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Mirrors > Home > ILE Home > Th. List > imi | GIF version |
Description: The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.) |
Ref | Expression |
---|---|
imi | ⊢ (ℑ‘i) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 7908 | . . . . . 6 ⊢ i ∈ ℂ | |
2 | ax-1cn 7906 | . . . . . 6 ⊢ 1 ∈ ℂ | |
3 | 1, 2 | mulcli 7964 | . . . . 5 ⊢ (i · 1) ∈ ℂ |
4 | 3 | addid2i 8102 | . . . 4 ⊢ (0 + (i · 1)) = (i · 1) |
5 | 4 | eqcomi 2181 | . . 3 ⊢ (i · 1) = (0 + (i · 1)) |
6 | 5 | fveq2i 5520 | . 2 ⊢ (ℑ‘(i · 1)) = (ℑ‘(0 + (i · 1))) |
7 | 1 | mulid1i 7961 | . . 3 ⊢ (i · 1) = i |
8 | 7 | fveq2i 5520 | . 2 ⊢ (ℑ‘(i · 1)) = (ℑ‘i) |
9 | 0re 7959 | . . 3 ⊢ 0 ∈ ℝ | |
10 | 1re 7958 | . . 3 ⊢ 1 ∈ ℝ | |
11 | crim 10869 | . . 3 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → (ℑ‘(0 + (i · 1))) = 1) | |
12 | 9, 10, 11 | mp2an 426 | . 2 ⊢ (ℑ‘(0 + (i · 1))) = 1 |
13 | 6, 8, 12 | 3eqtr3i 2206 | 1 ⊢ (ℑ‘i) = 1 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 ‘cfv 5218 (class class class)co 5877 ℝcr 7812 0cc0 7813 1c1 7814 ici 7815 + caddc 7816 · cmul 7818 ℑcim 10852 |
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-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 |
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-rmo 2463 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-mpt 4068 df-id 4295 df-po 4298 df-iso 4299 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-2 8980 df-cj 10853 df-re 10854 df-im 10855 |
This theorem is referenced by: cji 10913 igz 12374 |
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