Theorem imo72b2lem0 38465

Description: Lemma for imo72b2 38475. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypotheses

Ref	Expression
imo72b2lem0.1	|- ( ph -> F : RR --> RR )
imo72b2lem0.2	|- ( ph -> G : RR --> RR )
imo72b2lem0.3	|- ( ph -> A e. RR )
imo72b2lem0.4	|- ( ph -> B e. RR )
imo72b2lem0.5	|- ( ph -> ( ( F ` ( A + B ) ) + ( F ` ( A - B ) ) ) = ( 2 x. ( ( F ` A ) x. ( G ` B ) ) ) )
imo72b2lem0.6	|- ( ph -> A. y e. RR ( abs ` ( F ` y ) ) <_ 1 )

Assertion

Ref	Expression
imo72b2lem0	|- ( ph -> ( ( abs ` ( F ` A ) ) x. ( abs ` ( G ` B ) ) ) <_ sup ( ( abs " ( F " RR ) ) , RR , < ) )

Distinct variable groups:   y, F   ph, y

Allowed substitution hints:   A( y)   B( y)   G( y)

Proof of Theorem imo72b2lem0

Dummy variables c x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imo72b2lem0.1	. . . . . . . 8 |- ( ph -> F : RR --> RR )
2		imo72b2lem0.3	. . . . . . . 8 |- ( ph -> A e. RR )
3	1, 2	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( F ` A ) e. RR )
4	3	recnd 10068	. . . . . 6 |- ( ph -> ( F ` A ) e. CC )
5	4	idi 2	. . . . 5 |- ( ph -> ( F ` A ) e. CC )
6		imo72b2lem0.2	. . . . . . . 8 |- ( ph -> G : RR --> RR )
7		imo72b2lem0.4	. . . . . . . 8 |- ( ph -> B e. RR )
8	6, 7	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( G ` B ) e. RR )
9	8	recnd 10068	. . . . . 6 |- ( ph -> ( G ` B ) e. CC )
10	9	idi 2	. . . . 5 |- ( ph -> ( G ` B ) e. CC )
11	5, 10	mulcld 10060	. . . 4 |- ( ph -> ( ( F ` A ) x. ( G ` B ) ) e. CC )
12	11	abscld 14175	. . 3 |- ( ph -> ( abs ` ( ( F ` A ) x. ( G ` B ) ) ) e. RR )
13		imaco 5640	. . . . . 6 |- ( ( abs o. F ) " RR ) = ( abs " ( F " RR ) )
14	13	eqcomi 2631	. . . . 5 |- ( abs " ( F " RR ) ) = ( ( abs o. F ) " RR )
15		imassrn 5477	. . . . . . 7 |- ( ( abs o. F ) " RR ) C_ ran ( abs o. F )
16	15	a1i 11	. . . . . 6 |- ( ph -> ( ( abs o. F ) " RR ) C_ ran ( abs o. F ) )
17		absf 14077	. . . . . . . . . 10 |- abs : CC --> RR
18	17	a1i 11	. . . . . . . . 9 |- ( ph -> abs : CC --> RR )
19		ax-resscn 9993	. . . . . . . . . 10 |- RR C_ CC
20	19	a1i 11	. . . . . . . . 9 |- ( ph -> RR C_ CC )
21	18, 20	fssresd 6071	. . . . . . . 8 |- ( ph -> ( abs |` RR ) : RR --> RR )
22	1, 21	fco2d 38461	. . . . . . 7 |- ( ph -> ( abs o. F ) : RR --> RR )
23		frn 6053	. . . . . . 7 |- ( ( abs o. F ) : RR --> RR -> ran ( abs o. F ) C_ RR )
24	22, 23	syl 17	. . . . . 6 |- ( ph -> ran ( abs o. F ) C_ RR )
25	16, 24	sstrd 3613	. . . . 5 |- ( ph -> ( ( abs o. F ) " RR ) C_ RR )
26	14, 25	syl5eqss 3649	. . . 4 |- ( ph -> ( abs " ( F " RR ) ) C_ RR )
27		0re 10040	. . . . . . . . . 10 |- 0 e. RR
28	27	ne0ii 3923	. . . . . . . . 9 |- RR =/= (/)
29	28	a1i 11	. . . . . . . 8 |- ( ph -> RR =/= (/) )
30	29, 22	wnefimgd 38460	. . . . . . 7 |- ( ph -> ( ( abs o. F ) " RR ) =/= (/) )
31	30	necomd 2849	. . . . . 6 |- ( ph -> (/) =/= ( ( abs o. F ) " RR ) )
32	14	a1i 11	. . . . . 6 |- ( ph -> ( abs " ( F " RR ) ) = ( ( abs o. F ) " RR ) )
33	31, 32	neeqtrrd 2868	. . . . 5 |- ( ph -> (/) =/= ( abs " ( F " RR ) ) )
34	33	necomd 2849	. . . 4 |- ( ph -> ( abs " ( F " RR ) ) =/= (/) )
35		1red 10055	. . . . 5 |- ( ph -> 1 e. RR )
36		simpr 477	. . . . . . 7 |- ( ( ph /\ c = 1 ) -> c = 1 )
37	36	breq2d 4665	. . . . . 6 |- ( ( ph /\ c = 1 ) -> ( x <_ c <-> x <_ 1 ) )
38	37	ralbidv 2986	. . . . 5 |- ( ( ph /\ c = 1 ) -> ( A. x e. ( abs " ( F " RR ) ) x <_ c <-> A. x e. ( abs " ( F " RR ) ) x <_ 1 ) )
39		imo72b2lem0.6	. . . . . 6 |- ( ph -> A. y e. RR ( abs ` ( F ` y ) ) <_ 1 )
40	1, 39	extoimad 38464	. . . . 5 |- ( ph -> A. x e. ( abs " ( F " RR ) ) x <_ 1 )
41	35, 38, 40	rspcedvd 3317	. . . 4 |- ( ph -> E. c e. RR A. x e. ( abs " ( F " RR ) ) x <_ c )
42	26, 34, 41	suprcld 10986	. . 3 |- ( ph -> sup ( ( abs " ( F " RR ) ) , RR , < ) e. RR )
43		2re 11090	. . . 4 |- 2 e. RR
44	43	a1i 11	. . 3 |- ( ph -> 2 e. RR )
45		imo72b2lem0.5	. . . . . . . . 9 |- ( ph -> ( ( F ` ( A + B ) ) + ( F ` ( A - B ) ) ) = ( 2 x. ( ( F ` A ) x. ( G ` B ) ) ) )
46	45	idi 2	. . . . . . . 8 |- ( ph -> ( ( F ` ( A + B ) ) + ( F ` ( A - B ) ) ) = ( 2 x. ( ( F ` A ) x. ( G ` B ) ) ) )
47	46	fveq2d 6195	. . . . . . 7 |- ( ph -> ( abs ` ( ( F ` ( A + B ) ) + ( F ` ( A - B ) ) ) ) = ( abs ` ( 2 x. ( ( F ` A ) x. ( G ` B ) ) ) ) )
48		2cnd 11093	. . . . . . . . 9 |- ( ph -> 2 e. CC )
49	48, 11	mulcld 10060	. . . . . . . 8 |- ( ph -> ( 2 x. ( ( F ` A ) x. ( G ` B ) ) ) e. CC )
50	49	abscld 14175	. . . . . . 7 |- ( ph -> ( abs ` ( 2 x. ( ( F ` A ) x. ( G ` B ) ) ) ) e. RR )
51	47, 50	eqeltrd 2701	. . . . . 6 |- ( ph -> ( abs ` ( ( F ` ( A + B ) ) + ( F ` ( A - B ) ) ) ) e. RR )
52	1	idi 2	. . . . . . . . . 10 |- ( ph -> F : RR --> RR )
53	2	idi 2	. . . . . . . . . . 11 |- ( ph -> A e. RR )
54	7	idi 2	. . . . . . . . . . 11 |- ( ph -> B e. RR )
55	53, 54	readdcld 10069	. . . . . . . . . 10 |- ( ph -> ( A + B ) e. RR )
56	52, 55	ffvelrnd 6360	. . . . . . . . 9 |- ( ph -> ( F ` ( A + B ) ) e. RR )
57	56	recnd 10068	. . . . . . . 8 |- ( ph -> ( F ` ( A + B ) ) e. CC )
58	57	abscld 14175	. . . . . . 7 |- ( ph -> ( abs ` ( F ` ( A + B ) ) ) e. RR )
59	53, 54	resubcld 10458	. . . . . . . . . 10 |- ( ph -> ( A - B ) e. RR )
60	52, 59	ffvelrnd 6360	. . . . . . . . 9 |- ( ph -> ( F ` ( A - B ) ) e. RR )
61	60	recnd 10068	. . . . . . . 8 |- ( ph -> ( F ` ( A - B ) ) e. CC )
62	61	abscld 14175	. . . . . . 7 |- ( ph -> ( abs ` ( F ` ( A - B ) ) ) e. RR )
63	58, 62	readdcld 10069	. . . . . 6 |- ( ph -> ( ( abs ` ( F ` ( A + B ) ) ) + ( abs ` ( F ` ( A - B ) ) ) ) e. RR )
64	44, 42	remulcld 10070	. . . . . 6 |- ( ph -> ( 2 x. sup ( ( abs " ( F " RR ) ) , RR , < ) ) e. RR )
65	57, 61	abstrid 14195	. . . . . 6 |- ( ph -> ( abs ` ( ( F ` ( A + B ) ) + ( F ` ( A - B ) ) ) ) <_ ( ( abs ` ( F ` ( A + B ) ) ) + ( abs ` ( F ` ( A - B ) ) ) ) )
66	1, 55	fvco3d 38462	. . . . . . . . . 10 |- ( ph -> ( ( abs o. F ) ` ( A + B ) ) = ( abs ` ( F ` ( A + B ) ) ) )
67	55, 22	wfximgfd 38463	. . . . . . . . . . 11 |- ( ph -> ( ( abs o. F ) ` ( A + B ) ) e. ( ( abs o. F ) " RR ) )
68	32	idi 2	. . . . . . . . . . 11 |- ( ph -> ( abs " ( F " RR ) ) = ( ( abs o. F ) " RR ) )
69	67, 68	eleqtrrd 2704	. . . . . . . . . 10 |- ( ph -> ( ( abs o. F ) ` ( A + B ) ) e. ( abs " ( F " RR ) ) )
70	66, 69	eqeltrrd 2702	. . . . . . . . 9 |- ( ph -> ( abs ` ( F ` ( A + B ) ) ) e. ( abs " ( F " RR ) ) )
71	26, 34, 41, 70	suprubd 10985	. . . . . . . 8 |- ( ph -> ( abs ` ( F ` ( A + B ) ) ) <_ sup ( ( abs " ( F " RR ) ) , RR , < ) )
72	1, 59	fvco3d 38462	. . . . . . . . . 10 |- ( ph -> ( ( abs o. F ) ` ( A - B ) ) = ( abs ` ( F ` ( A - B ) ) ) )
73	59, 22	wfximgfd 38463	. . . . . . . . . . 11 |- ( ph -> ( ( abs o. F ) ` ( A - B ) ) e. ( ( abs o. F ) " RR ) )
74	73, 32	eleqtrrd 2704	. . . . . . . . . 10 |- ( ph -> ( ( abs o. F ) ` ( A - B ) ) e. ( abs " ( F " RR ) ) )
75	72, 74	eqeltrrd 2702	. . . . . . . . 9 |- ( ph -> ( abs ` ( F ` ( A - B ) ) ) e. ( abs " ( F " RR ) ) )
76	26, 34, 41, 75	suprubd 10985	. . . . . . . 8 |- ( ph -> ( abs ` ( F ` ( A - B ) ) ) <_ sup ( ( abs " ( F " RR ) ) , RR , < ) )
77	58, 62, 42, 42, 71, 76	le2addd 10646	. . . . . . 7 |- ( ph -> ( ( abs ` ( F ` ( A + B ) ) ) + ( abs ` ( F ` ( A - B ) ) ) ) <_ ( sup ( ( abs " ( F " RR ) ) , RR , < ) + sup ( ( abs " ( F " RR ) ) , RR , < ) ) )
78	42	recnd 10068	. . . . . . . . 9 |- ( ph -> sup ( ( abs " ( F " RR ) ) , RR , < ) e. CC )
79	78	2timesd 11275	. . . . . . . 8 |- ( ph -> ( 2 x. sup ( ( abs " ( F " RR ) ) , RR , < ) ) = ( sup ( ( abs " ( F " RR ) ) , RR , < ) + sup ( ( abs " ( F " RR ) ) , RR , < ) ) )
80	79	eqcomd 2628	. . . . . . 7 |- ( ph -> ( sup ( ( abs " ( F " RR ) ) , RR , < ) + sup ( ( abs " ( F " RR ) ) , RR , < ) ) = ( 2 x. sup ( ( abs " ( F " RR ) ) , RR , < ) ) )
81	80, 64	eqeltrd 2701	. . . . . . 7 |- ( ph -> ( sup ( ( abs " ( F " RR ) ) , RR , < ) + sup ( ( abs " ( F " RR ) ) , RR , < ) ) e. RR )
82	77, 80, 63, 81	leeq2d 38456	. . . . . 6 |- ( ph -> ( ( abs ` ( F ` ( A + B ) ) ) + ( abs ` ( F ` ( A - B ) ) ) ) <_ ( 2 x. sup ( ( abs " ( F " RR ) ) , RR , < ) ) )
83	51, 63, 64, 65, 82	letrd 10194	. . . . 5 |- ( ph -> ( abs ` ( ( F ` ( A + B ) ) + ( F ` ( A - B ) ) ) ) <_ ( 2 x. sup ( ( abs " ( F " RR ) ) , RR , < ) ) )
84	83, 47, 51, 64	leeq1d 38455	. . . 4 |- ( ph -> ( abs ` ( 2 x. ( ( F ` A ) x. ( G ` B ) ) ) ) <_ ( 2 x. sup ( ( abs " ( F " RR ) ) , RR , < ) ) )
85		0le2 11111	. . . . . 6 |- 0 <_ 2
86	85	a1i 11	. . . . 5 |- ( ph -> 0 <_ 2 )
87	3	idi 2	. . . . . 6 |- ( ph -> ( F ` A ) e. RR )
88	8	idi 2	. . . . . 6 |- ( ph -> ( G ` B ) e. RR )
89	87, 88	remulcld 10070	. . . . 5 |- ( ph -> ( ( F ` A ) x. ( G ` B ) ) e. RR )
90	86, 44, 89	absmulrposd 38457	. . . 4 |- ( ph -> ( abs ` ( 2 x. ( ( F ` A ) x. ( G ` B ) ) ) ) = ( 2 x. ( abs ` ( ( F ` A ) x. ( G ` B ) ) ) ) )
91	84, 90, 50, 64	leeq1d 38455	. . 3 |- ( ph -> ( 2 x. ( abs ` ( ( F ` A ) x. ( G ` B ) ) ) ) <_ ( 2 x. sup ( ( abs " ( F " RR ) ) , RR , < ) ) )
92		2pos 11112	. . . 4 |- 0 < 2
93	92	a1i 11	. . 3 |- ( ph -> 0 < 2 )
94	12, 42, 44, 91, 93	wwlemuld 38454	. 2 |- ( ph -> ( abs ` ( ( F ` A ) x. ( G ` B ) ) ) <_ sup ( ( abs " ( F " RR ) ) , RR , < ) )
95	4, 9	absmuld 14193	. . 3 |- ( ph -> ( abs ` ( ( F ` A ) x. ( G ` B ) ) ) = ( ( abs ` ( F ` A ) ) x. ( abs ` ( G ` B ) ) ) )
96	95	idi 2	. 2 |- ( ph -> ( abs ` ( ( F ` A ) x. ( G ` B ) ) ) = ( ( abs ` ( F ` A ) ) x. ( abs ` ( G ` B ) ) ) )
97	94, 96, 12, 42	leeq1d 38455	1 |- ( ph -> ( ( abs ` ( F ` A ) ) x. ( abs ` ( G ` B ) ) ) <_ sup ( ( abs " ( F " RR ) ) , RR , < ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ran crn 5115   "cima 5117   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   2c2 11070   abscabs 13974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976

This theorem is referenced by:  imo72b2  38475