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Mirrors > Home > ILE Home > Th. List > rereim | Unicode version |
Description: Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.) |
Ref | Expression |
---|---|
rereim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 518 | . . . . 5 | |
2 | 1 | recnd 7787 | . . . 4 |
3 | simplr 519 | . . . . 5 | |
4 | 3 | recnd 7787 | . . . 4 |
5 | simprr 521 | . . . . . . 7 | |
6 | 5 | eqcomd 2143 | . . . . . 6 |
7 | ax-icn 7708 | . . . . . . . . 9 | |
8 | 7 | a1i 9 | . . . . . . . 8 |
9 | simprl 520 | . . . . . . . . 9 | |
10 | 9 | recnd 7787 | . . . . . . . 8 |
11 | 8, 10 | mulcld 7779 | . . . . . . 7 |
12 | 2, 4, 11 | subaddd 8084 | . . . . . 6 |
13 | 6, 12 | mpbird 166 | . . . . 5 |
14 | 1, 3 | resubcld 8136 | . . . . . . . . 9 |
15 | 13, 14 | eqeltrrd 2215 | . . . . . . . 8 |
16 | rimul 8340 | . . . . . . . 8 | |
17 | 9, 15, 16 | syl2anc 408 | . . . . . . 7 |
18 | 17 | oveq2d 5783 | . . . . . 6 |
19 | 7 | mul01i 8146 | . . . . . 6 |
20 | 18, 19 | syl6eq 2186 | . . . . 5 |
21 | 13, 20 | eqtrd 2170 | . . . 4 |
22 | 2, 4, 21 | subeq0d 8074 | . . 3 |
23 | 22 | eqcomd 2143 | . 2 |
24 | 23, 17 | jca 304 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 (class class class)co 5767 cc 7611 cr 7612 cc0 7613 ci 7615 caddc 7616 cmul 7618 cmin 7926 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-sub 7928 df-neg 7929 df-reap 8330 |
This theorem is referenced by: apreap 8342 |
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