . . . . 6 ⊢ (i · i) ∈ ℝ |
11 | 4, 10, 5 | ltadd1i 11844 | . . . . 5 ⊢ (0 < (i · i) ↔ (0 + 1) < ((i · i) + 1)) |
12 | ax-1cn 11242 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
13 | 12 | addlidi 11478 | . . . . . 6 ⊢ (0 + 1) = 1 |
14 | ax-i2m1 11252 | . . . . . 6 ⊢ ((i · i) + 1) = 0 | |
15 | 13, 14 | breq12i 5175 | . . . . 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 11810 | . . . . 5 ⊢ ((i ∈ ℝ ∧ i ≠ 0) → 0 < (i · i)) | |
19 | 18 | ex 412 | . . . 4 ⊢ (i ∈ ℝ → (i ≠ 0 → 0 < (i · i))) |
20 | 19 | necon1bd 2964 | . . 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 1537 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 (class class class)co 7448 ℝcr 11183 0cc0 11184 1c1 11185 ici 11186 + caddc 11187 · cmul 11189 < clt 11324 -cneg 11521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 |
This theorem is referenced by: rimul 12284 nthruc 16300 areacirclem4 37671 ine1 42303 itrere 42307 sqrtnegnre 47222 requad01 47495 |
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