Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 8509 | . . . 4 ⊢ 1 # 0 | |
2 | 1 | olci 727 | . . 3 ⊢ (0 # 0 ∨ 1 # 0) |
3 | 0re 7920 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | 1re 7919 | . . . 4 ⊢ 1 ∈ ℝ | |
5 | apreim 8522 | . . . 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 425 | . . 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 7869 | . . . . 5 ⊢ i ∈ ℂ | |
9 | 8 | mulid1i 7922 | . . . 4 ⊢ (i · 1) = i |
10 | 9 | oveq2i 5864 | . . 3 ⊢ (0 + (i · 1)) = (0 + i) |
11 | 8 | addid2i 8062 | . . 3 ⊢ (0 + i) = i |
12 | 10, 11 | eqtri 2191 | . 2 ⊢ (0 + (i · 1)) = i |
13 | it0e0 9099 | . . . 4 ⊢ (i · 0) = 0 | |
14 | 13 | oveq2i 5864 | . . 3 ⊢ (0 + (i · 0)) = (0 + 0) |
15 | 00id 8060 | . . 3 ⊢ (0 + 0) = 0 | |
16 | 14, 15 | eqtri 2191 | . 2 ⊢ (0 + (i · 0)) = 0 |
17 | 7, 12, 16 | 3brtr3i 4018 | 1 ⊢ i # 0 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 703 ∈ wcel 2141 class class class wbr 3989 (class class class)co 5853 ℝcr 7773 0cc0 7774 1c1 7775 ici 7776 + caddc 7777 · cmul 7779 # cap 8500 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-ltxr 7959 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 |
This theorem is referenced by: 2muliap0 9102 irec 10575 iexpcyc 10580 imval 10814 imre 10815 reim 10816 crim 10822 cjreb 10830 tanval2ap 11676 tanval3ap 11677 efival 11695 |
Copyright terms: Public domain | W3C validator |