Theorem conjmulap 8625

Description: Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)

Assertion

Ref	Expression
conjmulap	|- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( ( 1 / P ) + ( 1 / Q ) ) = 1 <-> ( ( P - 1 ) x. ( Q - 1 ) ) = 1 ) )

Proof of Theorem conjmulap

Step	Hyp	Ref	Expression
1		simpll 519	. . . . . . 7 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> P e. CC )
2		simprl 521	. . . . . . 7 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> Q e. CC )
3		recclap 8575	. . . . . . . 8 |- ( ( P e. CC /\ P # 0 ) -> ( 1 / P ) e. CC )
4	3	adantr 274	. . . . . . 7 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( 1 / P ) e. CC )
5	1, 2, 4	mul32d 8051	. . . . . 6 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( 1 / P ) ) = ( ( P x. ( 1 / P ) ) x. Q ) )
6		recidap 8582	. . . . . . . 8 |- ( ( P e. CC /\ P # 0 ) -> ( P x. ( 1 / P ) ) = 1 )
7	6	oveq1d 5857	. . . . . . 7 |- ( ( P e. CC /\ P # 0 ) -> ( ( P x. ( 1 / P ) ) x. Q ) = ( 1 x. Q ) )
8	7	adantr 274	. . . . . 6 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. ( 1 / P ) ) x. Q ) = ( 1 x. Q ) )
9		mulid2 7897	. . . . . . 7 |- ( Q e. CC -> ( 1 x. Q ) = Q )
10	9	ad2antrl 482	. . . . . 6 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( 1 x. Q ) = Q )
11	5, 8, 10	3eqtrd 2202	. . . . 5 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( 1 / P ) ) = Q )
12		recclap 8575	. . . . . . . 8 |- ( ( Q e. CC /\ Q # 0 ) -> ( 1 / Q ) e. CC )
13	12	adantl 275	. . . . . . 7 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( 1 / Q ) e. CC )
14	1, 2, 13	mulassd 7922	. . . . . 6 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( 1 / Q ) ) = ( P x. ( Q x. ( 1 / Q ) ) ) )
15		recidap 8582	. . . . . . . 8 |- ( ( Q e. CC /\ Q # 0 ) -> ( Q x. ( 1 / Q ) ) = 1 )
16	15	oveq2d 5858	. . . . . . 7 |- ( ( Q e. CC /\ Q # 0 ) -> ( P x. ( Q x. ( 1 / Q ) ) ) = ( P x. 1 ) )
17	16	adantl 275	. . . . . 6 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P x. ( Q x. ( 1 / Q ) ) ) = ( P x. 1 ) )
18		mulid1 7896	. . . . . . 7 |- ( P e. CC -> ( P x. 1 ) = P )
19	18	ad2antrr 480	. . . . . 6 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P x. 1 ) = P )
20	14, 17, 19	3eqtrd 2202	. . . . 5 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( 1 / Q ) ) = P )
21	11, 20	oveq12d 5860	. . . 4 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( ( P x. Q ) x. ( 1 / P ) ) + ( ( P x. Q ) x. ( 1 / Q ) ) ) = ( Q + P ) )
22		mulcl 7880	. . . . . 6 |- ( ( P e. CC /\ Q e. CC ) -> ( P x. Q ) e. CC )
23	22	ad2ant2r 501	. . . . 5 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P x. Q ) e. CC )
24	23, 4, 13	adddid 7923	. . . 4 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( ( 1 / P ) + ( 1 / Q ) ) ) = ( ( ( P x. Q ) x. ( 1 / P ) ) + ( ( P x. Q ) x. ( 1 / Q ) ) ) )
25		addcom 8035	. . . . 5 |- ( ( P e. CC /\ Q e. CC ) -> ( P + Q ) = ( Q + P ) )
26	25	ad2ant2r 501	. . . 4 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P + Q ) = ( Q + P ) )
27	21, 24, 26	3eqtr4d 2208	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( ( 1 / P ) + ( 1 / Q ) ) ) = ( P + Q ) )
28	22	mulid1d 7916	. . . 4 |- ( ( P e. CC /\ Q e. CC ) -> ( ( P x. Q ) x. 1 ) = ( P x. Q ) )
29	28	ad2ant2r 501	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. 1 ) = ( P x. Q ) )
30	27, 29	eqeq12d 2180	. 2 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( ( P x. Q ) x. ( ( 1 / P ) + ( 1 / Q ) ) ) = ( ( P x. Q ) x. 1 ) <-> ( P + Q ) = ( P x. Q ) ) )
31		addcl 7878	. . . 4 |- ( ( ( 1 / P ) e. CC /\ ( 1 / Q ) e. CC ) -> ( ( 1 / P ) + ( 1 / Q ) ) e. CC )
32	3, 12, 31	syl2an 287	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( 1 / P ) + ( 1 / Q ) ) e. CC )
33		mulap0 8551	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P x. Q ) # 0 )
34		ax-1cn 7846	. . . 4 |- 1 e. CC
35		mulcanap 8562	. . . 4 |- ( ( ( ( 1 / P ) + ( 1 / Q ) ) e. CC /\ 1 e. CC /\ ( ( P x. Q ) e. CC /\ ( P x. Q ) # 0 ) ) -> ( ( ( P x. Q ) x. ( ( 1 / P ) + ( 1 / Q ) ) ) = ( ( P x. Q ) x. 1 ) <-> ( ( 1 / P ) + ( 1 / Q ) ) = 1 ) )
36	34, 35	mp3an2 1315	. . 3 |- ( ( ( ( 1 / P ) + ( 1 / Q ) ) e. CC /\ ( ( P x. Q ) e. CC /\ ( P x. Q ) # 0 ) ) -> ( ( ( P x. Q ) x. ( ( 1 / P ) + ( 1 / Q ) ) ) = ( ( P x. Q ) x. 1 ) <-> ( ( 1 / P ) + ( 1 / Q ) ) = 1 ) )
37	32, 23, 33, 36	syl12anc 1226	. 2 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( ( P x. Q ) x. ( ( 1 / P ) + ( 1 / Q ) ) ) = ( ( P x. Q ) x. 1 ) <-> ( ( 1 / P ) + ( 1 / Q ) ) = 1 ) )
38		eqcom 2167	. . . 4 |- ( ( P + Q ) = ( P x. Q ) <-> ( P x. Q ) = ( P + Q ) )
39		muleqadd 8565	. . . 4 |- ( ( P e. CC /\ Q e. CC ) -> ( ( P x. Q ) = ( P + Q ) <-> ( ( P - 1 ) x. ( Q - 1 ) ) = 1 ) )
40	38, 39	syl5bb 191	. . 3 |- ( ( P e. CC /\ Q e. CC ) -> ( ( P + Q ) = ( P x. Q ) <-> ( ( P - 1 ) x. ( Q - 1 ) ) = 1 ) )
41	40	ad2ant2r 501	. 2 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P + Q ) = ( P x. Q ) <-> ( ( P - 1 ) x. ( Q - 1 ) ) = 1 ) )
42	30, 37, 41	3bitr3d 217	1 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( ( 1 / P ) + ( 1 / Q ) ) = 1 <-> ( ( P - 1 ) x. ( Q - 1 ) ) = 1 ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   - cmin 8069   # cap 8479   / cdiv 8568

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569

This theorem is referenced by: (None)