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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 10594 | . . . 4 ⊢ 1 ≠ 0 | |
2 | 1 | neii 3015 | . . 3 ⊢ ¬ 1 = 0 |
3 | oveq2 7153 | . . . . . 6 ⊢ (i = 0 → (i · i) = (i · 0)) | |
4 | ax-icn 10584 | . . . . . . 7 ⊢ i ∈ ℂ | |
5 | 4 | mul01i 10818 | . . . . . 6 ⊢ (i · 0) = 0 |
6 | 3, 5 | syl6req 2870 | . . . . 5 ⊢ (i = 0 → 0 = (i · i)) |
7 | 6 | oveq1d 7160 | . . . 4 ⊢ (i = 0 → (0 + 1) = ((i · i) + 1)) |
8 | ax-1cn 10583 | . . . . 5 ⊢ 1 ∈ ℂ | |
9 | 8 | addid2i 10816 | . . . 4 ⊢ (0 + 1) = 1 |
10 | ax-i2m1 10593 | . . . 4 ⊢ ((i · i) + 1) = 0 | |
11 | 7, 9, 10 | 3eqtr3g 2876 | . . 3 ⊢ (i = 0 → 1 = 0) |
12 | 2, 11 | mto 198 | . 2 ⊢ ¬ i = 0 |
13 | 12 | neir 3016 | 1 ⊢ i ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ≠ wne 3013 (class class class)co 7145 0cc0 10525 1c1 10526 ici 10527 + caddc 10528 · cmul 10530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 |
This theorem is referenced by: inelr 11616 2muline0 11849 irec 13552 iexpcyc 13557 imre 14455 reim 14456 crim 14462 cjreb 14470 cnpart 14587 tanval2 15474 tanval3 15475 efival 15493 sinhval 15495 retanhcl 15500 tanhlt1 15501 tanhbnd 15502 itgz 24308 ibl0 24314 iblcnlem1 24315 itgcnlem 24317 iblss 24332 iblss2 24333 itgss 24339 itgeqa 24341 iblconst 24345 iblabsr 24357 iblmulc2 24358 itgsplit 24363 dvsincos 24505 efeq1 25040 tanregt0 25050 efif1olem4 25056 eflogeq 25112 cxpsqrtlem 25212 root1eq1 25263 ang180lem1 25314 ang180lem2 25315 ang180lem3 25316 atandm2 25382 2efiatan 25423 atantan 25428 dvatan 25440 atantayl2 25443 log2cnv 25449 ccfldextdgrr 30956 itgexpif 31776 logi 32863 iexpire 32864 iblmulc2nc 34838 ftc1anclem6 34853 proot1ex 39679 iblsplit 42127 sinh-conventional 44766 |
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