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Mirrors > Home > ILE Home > Th. List > conjmulap | Unicode version |
Description: Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.) |
Ref | Expression |
---|---|
conjmulap | # # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 519 | . . . . . . 7 # # | |
2 | simprl 521 | . . . . . . 7 # # | |
3 | recclap 8566 | . . . . . . . 8 # | |
4 | 3 | adantr 274 | . . . . . . 7 # # |
5 | 1, 2, 4 | mul32d 8042 | . . . . . 6 # # |
6 | recidap 8573 | . . . . . . . 8 # | |
7 | 6 | oveq1d 5851 | . . . . . . 7 # |
8 | 7 | adantr 274 | . . . . . 6 # # |
9 | mulid2 7888 | . . . . . . 7 | |
10 | 9 | ad2antrl 482 | . . . . . 6 # # |
11 | 5, 8, 10 | 3eqtrd 2201 | . . . . 5 # # |
12 | recclap 8566 | . . . . . . . 8 # | |
13 | 12 | adantl 275 | . . . . . . 7 # # |
14 | 1, 2, 13 | mulassd 7913 | . . . . . 6 # # |
15 | recidap 8573 | . . . . . . . 8 # | |
16 | 15 | oveq2d 5852 | . . . . . . 7 # |
17 | 16 | adantl 275 | . . . . . 6 # # |
18 | mulid1 7887 | . . . . . . 7 | |
19 | 18 | ad2antrr 480 | . . . . . 6 # # |
20 | 14, 17, 19 | 3eqtrd 2201 | . . . . 5 # # |
21 | 11, 20 | oveq12d 5854 | . . . 4 # # |
22 | mulcl 7871 | . . . . . 6 | |
23 | 22 | ad2ant2r 501 | . . . . 5 # # |
24 | 23, 4, 13 | adddid 7914 | . . . 4 # # |
25 | addcom 8026 | . . . . 5 | |
26 | 25 | ad2ant2r 501 | . . . 4 # # |
27 | 21, 24, 26 | 3eqtr4d 2207 | . . 3 # # |
28 | 22 | mulid1d 7907 | . . . 4 |
29 | 28 | ad2ant2r 501 | . . 3 # # |
30 | 27, 29 | eqeq12d 2179 | . 2 # # |
31 | addcl 7869 | . . . 4 | |
32 | 3, 12, 31 | syl2an 287 | . . 3 # # |
33 | mulap0 8542 | . . 3 # # # | |
34 | ax-1cn 7837 | . . . 4 | |
35 | mulcanap 8553 | . . . 4 # | |
36 | 34, 35 | mp3an2 1314 | . . 3 # |
37 | 32, 23, 33, 36 | syl12anc 1225 | . 2 # # |
38 | eqcom 2166 | . . . 4 | |
39 | muleqadd 8556 | . . . 4 | |
40 | 38, 39 | syl5bb 191 | . . 3 |
41 | 40 | ad2ant2r 501 | . 2 # # |
42 | 30, 37, 41 | 3bitr3d 217 | 1 # # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 class class class wbr 3976 (class class class)co 5836 cc 7742 cc0 7744 c1 7745 caddc 7747 cmul 7749 cmin 8060 # cap 8470 cdiv 8559 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 |
This theorem is referenced by: (None) |
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