Theorem conjmulap 8775

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 527	. . . . . . 7 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> P e. CC )
2		simprl 529	. . . . . . 7 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> Q e. CC )
3		recclap 8725	. . . . . . . 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 8198	. . . . . 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 8732	. . . . . . . 8 |- ( ( P e. CC /\ P # 0 ) -> ( P x. ( 1 / P ) ) = 1 )
7	6	oveq1d 5940	. . . . . . 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 8043	. . . . . . 7 |- ( Q e. CC -> ( 1 x. Q ) = Q )
10	9	ad2antrl 490	. . . . . 6 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( 1 x. Q ) = Q )
11	5, 8, 10	3eqtrd 2233	. . . . 5 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( 1 / P ) ) = Q )
12		recclap 8725	. . . . . . . 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 8069	. . . . . 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 8732	. . . . . . . 8 |- ( ( Q e. CC /\ Q # 0 ) -> ( Q x. ( 1 / Q ) ) = 1 )
16	15	oveq2d 5941	. . . . . . 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 8042	. . . . . . 7 |- ( P e. CC -> ( P x. 1 ) = P )
19	18	ad2antrr 488	. . . . . 6 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P x. 1 ) = P )
20	14, 17, 19	3eqtrd 2233	. . . . 5 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( 1 / Q ) ) = P )
21	11, 20	oveq12d 5943	. . . 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 8025	. . . . . 6 |- ( ( P e. CC /\ Q e. CC ) -> ( P x. Q ) e. CC )
23	22	ad2ant2r 509	. . . . 5 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P x. Q ) e. CC )
24	23, 4, 13	adddid 8070	. . . 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 8182	. . . . 5 |- ( ( P e. CC /\ Q e. CC ) -> ( P + Q ) = ( Q + P ) )
26	25	ad2ant2r 509	. . . 4 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P + Q ) = ( Q + P ) )
27	21, 24, 26	3eqtr4d 2239	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( ( 1 / P ) + ( 1 / Q ) ) ) = ( P + Q ) )
28	22	mulridd 8062	. . . 4 |- ( ( P e. CC /\ Q e. CC ) -> ( ( P x. Q ) x. 1 ) = ( P x. Q ) )
29	28	ad2ant2r 509	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. 1 ) = ( P x. Q ) )
30	27, 29	eqeq12d 2211	. 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 8023	. . . 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 8700	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P x. Q ) # 0 )
34		ax-1cn 7991	. . . 4 |- 1 e. CC
35		mulcanap 8711	. . . 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 1336	. . 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 1247	. 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 2198	. . . 4 |- ( ( P + Q ) = ( P x. Q ) <-> ( P x. Q ) = ( P + Q ) )
39		muleqadd 8714	. . . 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 509	. 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7896   0cc0 7898   1c1 7899   + caddc 7901   x. cmul 7903   - cmin 8216   # cap 8627   / cdiv 8718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628  df-div 8719

This theorem is referenced by: (None)