Theorem efcnlem1 7376

Description: Lemma for efcn 7380.

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 5317	. . . . . . . 8 |- (X + Y) e. RR
5	4	recn 5297	. . . . . . 7 |- (X + Y) e. CC
6	5	mulid1 5315	. . . . . 6 |- ((X + Y) x. 1) = (X + Y)
7	6	breq2i 2623	. . . . 5 |- (Y < ((X + Y) x. 1) <-> Y < (X + Y))
8		1re 5418	. . . . . 6 |- 1 e. RR
9		efcnlem1.5	. . . . . . . 8 |- 0 < Y
10	2, 3, 1, 9	addgt0i 5585	. . . . . . 7 |- 0 < (X + Y)
11		ltdivmult 5829	. . . . . . 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 915	. . . . 5 |- ((Y / (X + Y)) < 1 <-> Y < ((X + Y) x. 1))
14		ltaddpos2t 5635	. . . . . 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 5601	. . . . 5 |- (X + Y) =/= 0
20	3, 4, 19	redivcl 5764	. . . 4 |- (Y / (X + Y)) e. RR
21	18, 20, 8	lttr 5569	. . 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 5424	. . . . . 6 |- 0 e. RR
24	23, 18, 8	ltaddsub 5623	. . . . 5 |- ((0 + A) < 1 <-> 0 < (1 - A))
25	18	recn 5297	. . . . . . 7 |- A e. CC
26	25	addid2 5314	. . . . . 6 |- (0 + A) = A
27	26	breq1i 2622	. . . . 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 5422	. . . 4 |- (1 - A) e. RR
31	30	gt0ne0 5595	. . 3 |- (0 < (1 - A) -> (1 - A) =/= 0)
32	29, 31	syl 10	. 2 |- (A < (Y / (X + Y)) -> (1 - A) =/= 0)
33	2	recn 5297	. . . . 5 |- X e. CC
34	30	recn 5297	. . . . 5 |- (1 - A) e. CC
35	33, 25, 34	divassz 5718	. . . 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 5791	. . . . . . . . . . 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 5297	. . . . . . . . . 10 |- Y e. CC
41	25, 33, 40	adddi 5309	. . . . . . . . 9 |- (A x. (X + Y)) = ((A x. X) + (A x. Y))
42	39, 41	syl5eqbrr 2645	. . . . . . . 8 |- (A < (Y / (X + Y)) -> ((A x. X) + (A x. Y)) < Y)
43	40	mulid2 5316	. . . . . . . 8 |- (1 x. Y) = Y
44	42, 43	syl6breqr 2651	. . . . . . 7 |- (A < (Y / (X + Y)) -> ((A x. X) + (A x. Y)) < (1 x. Y))
45	18, 2	remulcl 5318	. . . . . . . 8 |- (A x. X) e. RR
46	18, 3	remulcl 5318	. . . . . . . 8 |- (A x. Y) e. RR
47	8, 3	remulcl 5318	. . . . . . . 8 |- (1 x. Y) e. RR
48	45, 46, 47	ltaddsub 5623	. . . . . . 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 5252	. . . . . . 7 |- 1 e. CC
51	50, 25, 40	subdir 5413	. . . . . 6 |- ((1 - A) x. Y) = ((1 x. Y) - (A x. Y))
52	49, 51	syl6breqr 2651	. . . . 5 |- (A < (Y / (X + Y)) -> (A x. X) < ((1 - A) x. Y))
53	25, 33	mulcom 5306	. . . . 5 |- (A x. X) = (X x. A)
54	52, 53	syl5eqbrr 2645	. . . 4 |- (A < (Y / (X + Y)) -> (X x. A) < ((1 - A) x. Y))
55	2, 18	remulcl 5318	. . . . . . 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 5829	. . . . . 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 2633	. 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 955   e. wcel 957   =/= wne 1583   class class class wbr 2615  (class class class)co 3958  RRcr 5216  0cc0 5217  1c1 5218   + caddc 5220   x. cmul 5222   - cmin 5275   / cdiv 5277   < clt 5469

This theorem is referenced by:  efcnlem2 7377

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-ltr 5153  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-1 5225  df-i 5226  df-r 5227  df-plus 5228  df-mul 5229  df-lt 5230  df-sub 5339  df-neg 5341  df-pnf 5470  df-mnf 5471  df-xr 5472  df-ltxr 5473  df-le 5474  df-div 5682

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