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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 11215 | . . . 4 ⊢ 1 ≠ 0 | |
2 | 1 | neii 2932 | . . 3 ⊢ ¬ 1 = 0 |
3 | oveq2 7421 | . . . . . 6 ⊢ (i = 0 → (i · i) = (i · 0)) | |
4 | ax-icn 11205 | . . . . . . 7 ⊢ i ∈ ℂ | |
5 | 4 | mul01i 11442 | . . . . . 6 ⊢ (i · 0) = 0 |
6 | 3, 5 | eqtr2di 2783 | . . . . 5 ⊢ (i = 0 → 0 = (i · i)) |
7 | 6 | oveq1d 7428 | . . . 4 ⊢ (i = 0 → (0 + 1) = ((i · i) + 1)) |
8 | ax-1cn 11204 | . . . . 5 ⊢ 1 ∈ ℂ | |
9 | 8 | addlidi 11440 | . . . 4 ⊢ (0 + 1) = 1 |
10 | ax-i2m1 11214 | . . . 4 ⊢ ((i · i) + 1) = 0 | |
11 | 7, 9, 10 | 3eqtr3g 2789 | . . 3 ⊢ (i = 0 → 1 = 0) |
12 | 2, 11 | mto 196 | . 2 ⊢ ¬ i = 0 |
13 | 12 | neir 2933 | 1 ⊢ i ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ≠ wne 2930 (class class class)co 7413 0cc0 11146 1c1 11147 ici 11148 + caddc 11149 · cmul 11151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7416 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-ltxr 11291 |
This theorem is referenced by: inelr 12245 2muline0 12479 irec 14210 iexpcyc 14216 imre 15105 reim 15106 crim 15112 cjreb 15120 cnpart 15237 tanval2 16127 tanval3 16128 efival 16146 sinhval 16148 retanhcl 16153 tanhlt1 16154 tanhbnd 16155 itgz 25795 ibl0 25801 iblcnlem1 25802 itgcnlem 25804 iblss 25819 iblss2 25820 itgss 25826 itgeqa 25828 iblconst 25832 iblabsr 25844 iblmulc2 25845 itgsplit 25850 dvsincos 25998 efeq1 26549 tanregt0 26560 efif1olem4 26566 logi 26608 eflogeq 26623 cxpsqrtlem 26723 root1eq1 26777 ang180lem1 26831 ang180lem2 26832 ang180lem3 26833 atandm2 26899 2efiatan 26940 atantan 26945 dvatan 26957 atantayl2 26960 log2cnv 26966 ccfldextdgrr 33561 constrelextdg2 33616 itgexpif 34462 iexpire 35567 iblmulc2nc 37396 ftc1anclem6 37409 ef11d 42063 cxpi11d 42067 proot1ex 42895 iblsplit 45620 sinh-conventional 48518 |
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