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Mirrors > Home > ILE Home > Th. List > imadd | Unicode version |
Description: Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
Ref | Expression |
---|---|
imadd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recl 10810 | . . . . . . 7 | |
2 | 1 | adantr 274 | . . . . . 6 |
3 | 2 | recnd 7941 | . . . . 5 |
4 | ax-icn 7862 | . . . . . 6 | |
5 | imcl 10811 | . . . . . . . 8 | |
6 | 5 | adantr 274 | . . . . . . 7 |
7 | 6 | recnd 7941 | . . . . . 6 |
8 | mulcl 7894 | . . . . . 6 | |
9 | 4, 7, 8 | sylancr 412 | . . . . 5 |
10 | recl 10810 | . . . . . . 7 | |
11 | 10 | adantl 275 | . . . . . 6 |
12 | 11 | recnd 7941 | . . . . 5 |
13 | imcl 10811 | . . . . . . . 8 | |
14 | 13 | adantl 275 | . . . . . . 7 |
15 | 14 | recnd 7941 | . . . . . 6 |
16 | mulcl 7894 | . . . . . 6 | |
17 | 4, 15, 16 | sylancr 412 | . . . . 5 |
18 | 3, 9, 12, 17 | add4d 8081 | . . . 4 |
19 | replim 10816 | . . . . 5 | |
20 | replim 10816 | . . . . 5 | |
21 | 19, 20 | oveqan12d 5870 | . . . 4 |
22 | 4 | a1i 9 | . . . . . 6 |
23 | 22, 7, 15 | adddid 7937 | . . . . 5 |
24 | 23 | oveq2d 5867 | . . . 4 |
25 | 18, 21, 24 | 3eqtr4d 2213 | . . 3 |
26 | 25 | fveq2d 5498 | . 2 |
27 | readdcl 7893 | . . . 4 | |
28 | 1, 10, 27 | syl2an 287 | . . 3 |
29 | readdcl 7893 | . . . 4 | |
30 | 5, 13, 29 | syl2an 287 | . . 3 |
31 | crim 10815 | . . 3 | |
32 | 28, 30, 31 | syl2anc 409 | . 2 |
33 | 26, 32 | eqtrd 2203 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 cfv 5196 (class class class)co 5851 cc 7765 cr 7766 ci 7769 caddc 7770 cmul 7772 cre 10797 cim 10798 |
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 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 |
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-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-2 8930 df-cj 10799 df-re 10800 df-im 10801 |
This theorem is referenced by: imsub 10835 cjadd 10841 imaddi 10886 imaddd 10917 fsumim 11429 gzaddcl 12322 |
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