Theorem cosh111lem1 10068

Description: Lemma for cosh111 10071.

Hypotheses

Ref	Expression
cosh111lem1.1	|- A e. RR
cosh111lem1.2	|- B e. RR
cosh111lem1.3	|- 0 <_ A
cosh111lem1.4	|- 0 <_ B
cosh111lem1.5	|- A < pi
cosh111lem1.6	|- B < pi

Assertion

Ref	Expression
cosh111lem1	|- (A < B -> (cos` B) < (cos` A))

Proof of Theorem cosh111lem1

Step	Hyp	Ref	Expression
1		cosh111lem1.3	. . . . . . 7 |- 0 <_ A
2		0re 6603	. . . . . . . 8 |- 0 e. RR
3		cosh111lem1.1	. . . . . . . 8 |- A e. RR
4		cosh111lem1.2	. . . . . . . 8 |- B e. RR
5	2, 3, 4	lelttri 6761	. . . . . . 7 |- ((0 <_ A /\ A < B) -> 0 < B)
6	1, 5	mpan 759	. . . . . 6 |- (A < B -> 0 < B)
7		addgtge0 6835	. . . . . . . . 9 |- (((B e. RR /\ A e. RR) /\ (0 < B /\ 0 <_ A)) -> 0 < (B + A))
8	4, 3, 7	mpanl12 773	. . . . . . . 8 |- ((0 < B /\ 0 <_ A) -> 0 < (B + A))
9	1, 8	mpan2 760	. . . . . . 7 |- (0 < B -> 0 < (B + A))
10	4, 3	readdcli 6487	. . . . . . . 8 |- (B + A) e. RR
11		2re 7163	. . . . . . . 8 |- 2 e. RR
12		2pos 7173	. . . . . . . 8 |- 0 < 2
13	10, 11, 12	divgt0i2i 7041	. . . . . . 7 |- (0 < (B + A) -> 0 < ((B + A) / 2))
14	9, 13	syl 12	. . . . . 6 |- (0 < B -> 0 < ((B + A) / 2))
15		2ne0 7174	. . . . . . . 8 |- 2 =/= 0
16	10, 11, 15	redivcli 6976	. . . . . . 7 |- ((B + A) / 2) e. RR
17		cosh111lem1.6	. . . . . . . . . . 11 |- B < pi
18		cosh111lem1.5	. . . . . . . . . . 11 |- A < pi
19		pire 10026	. . . . . . . . . . . 12 |- pi e. RR
20	4, 3, 19, 19	lt2addi 6773	. . . . . . . . . . 11 |- ((B < pi /\ A < pi) -> (B + A) < (pi + pi))
21	17, 18, 20	mp2an 761	. . . . . . . . . 10 |- (B + A) < (pi + pi)
22	19	recni 6467	. . . . . . . . . . 11 |- pi e. CC
23	22	2timesi 7187	. . . . . . . . . 10 |- (2 x. pi) = (pi + pi)
24	21, 23	breqtrri 3362	. . . . . . . . 9 |- (B + A) < (2 x. pi)
25	11, 19	remulcli 6488	. . . . . . . . . 10 |- (2 x. pi) e. RR
26	10, 25, 11, 12	ltdiv1ii 7001	. . . . . . . . 9 |- ((B + A) < (2 x. pi) <-> ((B + A) / 2) < ((2 x. pi) / 2))
27	24, 26	mpbi 206	. . . . . . . 8 |- ((B + A) / 2) < ((2 x. pi) / 2)
28		2cn 7164	. . . . . . . . 9 |- 2 e. CC
29	22, 28, 15	divcan3i 6934	. . . . . . . 8 |- ((2 x. pi) / 2) = pi
30	27, 29	breqtri 3360	. . . . . . 7 |- ((B + A) / 2) < pi
31		elioo2 7546	. . . . . . . . . 10 |- ((0 e. RR* /\ pi e. RR*) -> (((B + A) / 2) e. (0(,)pi) <-> (((B + A) / 2) e. RR /\ 0 < ((B + A) / 2) /\ ((B + A) / 2) < pi)))
32		rexr 6668	. . . . . . . . . 10 |- (0 e. RR -> 0 e. RR*)
33		rexr 6668	. . . . . . . . . 10 |- (pi e. RR -> pi e. RR*)
34	31, 32, 33	syl2an 503	. . . . . . . . 9 |- ((0 e. RR /\ pi e. RR) -> (((B + A) / 2) e. (0(,)pi) <-> (((B + A) / 2) e. RR /\ 0 < ((B + A) / 2) /\ ((B + A) / 2) < pi)))
35	2, 19, 34	mp2an 761	. . . . . . . 8 |- (((B + A) / 2) e. (0(,)pi) <-> (((B + A) / 2) e. RR /\ 0 < ((B + A) / 2) /\ ((B + A) / 2) < pi))
36		sinq12gt0t 10057	. . . . . . . 8 |- (((B + A) / 2) e. (0(,)pi) -> 0 < (sin` ((B + A) / 2)))
37	35, 36	sylbir 218	. . . . . . 7 |- ((((B + A) / 2) e. RR /\ 0 < ((B + A) / 2) /\ ((B + A) / 2) < pi) -> 0 < (sin` ((B + A) / 2)))
38	16, 30, 37	mp3an13 1182	. . . . . 6 |- (0 < ((B + A) / 2) -> 0 < (sin` ((B + A) / 2)))
39	6, 14, 38	3syl 24	. . . . 5 |- (A < B -> 0 < (sin` ((B + A) / 2)))
40	3, 4	posdifi 6854	. . . . . . 7 |- (A < B <-> 0 < (B - A))
41	4, 3	resubcli 6602	. . . . . . . 8 |- (B - A) e. RR
42		gt0div 7035	. . . . . . . 8 |- (((B - A) e. RR /\ 2 e. RR /\ 0 < 2) -> (0 < (B - A) <-> 0 < ((B - A) / 2)))
43	41, 11, 12, 42	mp3an 1191	. . . . . . 7 |- (0 < (B - A) <-> 0 < ((B - A) / 2))
44	40, 43	bitri 190	. . . . . 6 |- (A < B <-> 0 < ((B - A) / 2))
45	41, 11, 15	redivcli 6976	. . . . . . 7 |- ((B - A) / 2) e. RR
46	22	addid2i 6484	. . . . . . . . . . . . 13 |- (0 + pi) = pi
47	2, 3, 19	leadd1i 6767	. . . . . . . . . . . . . 14 |- (0 <_ A <-> (0 + pi) <_ (A + pi))
48	1, 47	mpbi 206	. . . . . . . . . . . . 13 |- (0 + pi) <_ (A + pi)
49	46, 48	eqbrtrri 3358	. . . . . . . . . . . 12 |- pi <_ (A + pi)
50	3, 19	readdcli 6487	. . . . . . . . . . . . 13 |- (A + pi) e. RR
51	4, 19, 50	ltletri 6762	. . . . . . . . . . . 12 |- ((B < pi /\ pi <_ (A + pi)) -> B < (A + pi))
52	17, 49, 51	mp2an 761	. . . . . . . . . . 11 |- B < (A + pi)
53	4, 3, 19	ltsubadd2i 6821	. . . . . . . . . . 11 |- ((B - A) < pi <-> B < (A + pi))
54	52, 53	mpbir 207	. . . . . . . . . 10 |- (B - A) < pi
55		1lt2 7212	. . . . . . . . . . 11 |- 1 < 2
56		pipos 10027	. . . . . . . . . . . 12 |- 0 < pi
57		ltmulgt12 7029	. . . . . . . . . . . 12 |- ((pi e. RR /\ 2 e. RR /\ 0 < pi) -> (1 < 2 <-> pi < (2 x. pi)))
58	19, 11, 56, 57	mp3an 1191	. . . . . . . . . . 11 |- (1 < 2 <-> pi < (2 x. pi))
59	55, 58	mpbi 206	. . . . . . . . . 10 |- pi < (2 x. pi)
60	41, 19, 25	lttri 6760	. . . . . . . . . 10 |- (((B - A) < pi /\ pi < (2 x. pi)) -> (B - A) < (2 x. pi))
61	54, 59, 60	mp2an 761	. . . . . . . . 9 |- (B - A) < (2 x. pi)
62	41, 25, 11, 12	ltdiv1ii 7001	. . . . . . . . 9 |- ((B - A) < (2 x. pi) <-> ((B - A) / 2) < ((2 x. pi) / 2))
63	61, 62	mpbi 206	. . . . . . . 8 |- ((B - A) / 2) < ((2 x. pi) / 2)
64	63, 29	breqtri 3360	. . . . . . 7 |- ((B - A) / 2) < pi
65		elioo2 7546	. . . . . . . . . 10 |- ((0 e. RR* /\ pi e. RR*) -> (((B - A) / 2) e. (0(,)pi) <-> (((B - A) / 2) e. RR /\ 0 < ((B - A) / 2) /\ ((B - A) / 2) < pi)))
66	65, 32, 33	syl2an 503	. . . . . . . . 9 |- ((0 e. RR /\ pi e. RR) -> (((B - A) / 2) e. (0(,)pi) <-> (((B - A) / 2) e. RR /\ 0 < ((B - A) / 2) /\ ((B - A) / 2) < pi)))
67	2, 19, 66	mp2an 761	. . . . . . . 8 |- (((B - A) / 2) e. (0(,)pi) <-> (((B - A) / 2) e. RR /\ 0 < ((B - A) / 2) /\ ((B - A) / 2) < pi))
68		sinq12gt0t 10057	. . . . . . . 8 |- (((B - A) / 2) e. (0(,)pi) -> 0 < (sin` ((B - A) / 2)))
69	67, 68	sylbir 218	. . . . . . 7 |- ((((B - A) / 2) e. RR /\ 0 < ((B - A) / 2) /\ ((B - A) / 2) < pi) -> 0 < (sin` ((B - A) / 2)))
70	45, 64, 69	mp3an13 1182	. . . . . 6 |- (0 < ((B - A) / 2) -> 0 < (sin` ((B - A) / 2)))
71	44, 70	sylbi 216	. . . . 5 |- (A < B -> 0 < (sin` ((B - A) / 2)))
72		resincl 8703	. . . . . . 7 |- (((B + A) / 2) e. RR -> (sin` ((B + A) / 2)) e. RR)
73	16, 72	ax-mp 7	. . . . . 6 |- (sin` ((B + A) / 2)) e. RR
74		resincl 8703	. . . . . . 7 |- (((B - A) / 2) e. RR -> (sin` ((B - A) / 2)) e. RR)
75	45, 74	ax-mp 7	. . . . . 6 |- (sin` ((B - A) / 2)) e. RR
76	73, 75	mulgt0i 6786	. . . . 5 |- ((0 < (sin` ((B + A) / 2)) /\ 0 < (sin` ((B - A) / 2))) -> 0 < ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2))))
77	39, 71, 76	syl11anc 524	. . . 4 |- (A < B -> 0 < ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2))))
78	73, 75	remulcli 6488	. . . . . 6 |- ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2))) e. RR
79	11, 78	mulgt0i 6786	. . . . 5 |- ((0 < 2 /\ 0 < ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2)))) -> 0 < (2 x. ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2)))))
80	12, 79	mpan 759	. . . 4 |- (0 < ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2))) -> 0 < (2 x. ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2)))))
81	77, 80	syl 12	. . 3 |- (A < B -> 0 < (2 x. ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2)))))
82	4	recni 6467	. . . . . . 7 |- B e. CC
83	3	recni 6467	. . . . . . 7 |- A e. CC
84		subcos 8725	. . . . . . 7 |- ((B e. CC /\ A e. CC) -> ((cos` B) - (cos` A)) = (-u2 x. ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2)))))
85	82, 83, 84	mp2an 761	. . . . . 6 |- ((cos` B) - (cos` A)) = (-u2 x. ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2))))
86	78	recni 6467	. . . . . . 7 |- ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2))) e. CC
87	28, 86	mulneg1i 6608	. . . . . 6 |- (-u2 x. ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2)))) = -u(2 x. ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2))))
88	85, 87	eqtri 1908	. . . . 5 |- ((cos` B) - (cos` A)) = -u(2 x. ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2))))
89		recoscl 8704	. . . . . . . . 9 |- (B e. RR -> (cos` B) e. RR)
90	4, 89	ax-mp 7	. . . . . . . 8 |- (cos` B) e. RR
91	90	recni 6467	. . . . . . 7 |- (cos` B) e. CC
92		recoscl 8704	. . . . . . . . 9 |- (A e. RR -> (cos` A) e. RR)
93	3, 92	ax-mp 7	. . . . . . . 8 |- (cos` A) e. RR
94	93	recni 6467	. . . . . . 7 |- (cos` A) e. CC
95	91, 94	subcli 6523	. . . . . 6 |- ((cos` B) - (cos` A)) e. CC
96	28, 86	mulcli 6474	. . . . . 6 |- (2 x. ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2)))) e. CC
97	95, 96	negcon2i 6568	. . . . 5 |- (((cos` B) - (cos` A)) = -u(2 x. ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2)))) <-> (2 x. ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2)))) = -u((cos` B) - (cos` A)))
98	88, 97	mpbi 206	. . . 4 |- (2 x. ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2)))) = -u((cos` B) - (cos` A))
99	91, 94	negsubdi2i 6614	. . . 4 |- -u((cos` B) - (cos` A)) = ((cos` A) - (cos` B))
100	98, 99	eqtr2i 1909	. . 3 |- ((cos` A) - (cos` B)) = (2 x. ((sin` ((B + A) / 2)) x. (sin` ((B - A) / 2))))
101	81, 100	syl6breqr 3377	. 2 |- (A < B -> 0 < ((cos` A) - (cos` B)))
102	90, 93	posdifi 6854	. 2 |- ((cos` B) < (cos` A) <-> 0 < ((cos` A) - (cos` B)))
103	101, 102	sylibr 217	1 |- (A < B -> (cos` B) < (cos` A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447   <_ cle 6448  RR*cxr 6652   < clt 6653  2c2 7145  (,)cioo 7524  sincsin 8557  cosccos 8558  picpi 8559

This theorem is referenced by:  cosh111lem2 10069

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-5 7157  df-6 7158  df-7 7159  df-8 7160  df-9 7161  df-rp 7232  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-ioc 7529  df-ico 7530  df-icc 7531  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-fac 8184  df-bc 8209  df-clim 8235  df-sum 8240  df-cncf 8525  df-ef 8560  df-sin 8562  df-cos 8563  df-pi 8564  df-top 8861  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200

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