Theorem conjmult 5761

Description: Two numbers whose reciprocals add to 1 are called "conjugates" and satisfy this relationship. Equation 5 of [Kreyszig] p. 12.

Assertion

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

Proof of Theorem conjmult

Step	Hyp	Ref	Expression
1		mul23t 5399	. . . . . . 7 |- ((P e. CC /\ Q e. CC /\ (1 / P) e. CC) -> ((P x. Q) x. (1 / P)) = ((P x. (1 / P)) x. Q))
2		simpll 412	. . . . . . 7 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> P e. CC)
3		simprl 414	. . . . . . 7 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> Q e. CC)
4		recclt 5692	. . . . . . . 8 |- ((P e. CC /\ P =/= 0) -> (1 / P) e. CC)
5	4	adantr 389	. . . . . . 7 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> (1 / P) e. CC)
6	1, 2, 3, 5	syl3anc 857	. . . . . 6 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> ((P x. Q) x. (1 / P)) = ((P x. (1 / P)) x. Q))
7		recidt 5706	. . . . . . . 8 |- ((P e. CC /\ P =/= 0) -> (P x. (1 / P)) = 1)
8	7	opreq1d 3966	. . . . . . 7 |- ((P e. CC /\ P =/= 0) -> ((P x. (1 / P)) x. Q) = (1 x. Q))
9	8	adantr 389	. . . . . 6 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> ((P x. (1 / P)) x. Q) = (1 x. Q))
10		mulid2t 5397	. . . . . . 7 |- (Q e. CC -> (1 x. Q) = Q)
11	10	ad2antrl 406	. . . . . 6 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> (1 x. Q) = Q)
12	6, 9, 11	3eqtrd 1508	. . . . 5 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> ((P x. Q) x. (1 / P)) = Q)
13		axmulass 5258	. . . . . . 7 |- ((P e. CC /\ Q e. CC /\ (1 / Q) e. CC) -> ((P x. Q) x. (1 / Q)) = (P x. (Q x. (1 / Q))))
14		recclt 5692	. . . . . . . 8 |- ((Q e. CC /\ Q =/= 0) -> (1 / Q) e. CC)
15	14	adantl 388	. . . . . . 7 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> (1 / Q) e. CC)
16	13, 2, 3, 15	syl3anc 857	. . . . . 6 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> ((P x. Q) x. (1 / Q)) = (P x. (Q x. (1 / Q))))
17		recidt 5706	. . . . . . . 8 |- ((Q e. CC /\ Q =/= 0) -> (Q x. (1 / Q)) = 1)
18	17	opreq2d 3967	. . . . . . 7 |- ((Q e. CC /\ Q =/= 0) -> (P x. (Q x. (1 / Q))) = (P x. 1))
19	18	adantl 388	. . . . . 6 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> (P x. (Q x. (1 / Q))) = (P x. 1))
20		ax1id 5262	. . . . . . 7 |- (P e. CC -> (P x. 1) = P)
21	20	ad2antrr 404	. . . . . 6 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> (P x. 1) = P)
22	16, 19, 21	3eqtrd 1508	. . . . 5 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> ((P x. Q) x. (1 / Q)) = P)
23	12, 22	opreq12d 3969	. . . 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))
24		axdistr 5259	. . . . 5 |- (((P x. Q) e. CC /\ (1 / P) e. CC /\ (1 / Q) e. CC) -> ((P x. Q) x. ((1 / P) + (1 / Q))) = (((P x. Q) x. (1 / P)) + ((P x. Q) x. (1 / Q))))
25		axmulcl 5253	. . . . . 6 |- ((P e. CC /\ Q e. CC) -> (P x. Q) e. CC)
26	25	ad2ant2r 409	. . . . 5 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> (P x. Q) e. CC)
27	24, 26, 5, 15	syl3anc 857	. . . 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))))
28		axaddcom 5255	. . . . 5 |- ((P e. CC /\ Q e. CC) -> (P + Q) = (Q + P))
29	28	ad2ant2r 409	. . . 4 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> (P + Q) = (Q + P))
30	23, 27, 29	3eqtr4d 1514	. . 3 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> ((P x. Q) x. ((1 / P) + (1 / Q))) = (P + Q))
31		ax1id 5262	. . . . 5 |- ((P x. Q) e. CC -> ((P x. Q) x. 1) = (P x. Q))
32	25, 31	syl 10	. . . 4 |- ((P e. CC /\ Q e. CC) -> ((P x. Q) x. 1) = (P x. Q))
33	32	ad2ant2r 409	. . 3 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> ((P x. Q) x. 1) = (P x. Q))
34	30, 33	eqeq12d 1486	. 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)))
35		ax1cn 5249	. . . 4 |- 1 e. CC
36		mulcant 5669	. . . 4 |- ((((P x. Q) e. CC /\ ((1 / P) + (1 / Q)) e. CC /\ 1 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	35, 36	mp3anl3 910	. . 3 |- ((((P x. Q) e. CC /\ ((1 / P) + (1 / 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))
38		axaddcl 5251	. . . . 5 |- (((1 / P) e. CC /\ (1 / Q) e. CC) -> ((1 / P) + (1 / Q)) e. CC)
39	38, 4, 14	syl2an 454	. . . 4 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> ((1 / P) + (1 / Q)) e. CC)
40	26, 39	jca 288	. . 3 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> ((P x. Q) e. CC /\ ((1 / P) + (1 / Q)) e. CC))
41		muln0t 5675	. . 3 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> (P x. Q) =/= 0)
42	37, 40, 41	sylanc 471	. 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))
43		muleqaddt 5677	. . . 4 |- ((P e. CC /\ Q e. CC) -> ((P x. Q) = (P + Q) <-> ((P - 1) x. (Q - 1)) = 1))
44		eqcom 1474	. . . 4 |- ((P + Q) = (P x. Q) <-> (P x. Q) = (P + Q))
45	43, 44	syl5bb 531	. . 3 |- ((P e. CC /\ Q e. CC) -> ((P + Q) = (P x. Q) <-> ((P - 1) x. (Q - 1)) = 1))
46	45	ad2ant2r 409	. 2 |- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> ((P + Q) = (P x. Q) <-> ((P - 1) x. (Q - 1)) = 1))
47	34, 42, 46	3bitr3d 547	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 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956   =/= wne 1582  (class class class)co 3954  CCcc 5212  0cc0 5214  1c1 5215   + caddc 5217   x. cmul 5219   - cmin 5272   / cdiv 5274

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 146