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Mirrors > Home > ILE Home > Th. List > iap0 | GIF version |
Description: The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) |
Ref | Expression |
---|---|
iap0 | ⊢ i # 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ap0 7690 | . . . 4 ⊢ 1 # 0 | |
2 | 1 | olci 683 | . . 3 ⊢ (0 # 0 ∨ 1 # 0) |
3 | 0re 7119 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | 1re 7118 | . . . 4 ⊢ 1 ∈ ℝ | |
5 | apreim 7703 | . . . 4 ⊢ (((0 ∈ ℝ ∧ 1 ∈ ℝ) ∧ (0 ∈ ℝ ∧ 0 ∈ ℝ)) → ((0 + (i · 1)) # (0 + (i · 0)) ↔ (0 # 0 ∨ 1 # 0))) | |
6 | 3, 4, 3, 3, 5 | mp4an 417 | . . 3 ⊢ ((0 + (i · 1)) # (0 + (i · 0)) ↔ (0 # 0 ∨ 1 # 0)) |
7 | 2, 6 | mpbir 144 | . 2 ⊢ (0 + (i · 1)) # (0 + (i · 0)) |
8 | ax-icn 7071 | . . . . 5 ⊢ i ∈ ℂ | |
9 | 8 | mulid1i 7121 | . . . 4 ⊢ (i · 1) = i |
10 | 9 | oveq2i 5543 | . . 3 ⊢ (0 + (i · 1)) = (0 + i) |
11 | 8 | addid2i 7251 | . . 3 ⊢ (0 + i) = i |
12 | 10, 11 | eqtri 2101 | . 2 ⊢ (0 + (i · 1)) = i |
13 | it0e0 8252 | . . . 4 ⊢ (i · 0) = 0 | |
14 | 13 | oveq2i 5543 | . . 3 ⊢ (0 + (i · 0)) = (0 + 0) |
15 | 00id 7249 | . . 3 ⊢ (0 + 0) = 0 | |
16 | 14, 15 | eqtri 2101 | . 2 ⊢ (0 + (i · 0)) = 0 |
17 | 7, 12, 16 | 3brtr3i 3812 | 1 ⊢ i # 0 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∨ wo 661 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 ℝcr 6980 0cc0 6981 1c1 6982 ici 6983 + caddc 6984 · cmul 6986 # cap 7681 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 |
This theorem is referenced by: 2muliap0 8255 irec 9574 iexpcyc 9579 imval 9737 imre 9738 reim 9739 crim 9745 cjreb 9753 |
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