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Mirrors > Home > MPE Home > Th. List > ine0 | Structured version Visualization version 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 | ax-1ne0 9884 | . . . 4 ⊢ 1 ≠ 0 | |
2 | 1 | neii 2784 | . . 3 ⊢ ¬ 1 = 0 |
3 | oveq2 6557 | . . . . . 6 ⊢ (i = 0 → (i · i) = (i · 0)) | |
4 | ax-icn 9874 | . . . . . . 7 ⊢ i ∈ ℂ | |
5 | 4 | mul01i 10105 | . . . . . 6 ⊢ (i · 0) = 0 |
6 | 3, 5 | syl6req 2661 | . . . . 5 ⊢ (i = 0 → 0 = (i · i)) |
7 | 6 | oveq1d 6564 | . . . 4 ⊢ (i = 0 → (0 + 1) = ((i · i) + 1)) |
8 | ax-1cn 9873 | . . . . 5 ⊢ 1 ∈ ℂ | |
9 | 8 | addid2i 10103 | . . . 4 ⊢ (0 + 1) = 1 |
10 | ax-i2m1 9883 | . . . 4 ⊢ ((i · i) + 1) = 0 | |
11 | 7, 9, 10 | 3eqtr3g 2667 | . . 3 ⊢ (i = 0 → 1 = 0) |
12 | 2, 11 | mto 187 | . 2 ⊢ ¬ i = 0 |
13 | 12 | neir 2785 | 1 ⊢ i ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ≠ wne 2780 (class class class)co 6549 0cc0 9815 1c1 9816 ici 9817 + caddc 9818 · cmul 9820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 |
This theorem is referenced by: inelr 10887 2muline0 11133 irec 12826 iexpcyc 12831 imre 13696 reim 13697 crim 13703 cjreb 13711 cnpart 13828 tanval2 14702 tanval3 14703 efival 14721 sinhval 14723 retanhcl 14728 tanhlt1 14729 tanhbnd 14730 itgz 23353 ibl0 23359 iblcnlem1 23360 itgcnlem 23362 iblss 23377 iblss2 23378 itgss 23384 itgeqa 23386 iblconst 23390 iblabsr 23402 iblmulc2 23403 itgsplit 23408 dvsincos 23548 efeq1 24079 tanregt0 24089 efif1olem4 24095 eflogeq 24152 cxpsqrtlem 24248 root1eq1 24296 ang180lem1 24339 ang180lem2 24340 ang180lem3 24341 atandm2 24404 2efiatan 24445 atantan 24450 dvatan 24462 atantayl2 24465 log2cnv 24471 logi 30873 iexpire 30874 iblmulc2nc 32645 ftc1anclem6 32660 proot1ex 36798 iblsplit 38858 sinh-conventional 42279 |
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