Theorem conjmulap 9059

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 531	. . . . . . 7 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> P e. CC )
2		simprl 535	. . . . . . 7 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> Q e. CC )
3		recclap 9009	. . . . . . . 8 |- ( ( P e. CC /\ P # 0 ) -> ( 1 / P ) e. CC )
4	3	adantr 276	. . . . . . 7 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( 1 / P ) e. CC )
5	1, 2, 4	mul32d 8479	. . . . . 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 9016	. . . . . . . 8 |- ( ( P e. CC /\ P # 0 ) -> ( P x. ( 1 / P ) ) = 1 )
7	6	oveq1d 6100	. . . . . . 7 |- ( ( P e. CC /\ P # 0 ) -> ( ( P x. ( 1 / P ) ) x. Q ) = ( 1 x. Q ) )
8	7	adantr 276	. . . . . 6 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. ( 1 / P ) ) x. Q ) = ( 1 x. Q ) )
9		mullid 8324	. . . . . . 7 |- ( Q e. CC -> ( 1 x. Q ) = Q )
10	9	ad2antrl 494	. . . . . 6 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( 1 x. Q ) = Q )
11	5, 8, 10	3eqtrd 2275	. . . . 5 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( 1 / P ) ) = Q )
12		recclap 9009	. . . . . . . 8 |- ( ( Q e. CC /\ Q # 0 ) -> ( 1 / Q ) e. CC )
13	12	adantl 277	. . . . . . 7 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( 1 / Q ) e. CC )
14	1, 2, 13	mulassd 8349	. . . . . 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 9016	. . . . . . . 8 |- ( ( Q e. CC /\ Q # 0 ) -> ( Q x. ( 1 / Q ) ) = 1 )
16	15	oveq2d 6101	. . . . . . 7 |- ( ( Q e. CC /\ Q # 0 ) -> ( P x. ( Q x. ( 1 / Q ) ) ) = ( P x. 1 ) )
17	16	adantl 277	. . . . . 6 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P x. ( Q x. ( 1 / Q ) ) ) = ( P x. 1 ) )
18		mulrid 8323	. . . . . . 7 |- ( P e. CC -> ( P x. 1 ) = P )
19	18	ad2antrr 492	. . . . . 6 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P x. 1 ) = P )
20	14, 17, 19	3eqtrd 2275	. . . . 5 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( 1 / Q ) ) = P )
21	11, 20	oveq12d 6103	. . . 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 8306	. . . . . 6 |- ( ( P e. CC /\ Q e. CC ) -> ( P x. Q ) e. CC )
23	22	ad2ant2r 513	. . . . 5 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P x. Q ) e. CC )
24	23, 4, 13	adddid 8350	. . . 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 8463	. . . . 5 |- ( ( P e. CC /\ Q e. CC ) -> ( P + Q ) = ( Q + P ) )
26	25	ad2ant2r 513	. . . 4 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P + Q ) = ( Q + P ) )
27	21, 24, 26	3eqtr4d 2281	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( ( 1 / P ) + ( 1 / Q ) ) ) = ( P + Q ) )
28	22	mulridd 8343	. . . 4 |- ( ( P e. CC /\ Q e. CC ) -> ( ( P x. Q ) x. 1 ) = ( P x. Q ) )
29	28	ad2ant2r 513	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. 1 ) = ( P x. Q ) )
30	27, 29	eqeq12d 2253	. 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 8304	. . . 4 |- ( ( ( 1 / P ) e. CC /\ ( 1 / Q ) e. CC ) -> ( ( 1 / P ) + ( 1 / Q ) ) e. CC )
32	3, 12, 31	syl2an 289	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( 1 / P ) + ( 1 / Q ) ) e. CC )
33		mulap0 8982	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P x. Q ) # 0 )
34		ax-1cn 8272	. . . 4 |- 1 e. CC
35		mulcanap 8993	. . . 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 1366	. . 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 1276	. 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 2240	. . . 4 |- ( ( P + Q ) = ( P x. Q ) <-> ( P x. Q ) = ( P + Q ) )
39		muleqadd 8998	. . . 4 |- ( ( P e. CC /\ Q e. CC ) -> ( ( P x. Q ) = ( P + Q ) <-> ( ( P - 1 ) x. ( Q - 1 ) ) = 1 ) )
40	38, 39	bitrid 192	. . 3 |- ( ( P e. CC /\ Q e. CC ) -> ( ( P + Q ) = ( P x. Q ) <-> ( ( P - 1 ) x. ( Q - 1 ) ) = 1 ) )
41	40	ad2ant2r 513	. 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 218	1 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( ( 1 / P ) + ( 1 / Q ) ) = 1 <-> ( ( P - 1 ) x. ( Q - 1 ) ) = 1 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   - cmin 8497   # cap 8909   / cdiv 9002

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003

This theorem is used by: (None)