Theorem conjmulap 8908

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 531	. . . . . . 7 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> Q e. CC )
3		recclap 8858	. . . . . . . 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 8331	. . . . . 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 8865	. . . . . . . 8 |- ( ( P e. CC /\ P # 0 ) -> ( P x. ( 1 / P ) ) = 1 )
7	6	oveq1d 6032	. . . . . . 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 8176	. . . . . . 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 2268	. . . . 5 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( 1 / P ) ) = Q )
12		recclap 8858	. . . . . . . 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 8202	. . . . . 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 8865	. . . . . . . 8 |- ( ( Q e. CC /\ Q # 0 ) -> ( Q x. ( 1 / Q ) ) = 1 )
16	15	oveq2d 6033	. . . . . . 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 8175	. . . . . . 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 2268	. . . . 5 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( 1 / Q ) ) = P )
21	11, 20	oveq12d 6035	. . . 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 8158	. . . . . 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 8203	. . . 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 8315	. . . . 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 2274	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( P x. Q ) x. ( ( 1 / P ) + ( 1 / Q ) ) ) = ( P + Q ) )
28	22	mulridd 8195	. . . 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 2246	. 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 8156	. . . 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 8833	. . 3 |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( P x. Q ) # 0 )
34		ax-1cn 8124	. . . 4 |- 1 e. CC
35		mulcanap 8844	. . . 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 1361	. . 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 1271	. 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 2233	. . . 4 |- ( ( P + Q ) = ( P x. Q ) <-> ( P x. Q ) = ( P + Q ) )
39		muleqadd 8847	. . . 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 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   - cmin 8349   # cap 8760   / cdiv 8851

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852

This theorem is referenced by: (None)