Theorem efcnlem1 7367

Description: Lemma for efcn 7371.

Hypotheses

Ref	Expression
efcnlem1.1	|- A e. RR
efcnlem1.2	|- X e. RR
efcnlem1.3	|- Y e. RR
efcnlem1.4	|- 0 < X
efcnlem1.5	|- 0 < Y

Assertion

Ref	Expression
efcnlem1	|- (A < (Y / (X + Y)) -> (A < 1 /\ (1 - A) =/= 0 /\ (X x. (A / (1 - A))) < Y))

Proof of Theorem efcnlem1

Step	Hyp	Ref	Expression
1		efcnlem1.4	. . . 4 |- 0 < X
2		efcnlem1.2	. . . . . . . . 9 |- X e. RR
3		efcnlem1.3	. . . . . . . . 9 |- Y e. RR
4	2, 3	readdcl 5314	. . . . . . . 8 |- (X + Y) e. RR
5	4	recn 5294	. . . . . . 7 |- (X + Y) e. CC
6	5	mulid1 5312	. . . . . 6 |- ((X + Y) x. 1) = (X + Y)
7	6	breq2i 2622	. . . . 5 |- (Y < ((X + Y) x. 1) <-> Y < (X + Y))
8		1re 5415	. . . . . 6 |- 1 e. RR
9		efcnlem1.5	. . . . . . . 8 |- 0 < Y
10	2, 3, 1, 9	addgt0i 5583	. . . . . . 7 |- 0 < (X + Y)
11		ltdivmult 5827	. . . . . . 7 |- (((Y e. RR /\ (X + Y) e. RR /\ 1 e. RR) /\ 0 < (X + Y)) -> ((Y / (X + Y)) < 1 <-> Y < ((X + Y) x. 1)))
12	10, 11	mpan2 695	. . . . . 6 |- ((Y e. RR /\ (X + Y) e. RR /\ 1 e. RR) -> ((Y / (X + Y)) < 1 <-> Y < ((X + Y) x. 1)))
13	3, 4, 8, 12	mp3an 914	. . . . 5 |- ((Y / (X + Y)) < 1 <-> Y < ((X + Y) x. 1))
14		ltaddpos2t 5633	. . . . . 6 |- ((X e. RR /\ Y e. RR) -> (0 < X <-> Y < (X + Y)))
15	2, 3, 14	mp2an 696	. . . . 5 |- (0 < X <-> Y < (X + Y))
16	7, 13, 15	3bitr4r 184	. . . 4 |- (0 < X <-> (Y / (X + Y)) < 1)
17	1, 16	mpbi 189	. . 3 |- (Y / (X + Y)) < 1
18		efcnlem1.1	. . . 4 |- A e. RR
19	4, 10	gt0ne0i 5599	. . . . 5 |- (X + Y) =/= 0
20	3, 4, 19	redivcl 5762	. . . 4 |- (Y / (X + Y)) e. RR
21	18, 20, 8	lttr 5567	. . 3 |- ((A < (Y / (X + Y)) /\ (Y / (X + Y)) < 1) -> A < 1)
22	17, 21	mpan2 695	. 2 |- (A < (Y / (X + Y)) -> A < 1)
23		0reALT 5421	. . . . . 6 |- 0 e. RR
24	23, 18, 8	ltaddsub 5621	. . . . 5 |- ((0 + A) < 1 <-> 0 < (1 - A))
25	18	recn 5294	. . . . . . 7 |- A e. CC
26	25	addid2 5311	. . . . . 6 |- (0 + A) = A
27	26	breq1i 2621	. . . . 5 |- ((0 + A) < 1 <-> A < 1)
28	24, 27	bitr3 175	. . . 4 |- (0 < (1 - A) <-> A < 1)
29	22, 28	sylibr 200	. . 3 |- (A < (Y / (X + Y)) -> 0 < (1 - A))
30	8, 18	resubcl 5419	. . . 4 |- (1 - A) e. RR
31	30	gt0ne0 5593	. . 3 |- (0 < (1 - A) -> (1 - A) =/= 0)
32	29, 31	syl 10	. 2 |- (A < (Y / (X + Y)) -> (1 - A) =/= 0)
33	2	recn 5294	. . . . 5 |- X e. CC
34	30	recn 5294	. . . . 5 |- (1 - A) e. CC
35	33, 25, 34	divassz 5716	. . . 4 |- ((1 - A) =/= 0 -> ((X x. A) / (1 - A)) = (X x. (A / (1 - A))))
36	32, 35	syl 10	. . 3 |- (A < (Y / (X + Y)) -> ((X x. A) / (1 - A)) = (X x. (A / (1 - A))))
37	18, 3, 4	ltmuldiv 5789	. . . . . . . . . . 11 |- (0 < (X + Y) -> ((A x. (X + Y)) < Y <-> A < (Y / (X + Y))))
38	10, 37	ax-mp 7	. . . . . . . . . 10 |- ((A x. (X + Y)) < Y <-> A < (Y / (X + Y)))
39	38	biimpr 152	. . . . . . . . 9 |- (A < (Y / (X + Y)) -> (A x. (X + Y)) < Y)
40	3	recn 5294	. . . . . . . . . 10 |- Y e. CC
41	25, 33, 40	adddi 5306	. . . . . . . . 9 |- (A x. (X + Y)) = ((A x. X) + (A x. Y))
42	39, 41	syl5eqbrr 2644	. . . . . . . 8 |- (A < (Y / (X + Y)) -> ((A x. X) + (A x. Y)) < Y)
43	40	mulid2 5313	. . . . . . . 8 |- (1 x. Y) = Y
44	42, 43	syl6breqr 2650	. . . . . . 7 |- (A < (Y / (X + Y)) -> ((A x. X) + (A x. Y)) < (1 x. Y))
45	18, 2	remulcl 5315	. . . . . . . 8 |- (A x. X) e. RR
46	18, 3	remulcl 5315	. . . . . . . 8 |- (A x. Y) e. RR
47	8, 3	remulcl 5315	. . . . . . . 8 |- (1 x. Y) e. RR
48	45, 46, 47	ltaddsub 5621	. . . . . . 7 |- (((A x. X) + (A x. Y)) < (1 x. Y) <-> (A x. X) < ((1 x. Y) - (A x. Y)))
49	44, 48	sylib 198	. . . . . 6 |- (A < (Y / (X + Y)) -> (A x. X) < ((1 x. Y) - (A x. Y)))
50		ax1cn 5249	. . . . . . 7 |- 1 e. CC
51	50, 25, 40	subdir 5410	. . . . . 6 |- ((1 - A) x. Y) = ((1 x. Y) - (A x. Y))
52	49, 51	syl6breqr 2650	. . . . 5 |- (A < (Y / (X + Y)) -> (A x. X) < ((1 - A) x. Y))
53	25, 33	mulcom 5303	. . . . 5 |- (A x. X) = (X x. A)
54	52, 53	syl5eqbrr 2644	. . . 4 |- (A < (Y / (X + Y)) -> (X x. A) < ((1 - A) x. Y))
55	2, 18	remulcl 5315	. . . . . . 7 |- (X x. A) e. RR
56	55, 30, 3	3pm3.2i 817	. . . . . 6 |- ((X x. A) e. RR /\ (1 - A) e. RR /\ Y e. RR)
57		ltdivmult 5827	. . . . . 6 |- ((((X x. A) e. RR /\ (1 - A) e. RR /\ Y e. RR) /\ 0 < (1 - A)) -> (((X x. A) / (1 - A)) < Y <-> (X x. A) < ((1 - A) x. Y)))
58	56, 57	mpan 694	. . . . 5 |- (0 < (1 - A) -> (((X x. A) / (1 - A)) < Y <-> (X x. A) < ((1 - A) x. Y)))
59	29, 58	syl 10	. . . 4 |- (A < (Y / (X + Y)) -> (((X x. A) / (1 - A)) < Y <-> (X x. A) < ((1 - A) x. Y)))
60	54, 59	mpbird 196	. . 3 |- (A < (Y / (X + Y)) -> ((X x. A) / (1 - A)) < Y)
61	36, 60	eqbrtrrd 2632	. 2 |- (A < (Y / (X + Y)) -> (X x. (A / (1 - A))) < Y)
62	22, 32, 61	3jca 818	1 |- (A < (Y / (X + Y)) -> (A < 1 /\ (1 - A) =/= 0 /\ (X x. (A / (1 - A))) < Y))

Syntax hints:   -> wi 3   <-> wb 146   /\ w3a 774   = wceq 954   e. wcel 956   =/= wne 1582   class class class wbr 2614  (class class class)co 3954  RRcr 5213  0cc0 5214  1c1 5215   + caddc 5217   x. cmul 5219   - cmin 5272   / cdiv 5274   < clt 5466

This theorem is referenced by:  efcnlem2 7368

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 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680

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