Theorem cnre2csqima 29957

Description: Image of a centered square by the canonical bijection from ( RR X. RR ) to CC. (Contributed by Thierry Arnoux, 27-Sep-2017.)

Hypothesis

Ref	Expression
cnre2csqima.1	|- F = ( x e. RR , y e. RR |-> ( x + ( _i x. y ) ) )

Assertion

Ref	Expression
cnre2csqima	|- ( ( X e. ( RR X. RR ) /\ Y e. ( RR X. RR ) /\ D e. RR+ ) -> ( Y e. ( ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) X. ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) -> ( ( abs ` ( Re ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D /\ ( abs ` ( Im ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   D( x, y)   F( x, y)   X( x, y)   Y( x, y)

Proof of Theorem cnre2csqima

Dummy variables z u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioossre 12235	. . 3 |- ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) C_ RR
2		ioossre 12235	. . 3 |- ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) C_ RR
3		xpinpreima2 29953	. . . 4 |- ( ( ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) C_ RR /\ ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) C_ RR ) -> ( ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) X. ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) = ( ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) )
4	3	eleq2d 2687	. . 3 |- ( ( ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) C_ RR /\ ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) C_ RR ) -> ( Y e. ( ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) X. ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) <-> Y e. ( ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) ) )
5	1, 2, 4	mp2an 708	. 2 |- ( Y e. ( ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) X. ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) <-> Y e. ( ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) )
6		elin 3796	. . 3 |- ( Y e. ( ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) <-> ( Y e. ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) /\ Y e. ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) )
7		simpl 473	. . . . . . . . . . 11 |- ( ( x e. RR /\ y e. RR ) -> x e. RR )
8	7	recnd 10068	. . . . . . . . . 10 |- ( ( x e. RR /\ y e. RR ) -> x e. CC )
9		ax-icn 9995	. . . . . . . . . . . 12 |- _i e. CC
10	9	a1i 11	. . . . . . . . . . 11 |- ( ( x e. RR /\ y e. RR ) -> _i e. CC )
11		simpr 477	. . . . . . . . . . . 12 |- ( ( x e. RR /\ y e. RR ) -> y e. RR )
12	11	recnd 10068	. . . . . . . . . . 11 |- ( ( x e. RR /\ y e. RR ) -> y e. CC )
13	10, 12	mulcld 10060	. . . . . . . . . 10 |- ( ( x e. RR /\ y e. RR ) -> ( _i x. y ) e. CC )
14	8, 13	addcld 10059	. . . . . . . . 9 |- ( ( x e. RR /\ y e. RR ) -> ( x + ( _i x. y ) ) e. CC )
15		reval 13846	. . . . . . . . 9 |- ( ( x + ( _i x. y ) ) e. CC -> ( Re ` ( x + ( _i x. y ) ) ) = ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) )
16	14, 15	syl 17	. . . . . . . 8 |- ( ( x e. RR /\ y e. RR ) -> ( Re ` ( x + ( _i x. y ) ) ) = ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) )
17		crre 13854	. . . . . . . 8 |- ( ( x e. RR /\ y e. RR ) -> ( Re ` ( x + ( _i x. y ) ) ) = x )
18	16, 17	eqtr3d 2658	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) = x )
19	18	mpt2eq3ia 6720	. . . . . 6 |- ( x e. RR , y e. RR |-> ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) ) = ( x e. RR , y e. RR |-> x )
20	14	adantl 482	. . . . . . . 8 |- ( ( T. /\ ( x e. RR /\ y e. RR ) ) -> ( x + ( _i x. y ) ) e. CC )
21		cnre2csqima.1	. . . . . . . . 9 |- F = ( x e. RR , y e. RR |-> ( x + ( _i x. y ) ) )
22	21	a1i 11	. . . . . . . 8 |- ( T. -> F = ( x e. RR , y e. RR |-> ( x + ( _i x. y ) ) ) )
23		df-re 13840	. . . . . . . . 9 |- Re = ( z e. CC |-> ( ( z + ( * ` z ) ) / 2 ) )
24	23	a1i 11	. . . . . . . 8 |- ( T. -> Re = ( z e. CC |-> ( ( z + ( * ` z ) ) / 2 ) ) )
25		id 22	. . . . . . . . . 10 |- ( z = ( x + ( _i x. y ) ) -> z = ( x + ( _i x. y ) ) )
26		fveq2 6191	. . . . . . . . . 10 |- ( z = ( x + ( _i x. y ) ) -> ( * ` z ) = ( * ` ( x + ( _i x. y ) ) ) )
27	25, 26	oveq12d 6668	. . . . . . . . 9 |- ( z = ( x + ( _i x. y ) ) -> ( z + ( * ` z ) ) = ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) )
28	27	oveq1d 6665	. . . . . . . 8 |- ( z = ( x + ( _i x. y ) ) -> ( ( z + ( * ` z ) ) / 2 ) = ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) )
29	20, 22, 24, 28	fmpt2co 7260	. . . . . . 7 |- ( T. -> ( Re o. F ) = ( x e. RR , y e. RR |-> ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) ) )
30	29	trud 1493	. . . . . 6 |- ( Re o. F ) = ( x e. RR , y e. RR |-> ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) )
31		df1stres 29481	. . . . . 6 |- ( 1st |` ( RR X. RR ) ) = ( x e. RR , y e. RR |-> x )
32	19, 30, 31	3eqtr4ri 2655	. . . . 5 |- ( 1st |` ( RR X. RR ) ) = ( Re o. F )
33	14	rgen2a 2977	. . . . . 6 |- A. x e. RR A. y e. RR ( x + ( _i x. y ) ) e. CC
34	21	fnmpt2 7238	. . . . . 6 |- ( A. x e. RR A. y e. RR ( x + ( _i x. y ) ) e. CC -> F Fn ( RR X. RR ) )
35	33, 34	ax-mp 5	. . . . 5 |- F Fn ( RR X. RR )
36		fo1st 7188	. . . . . 6 |- 1st : _V -onto-> _V
37		fofn 6117	. . . . . 6 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
38	36, 37	ax-mp 5	. . . . 5 |- 1st Fn _V
39		xp1st 7198	. . . . 5 |- ( z e. ( RR X. RR ) -> ( 1st ` z ) e. RR )
40	21	rnmpt2 6770	. . . . . . . 8 |- ran F = { z | E. x e. RR E. y e. RR z = ( x + ( _i x. y ) ) }
41		simpr 477	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ y e. RR ) /\ z = ( x + ( _i x. y ) ) ) -> z = ( x + ( _i x. y ) ) )
42	14	adantr 481	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ y e. RR ) /\ z = ( x + ( _i x. y ) ) ) -> ( x + ( _i x. y ) ) e. CC )
43	41, 42	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( ( x e. RR /\ y e. RR ) /\ z = ( x + ( _i x. y ) ) ) -> z e. CC )
44	43	ex 450	. . . . . . . . . 10 |- ( ( x e. RR /\ y e. RR ) -> ( z = ( x + ( _i x. y ) ) -> z e. CC ) )
45	44	rexlimivv 3036	. . . . . . . . 9 |- ( E. x e. RR E. y e. RR z = ( x + ( _i x. y ) ) -> z e. CC )
46	45	abssi 3677	. . . . . . . 8 |- { z | E. x e. RR E. y e. RR z = ( x + ( _i x. y ) ) } C_ CC
47	40, 46	eqsstri 3635	. . . . . . 7 |- ran F C_ CC
48		simpl 473	. . . . . . 7 |- ( ( z e. ran F /\ u e. ran F ) -> z e. ran F )
49	47, 48	sseldi 3601	. . . . . 6 |- ( ( z e. ran F /\ u e. ran F ) -> z e. CC )
50		simpr 477	. . . . . . 7 |- ( ( z e. ran F /\ u e. ran F ) -> u e. ran F )
51	47, 50	sseldi 3601	. . . . . 6 |- ( ( z e. ran F /\ u e. ran F ) -> u e. CC )
52	49, 51	resubd 13956	. . . . 5 |- ( ( z e. ran F /\ u e. ran F ) -> ( Re ` ( z - u ) ) = ( ( Re ` z ) - ( Re ` u ) ) )
53	32, 35, 38, 39, 52	cnre2csqlem 29956	. . . 4 |- ( ( X e. ( RR X. RR ) /\ Y e. ( RR X. RR ) /\ D e. RR+ ) -> ( Y e. ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) -> ( abs ` ( Re ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D ) )
54		imval 13847	. . . . . . . . 9 |- ( ( x + ( _i x. y ) ) e. CC -> ( Im ` ( x + ( _i x. y ) ) ) = ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) )
55	14, 54	syl 17	. . . . . . . 8 |- ( ( x e. RR /\ y e. RR ) -> ( Im ` ( x + ( _i x. y ) ) ) = ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) )
56		crim 13855	. . . . . . . 8 |- ( ( x e. RR /\ y e. RR ) -> ( Im ` ( x + ( _i x. y ) ) ) = y )
57	55, 56	eqtr3d 2658	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) = y )
58	57	mpt2eq3ia 6720	. . . . . 6 |- ( x e. RR , y e. RR |-> ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) ) = ( x e. RR , y e. RR |-> y )
59		df-im 13841	. . . . . . . . 9 |- Im = ( z e. CC |-> ( Re ` ( z / _i ) ) )
60	59	a1i 11	. . . . . . . 8 |- ( T. -> Im = ( z e. CC |-> ( Re ` ( z / _i ) ) ) )
61		oveq1 6657	. . . . . . . . 9 |- ( z = ( x + ( _i x. y ) ) -> ( z / _i ) = ( ( x + ( _i x. y ) ) / _i ) )
62	61	fveq2d 6195	. . . . . . . 8 |- ( z = ( x + ( _i x. y ) ) -> ( Re ` ( z / _i ) ) = ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) )
63	20, 22, 60, 62	fmpt2co 7260	. . . . . . 7 |- ( T. -> ( Im o. F ) = ( x e. RR , y e. RR |-> ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) ) )
64	63	trud 1493	. . . . . 6 |- ( Im o. F ) = ( x e. RR , y e. RR |-> ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) )
65		df2ndres 29482	. . . . . 6 |- ( 2nd |` ( RR X. RR ) ) = ( x e. RR , y e. RR |-> y )
66	58, 64, 65	3eqtr4ri 2655	. . . . 5 |- ( 2nd |` ( RR X. RR ) ) = ( Im o. F )
67		fo2nd 7189	. . . . . 6 |- 2nd : _V -onto-> _V
68		fofn 6117	. . . . . 6 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
69	67, 68	ax-mp 5	. . . . 5 |- 2nd Fn _V
70		xp2nd 7199	. . . . 5 |- ( z e. ( RR X. RR ) -> ( 2nd ` z ) e. RR )
71	49, 51	imsubd 13957	. . . . 5 |- ( ( z e. ran F /\ u e. ran F ) -> ( Im ` ( z - u ) ) = ( ( Im ` z ) - ( Im ` u ) ) )
72	66, 35, 69, 70, 71	cnre2csqlem 29956	. . . 4 |- ( ( X e. ( RR X. RR ) /\ Y e. ( RR X. RR ) /\ D e. RR+ ) -> ( Y e. ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) -> ( abs ` ( Im ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D ) )
73	53, 72	anim12d 586	. . 3 |- ( ( X e. ( RR X. RR ) /\ Y e. ( RR X. RR ) /\ D e. RR+ ) -> ( ( Y e. ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) /\ Y e. ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) -> ( ( abs ` ( Re ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D /\ ( abs ` ( Im ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D ) ) )
74	6, 73	syl5bi 232	. 2 |- ( ( X e. ( RR X. RR ) /\ Y e. ( RR X. RR ) /\ D e. RR+ ) -> ( Y e. ( ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) -> ( ( abs ` ( Re ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D /\ ( abs ` ( Im ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D ) ) )
75	5, 74	syl5bi 232	1 |- ( ( X e. ( RR X. RR ) /\ Y e. ( RR X. RR ) /\ D e. RR+ ) -> ( Y e. ( ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) X. ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) -> ( ( abs ` ( Re ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D /\ ( abs ` ( Im ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   CCcc 9934   RRcr 9935   _ici 9938   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   2c2 11070   RR+crp 11832   (,)cioo 12175   *ccj 13836   Recre 13837   Imcim 13838   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-1st 7168  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-ioo 12179  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:  tpr2rico  29958