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Mirrors > Home > MPE Home > Th. List > conjmul | Structured version Visualization version Unicode version |
Description: Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 12-Nov-2006.) |
Ref | Expression |
---|---|
conjmul |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 790 | . . . . . . 7 | |
2 | simprl 794 | . . . . . . 7 | |
3 | reccl 10692 | . . . . . . . 8 | |
4 | 3 | adantr 481 | . . . . . . 7 |
5 | 1, 2, 4 | mul32d 10246 | . . . . . 6 |
6 | recid 10699 | . . . . . . . 8 | |
7 | 6 | oveq1d 6665 | . . . . . . 7 |
8 | 7 | adantr 481 | . . . . . 6 |
9 | mulid2 10038 | . . . . . . 7 | |
10 | 9 | ad2antrl 764 | . . . . . 6 |
11 | 5, 8, 10 | 3eqtrd 2660 | . . . . 5 |
12 | reccl 10692 | . . . . . . . 8 | |
13 | 12 | adantl 482 | . . . . . . 7 |
14 | 1, 2, 13 | mulassd 10063 | . . . . . 6 |
15 | recid 10699 | . . . . . . . 8 | |
16 | 15 | oveq2d 6666 | . . . . . . 7 |
17 | 16 | adantl 482 | . . . . . 6 |
18 | mulid1 10037 | . . . . . . 7 | |
19 | 18 | ad2antrr 762 | . . . . . 6 |
20 | 14, 17, 19 | 3eqtrd 2660 | . . . . 5 |
21 | 11, 20 | oveq12d 6668 | . . . 4 |
22 | mulcl 10020 | . . . . . 6 | |
23 | 22 | ad2ant2r 783 | . . . . 5 |
24 | 23, 4, 13 | adddid 10064 | . . . 4 |
25 | addcom 10222 | . . . . 5 | |
26 | 25 | ad2ant2r 783 | . . . 4 |
27 | 21, 24, 26 | 3eqtr4d 2666 | . . 3 |
28 | 22 | mulid1d 10057 | . . . 4 |
29 | 28 | ad2ant2r 783 | . . 3 |
30 | 27, 29 | eqeq12d 2637 | . 2 |
31 | addcl 10018 | . . . 4 | |
32 | 3, 12, 31 | syl2an 494 | . . 3 |
33 | mulne0 10669 | . . 3 | |
34 | ax-1cn 9994 | . . . 4 | |
35 | mulcan 10664 | . . . 4 | |
36 | 34, 35 | mp3an2 1412 | . . 3 |
37 | 32, 23, 33, 36 | syl12anc 1324 | . 2 |
38 | eqcom 2629 | . . . 4 | |
39 | muleqadd 10671 | . . . 4 | |
40 | 38, 39 | syl5bb 272 | . . 3 |
41 | 40 | ad2ant2r 783 | . 2 |
42 | 30, 37, 41 | 3bitr3d 298 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wne 2794 (class class class)co 6650 cc 9934 cc0 9936 c1 9937 caddc 9939 cmul 9941 cmin 10266 cdiv 10684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 |
This theorem is referenced by: (None) |
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