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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 8549 | . . . 4 ⊢ 1 # 0 | |
2 | 1 | olci 732 | . . 3 ⊢ (0 # 0 ∨ 1 # 0) |
3 | 0re 7959 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | 1re 7958 | . . . 4 ⊢ 1 ∈ ℝ | |
5 | apreim 8562 | . . . 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 427 | . . 3 ⊢ ((0 + (i · 1)) # (0 + (i · 0)) ↔ (0 # 0 ∨ 1 # 0)) |
7 | 2, 6 | mpbir 146 | . 2 ⊢ (0 + (i · 1)) # (0 + (i · 0)) |
8 | ax-icn 7908 | . . . . 5 ⊢ i ∈ ℂ | |
9 | 8 | mulid1i 7961 | . . . 4 ⊢ (i · 1) = i |
10 | 9 | oveq2i 5888 | . . 3 ⊢ (0 + (i · 1)) = (0 + i) |
11 | 8 | addid2i 8102 | . . 3 ⊢ (0 + i) = i |
12 | 10, 11 | eqtri 2198 | . 2 ⊢ (0 + (i · 1)) = i |
13 | it0e0 9142 | . . . 4 ⊢ (i · 0) = 0 | |
14 | 13 | oveq2i 5888 | . . 3 ⊢ (0 + (i · 0)) = (0 + 0) |
15 | 00id 8100 | . . 3 ⊢ (0 + 0) = 0 | |
16 | 14, 15 | eqtri 2198 | . 2 ⊢ (0 + (i · 0)) = 0 |
17 | 7, 12, 16 | 3brtr3i 4034 | 1 ⊢ i # 0 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∨ wo 708 ∈ wcel 2148 class class class wbr 4005 (class class class)co 5877 ℝcr 7812 0cc0 7813 1c1 7814 ici 7815 + caddc 7816 · cmul 7818 # cap 8540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-ltxr 7999 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 |
This theorem is referenced by: 2muliap0 9145 irec 10622 iexpcyc 10627 imval 10861 imre 10862 reim 10863 crim 10869 cjreb 10877 tanval2ap 11723 tanval3ap 11724 efival 11742 |
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