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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 10043 | . . . 4 ⊢ 1 ≠ 0 | |
2 | 1 | neii 2825 | . . 3 ⊢ ¬ 1 = 0 |
3 | oveq2 6698 | . . . . . 6 ⊢ (i = 0 → (i · i) = (i · 0)) | |
4 | ax-icn 10033 | . . . . . . 7 ⊢ i ∈ ℂ | |
5 | 4 | mul01i 10264 | . . . . . 6 ⊢ (i · 0) = 0 |
6 | 3, 5 | syl6req 2702 | . . . . 5 ⊢ (i = 0 → 0 = (i · i)) |
7 | 6 | oveq1d 6705 | . . . 4 ⊢ (i = 0 → (0 + 1) = ((i · i) + 1)) |
8 | ax-1cn 10032 | . . . . 5 ⊢ 1 ∈ ℂ | |
9 | 8 | addid2i 10262 | . . . 4 ⊢ (0 + 1) = 1 |
10 | ax-i2m1 10042 | . . . 4 ⊢ ((i · i) + 1) = 0 | |
11 | 7, 9, 10 | 3eqtr3g 2708 | . . 3 ⊢ (i = 0 → 1 = 0) |
12 | 2, 11 | mto 188 | . 2 ⊢ ¬ i = 0 |
13 | 12 | neir 2826 | 1 ⊢ i ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ≠ wne 2823 (class class class)co 6690 0cc0 9974 1c1 9975 ici 9976 + caddc 9977 · cmul 9979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 |
This theorem is referenced by: inelr 11048 2muline0 11294 irec 13004 iexpcyc 13009 imre 13892 reim 13893 crim 13899 cjreb 13907 cnpart 14024 tanval2 14907 tanval3 14908 efival 14926 sinhval 14928 retanhcl 14933 tanhlt1 14934 tanhbnd 14935 itgz 23592 ibl0 23598 iblcnlem1 23599 itgcnlem 23601 iblss 23616 iblss2 23617 itgss 23623 itgeqa 23625 iblconst 23629 iblabsr 23641 iblmulc2 23642 itgsplit 23647 dvsincos 23789 efeq1 24320 tanregt0 24330 efif1olem4 24336 eflogeq 24393 cxpsqrtlem 24493 root1eq1 24541 ang180lem1 24584 ang180lem2 24585 ang180lem3 24586 atandm2 24649 2efiatan 24690 atantan 24695 dvatan 24707 atantayl2 24710 log2cnv 24716 itgexpif 30812 logi 31746 iexpire 31747 iblmulc2nc 33605 ftc1anclem6 33620 proot1ex 38096 iblsplit 40500 sinh-conventional 42808 |
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