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Mirrors > Home > ILE Home > Th. List > mulreim | Unicode version |
Description: Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.) |
Ref | Expression |
---|---|
mulreim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 524 | . . . 4 | |
2 | 1 | recnd 7948 | . . 3 |
3 | ax-icn 7869 | . . . . 5 | |
4 | 3 | a1i 9 | . . . 4 |
5 | simplr 525 | . . . . 5 | |
6 | 5 | recnd 7948 | . . . 4 |
7 | 4, 6 | mulcld 7940 | . . 3 |
8 | simprl 526 | . . . 4 | |
9 | 8 | recnd 7948 | . . 3 |
10 | simprr 527 | . . . . 5 | |
11 | 10 | recnd 7948 | . . . 4 |
12 | 4, 11 | mulcld 7940 | . . 3 |
13 | 2, 7, 9, 12 | muladdd 8335 | . 2 |
14 | 4, 11, 4, 6 | mul4d 8074 | . . . . . 6 |
15 | ixi 8502 | . . . . . . 7 | |
16 | 15 | oveq1i 5863 | . . . . . 6 |
17 | 14, 16 | eqtrdi 2219 | . . . . 5 |
18 | 11, 6 | mulcld 7940 | . . . . . 6 |
19 | 18 | mulm1d 8329 | . . . . 5 |
20 | 11, 6 | mulcomd 7941 | . . . . . 6 |
21 | 20 | negeqd 8114 | . . . . 5 |
22 | 17, 19, 21 | 3eqtrd 2207 | . . . 4 |
23 | 22 | oveq2d 5869 | . . 3 |
24 | 11, 2 | mulcld 7940 | . . . . . 6 |
25 | 4, 24 | mulcld 7940 | . . . . 5 |
26 | 9, 6 | mulcld 7940 | . . . . . 6 |
27 | 4, 26 | mulcld 7940 | . . . . 5 |
28 | 25, 27 | addcomd 8070 | . . . 4 |
29 | 2, 4, 11 | mul12d 8071 | . . . . . 6 |
30 | 2, 11 | mulcomd 7941 | . . . . . . 7 |
31 | 30 | oveq2d 5869 | . . . . . 6 |
32 | 29, 31 | eqtrd 2203 | . . . . 5 |
33 | 9, 4, 6 | mul12d 8071 | . . . . 5 |
34 | 32, 33 | oveq12d 5871 | . . . 4 |
35 | 4, 26, 24 | adddid 7944 | . . . 4 |
36 | 28, 34, 35 | 3eqtr4d 2213 | . . 3 |
37 | 23, 36 | oveq12d 5871 | . 2 |
38 | 13, 37 | eqtrd 2203 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 (class class class)co 5853 cc 7772 cr 7773 c1 7775 ci 7776 caddc 7777 cmul 7779 cneg 8091 |
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-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 ax-resscn 7866 ax-1cn 7867 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-cnre 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-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-sub 8092 df-neg 8093 |
This theorem is referenced by: mulext1 8531 |
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