Theorem conjmulap 8659

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 8609	. . . . . . . 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 8084	. . . . . 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 8616	. . . . . . . 8 |- ( ( P e. CC /\ P # 0 ) -> ( P x. ( 1 / P ) ) = 1 )
7	6	oveq1d 5880	. . . . . . 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		mulid2 7930	. . . . . . 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 2212	. . . . 5 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( 1 / P ) ) = Q )
12		recclap 8609	. . . . . . . 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 7955	. . . . . 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 8616	. . . . . . . 8 |- ( ( Q e. CC /\ Q # 0 ) -> ( Q x. ( 1 / Q ) ) = 1 )
16	15	oveq2d 5881	. . . . . . 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		mulid1 7929	. . . . . . 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 2212	. . . . 5 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( 1 / Q ) ) = P )
21	11, 20	oveq12d 5883	. . . 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 7913	. . . . . 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 7956	. . . 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 8068	. . . . 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 2218	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( ( 1 / P ) + ( 1 / Q ) ) ) = ( P + Q ) )
28	22	mulid1d 7949	. . . 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 2190	. 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 7911	. . . 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 8584	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P x. Q ) # 0 )
34		ax-1cn 7879	. . . 4 |- 1 e. CC
35		mulcanap 8595	. . . 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 1325	. . 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 1236	. 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 2177	. . . 4 |- ( ( P + Q ) = ( P x. Q ) <-> ( P x. Q ) = ( P + Q ) )
39		muleqadd 8598	. . . 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 1353   e. wcel 2146   class class class wbr 3998  (class class class)co 5865   CCcc 7784   0cc0 7786   1c1 7787   + caddc 7789   x. cmul 7791   - cmin 8102   # cap 8512   / cdiv 8602

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-po 4290  df-iso 4291  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603

This theorem is referenced by: (None)