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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 8465 | . . . 4 ⊢ 1 # 0 | |
2 | 1 | olci 722 | . . 3 ⊢ (0 # 0 ∨ 1 # 0) |
3 | 0re 7878 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | 1re 7877 | . . . 4 ⊢ 1 ∈ ℝ | |
5 | apreim 8478 | . . . 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 424 | . . 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 7827 | . . . . 5 ⊢ i ∈ ℂ | |
9 | 8 | mulid1i 7880 | . . . 4 ⊢ (i · 1) = i |
10 | 9 | oveq2i 5835 | . . 3 ⊢ (0 + (i · 1)) = (0 + i) |
11 | 8 | addid2i 8018 | . . 3 ⊢ (0 + i) = i |
12 | 10, 11 | eqtri 2178 | . 2 ⊢ (0 + (i · 1)) = i |
13 | it0e0 9054 | . . . 4 ⊢ (i · 0) = 0 | |
14 | 13 | oveq2i 5835 | . . 3 ⊢ (0 + (i · 0)) = (0 + 0) |
15 | 00id 8016 | . . 3 ⊢ (0 + 0) = 0 | |
16 | 14, 15 | eqtri 2178 | . 2 ⊢ (0 + (i · 0)) = 0 |
17 | 7, 12, 16 | 3brtr3i 3993 | 1 ⊢ i # 0 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 698 ∈ wcel 2128 class class class wbr 3965 (class class class)co 5824 ℝcr 7731 0cc0 7732 1c1 7733 ici 7734 + caddc 7735 · cmul 7737 # cap 8456 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-ltxr 7917 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 |
This theorem is referenced by: 2muliap0 9057 irec 10518 iexpcyc 10523 imval 10750 imre 10751 reim 10752 crim 10758 cjreb 10766 tanval2ap 11610 tanval3ap 11611 efival 11629 |
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