Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > reim0 | Structured version Visualization version GIF version |
Description: The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
Ref | Expression |
---|---|
reim0 | ⊢ (𝐴 ∈ ℝ → (ℑ‘𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 10630 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | it0e0 11862 | . . . . . 6 ⊢ (i · 0) = 0 | |
3 | 2 | oveq2i 7170 | . . . . 5 ⊢ (𝐴 + (i · 0)) = (𝐴 + 0) |
4 | addid1 10823 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) | |
5 | 3, 4 | syl5eq 2871 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 + (i · 0)) = 𝐴) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + (i · 0)) = 𝐴) |
7 | 6 | fveq2d 6677 | . 2 ⊢ (𝐴 ∈ ℝ → (ℑ‘(𝐴 + (i · 0))) = (ℑ‘𝐴)) |
8 | 0re 10646 | . . 3 ⊢ 0 ∈ ℝ | |
9 | crim 14477 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (ℑ‘(𝐴 + (i · 0))) = 0) | |
10 | 8, 9 | mpan2 689 | . 2 ⊢ (𝐴 ∈ ℝ → (ℑ‘(𝐴 + (i · 0))) = 0) |
11 | 7, 10 | eqtr3d 2861 | 1 ⊢ (𝐴 ∈ ℝ → (ℑ‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 ici 10542 + caddc 10543 · cmul 10545 ℑcim 14460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-2 11703 df-cj 14461 df-re 14462 df-im 14463 |
This theorem is referenced by: reim0b 14481 rereb 14482 remul2 14492 immul2 14499 im0 14515 im1 14517 reim0d 14587 sqrtneglem 14629 rlimrecl 14940 recld2 23425 relogrn 25148 logrnaddcl 25161 |
Copyright terms: Public domain | W3C validator |