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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 11224 | . . . 4 ⊢ 1 ≠ 0 | |
| 2 | 1 | neii 2942 | . . 3 ⊢ ¬ 1 = 0 |
| 3 | oveq2 7439 | . . . . . 6 ⊢ (i = 0 → (i · i) = (i · 0)) | |
| 4 | ax-icn 11214 | . . . . . . 7 ⊢ i ∈ ℂ | |
| 5 | 4 | mul01i 11451 | . . . . . 6 ⊢ (i · 0) = 0 |
| 6 | 3, 5 | eqtr2di 2794 | . . . . 5 ⊢ (i = 0 → 0 = (i · i)) |
| 7 | 6 | oveq1d 7446 | . . . 4 ⊢ (i = 0 → (0 + 1) = ((i · i) + 1)) |
| 8 | ax-1cn 11213 | . . . . 5 ⊢ 1 ∈ ℂ | |
| 9 | 8 | addlidi 11449 | . . . 4 ⊢ (0 + 1) = 1 |
| 10 | ax-i2m1 11223 | . . . 4 ⊢ ((i · i) + 1) = 0 | |
| 11 | 7, 9, 10 | 3eqtr3g 2800 | . . 3 ⊢ (i = 0 → 1 = 0) |
| 12 | 2, 11 | mto 197 | . 2 ⊢ ¬ i = 0 |
| 13 | 12 | neir 2943 | 1 ⊢ i ≠ 0 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ≠ wne 2940 (class class class)co 7431 0cc0 11155 1c1 11156 ici 11157 + caddc 11158 · cmul 11160 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 |
| This theorem is referenced by: inelr 12256 2muline0 12490 irec 14240 iexpcyc 14246 imre 15147 reim 15148 crim 15154 cjreb 15162 cnpart 15279 tanval2 16169 tanval3 16170 efival 16188 sinhval 16190 retanhcl 16195 tanhlt1 16196 tanhbnd 16197 itgz 25816 ibl0 25822 iblcnlem1 25823 itgcnlem 25825 iblss 25840 iblss2 25841 itgss 25847 itgeqa 25849 iblconst 25853 iblabsr 25865 iblmulc2 25866 itgsplit 25871 dvsincos 26019 efeq1 26570 tanregt0 26581 efif1olem4 26587 logi 26629 eflogeq 26644 cxpsqrtlem 26744 root1eq1 26798 ang180lem1 26852 ang180lem2 26853 ang180lem3 26854 atandm2 26920 2efiatan 26961 atantan 26966 dvatan 26978 atantayl2 26981 log2cnv 26987 ccfldextdgrr 33722 constrelextdg2 33788 itgexpif 34621 iexpire 35735 iblmulc2nc 37692 ftc1anclem6 37705 ef11d 42375 cxpi11d 42379 proot1ex 43208 iblsplit 45981 sinh-conventional 49258 |
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