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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 8352 | . . . 4 ⊢ 1 # 0 | |
2 | 1 | olci 721 | . . 3 ⊢ (0 # 0 ∨ 1 # 0) |
3 | 0re 7766 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | 1re 7765 | . . . 4 ⊢ 1 ∈ ℝ | |
5 | apreim 8365 | . . . 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 423 | . . 3 ⊢ ((0 + (i · 1)) # (0 + (i · 0)) ↔ (0 # 0 ∨ 1 # 0)) |
7 | 2, 6 | mpbir 145 | . 2 ⊢ (0 + (i · 1)) # (0 + (i · 0)) |
8 | ax-icn 7715 | . . . . 5 ⊢ i ∈ ℂ | |
9 | 8 | mulid1i 7768 | . . . 4 ⊢ (i · 1) = i |
10 | 9 | oveq2i 5785 | . . 3 ⊢ (0 + (i · 1)) = (0 + i) |
11 | 8 | addid2i 7905 | . . 3 ⊢ (0 + i) = i |
12 | 10, 11 | eqtri 2160 | . 2 ⊢ (0 + (i · 1)) = i |
13 | it0e0 8941 | . . . 4 ⊢ (i · 0) = 0 | |
14 | 13 | oveq2i 5785 | . . 3 ⊢ (0 + (i · 0)) = (0 + 0) |
15 | 00id 7903 | . . 3 ⊢ (0 + 0) = 0 | |
16 | 14, 15 | eqtri 2160 | . 2 ⊢ (0 + (i · 0)) = 0 |
17 | 7, 12, 16 | 3brtr3i 3957 | 1 ⊢ i # 0 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 697 ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 ℝcr 7619 0cc0 7620 1c1 7621 ici 7622 + caddc 7623 · cmul 7625 # cap 8343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 |
This theorem is referenced by: 2muliap0 8944 irec 10392 iexpcyc 10397 imval 10622 imre 10623 reim 10624 crim 10630 cjreb 10638 tanval2ap 11420 tanval3ap 11421 efival 11439 |
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