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Mirrors > Home > ILE Home > Th. List > ine0 | GIF version |
Description: The imaginary unit i is not zero. (Contributed by NM, 6-May-1999.) |
Ref | Expression |
---|---|
ine0 | ⊢ i ≠ 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7954 | . . . . 5 ⊢ 0 ∈ ℝ | |
2 | 0lt1 8080 | . . . . 5 ⊢ 0 < 1 | |
3 | 1, 2 | gtneii 8049 | . . . 4 ⊢ 1 ≠ 0 |
4 | 3 | neii 2349 | . . 3 ⊢ ¬ 1 = 0 |
5 | oveq2 5880 | . . . . . 6 ⊢ (i = 0 → (i · i) = (i · 0)) | |
6 | ax-icn 7903 | . . . . . . 7 ⊢ i ∈ ℂ | |
7 | 6 | mul01i 8344 | . . . . . 6 ⊢ (i · 0) = 0 |
8 | 5, 7 | eqtr2di 2227 | . . . . 5 ⊢ (i = 0 → 0 = (i · i)) |
9 | 8 | oveq1d 5887 | . . . 4 ⊢ (i = 0 → (0 + 1) = ((i · i) + 1)) |
10 | ax-1cn 7901 | . . . . 5 ⊢ 1 ∈ ℂ | |
11 | 10 | addid2i 8096 | . . . 4 ⊢ (0 + 1) = 1 |
12 | ax-i2m1 7913 | . . . 4 ⊢ ((i · i) + 1) = 0 | |
13 | 9, 11, 12 | 3eqtr3g 2233 | . . 3 ⊢ (i = 0 → 1 = 0) |
14 | 4, 13 | mto 662 | . 2 ⊢ ¬ i = 0 |
15 | 14 | neir 2350 | 1 ⊢ i ≠ 0 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ≠ wne 2347 (class class class)co 5872 0cc0 7808 1c1 7809 ici 7810 + caddc 7811 · cmul 7813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-cnre 7919 ax-pre-ltirr 7920 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5177 df-fun 5217 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-ltxr 7993 df-sub 8126 |
This theorem is referenced by: inelr 8537 |
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