Theorem iccshift 39744

Description: A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
iccshift.1	|- ( ph -> A e. RR )
iccshift.2	|- ( ph -> B e. RR )
iccshift.3	|- ( ph -> T e. RR )

Assertion

Ref	Expression
iccshift	|- ( ph -> ( ( A + T ) [,] ( B + T ) ) = { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } )

Distinct variable groups:   w, A, z   w, B, z   w, T, z   ph, z

Allowed substitution hint:   ph( w)

Proof of Theorem iccshift

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . . . . . 7 |- ( w = x -> ( w = ( z + T ) <-> x = ( z + T ) ) )
2	1	rexbidv 3052	. . . . . 6 |- ( w = x -> ( E. z e. ( A [,] B ) w = ( z + T ) <-> E. z e. ( A [,] B ) x = ( z + T ) ) )
3	2	elrab 3363	. . . . 5 |- ( x e. { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } <-> ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) )
4		simprr 796	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) -> E. z e. ( A [,] B ) x = ( z + T ) )
5		nfv 1843	. . . . . . . 8 |- F/ z ph
6		nfv 1843	. . . . . . . . 9 |- F/ z x e. CC
7		nfre1 3005	. . . . . . . . 9 |- F/ z E. z e. ( A [,] B ) x = ( z + T )
8	6, 7	nfan 1828	. . . . . . . 8 |- F/ z ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) )
9	5, 8	nfan 1828	. . . . . . 7 |- F/ z ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) )
10		nfv 1843	. . . . . . 7 |- F/ z x e. ( ( A + T ) [,] ( B + T ) )
11		simp3 1063	. . . . . . . . . 10 |- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> x = ( z + T ) )
12		iccshift.1	. . . . . . . . . . . . . . . 16 |- ( ph -> A e. RR )
13		iccshift.2	. . . . . . . . . . . . . . . 16 |- ( ph -> B e. RR )
14	12, 13	iccssred 39727	. . . . . . . . . . . . . . 15 |- ( ph -> ( A [,] B ) C_ RR )
15	14	sselda 3603	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. ( A [,] B ) ) -> z e. RR )
16		iccshift.3	. . . . . . . . . . . . . . 15 |- ( ph -> T e. RR )
17	16	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. ( A [,] B ) ) -> T e. RR )
18	15, 17	readdcld 10069	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( z + T ) e. RR )
19	12	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. ( A [,] B ) ) -> A e. RR )
20		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. ( A [,] B ) ) -> z e. ( A [,] B ) )
21	13	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. ( A [,] B ) ) -> B e. RR )
22		elicc2 12238	. . . . . . . . . . . . . . . . 17 |- ( ( A e. RR /\ B e. RR ) -> ( z e. ( A [,] B ) <-> ( z e. RR /\ A <_ z /\ z <_ B ) ) )
23	19, 21, 22	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( z e. ( A [,] B ) <-> ( z e. RR /\ A <_ z /\ z <_ B ) ) )
24	20, 23	mpbid 222	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( z e. RR /\ A <_ z /\ z <_ B ) )
25	24	simp2d 1074	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. ( A [,] B ) ) -> A <_ z )
26	19, 15, 17, 25	leadd1dd 10641	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( A + T ) <_ ( z + T ) )
27	24	simp3d 1075	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. ( A [,] B ) ) -> z <_ B )
28	15, 21, 17, 27	leadd1dd 10641	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( z + T ) <_ ( B + T ) )
29	18, 26, 28	3jca 1242	. . . . . . . . . . . 12 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( ( z + T ) e. RR /\ ( A + T ) <_ ( z + T ) /\ ( z + T ) <_ ( B + T ) ) )
30	29	3adant3 1081	. . . . . . . . . . 11 |- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( ( z + T ) e. RR /\ ( A + T ) <_ ( z + T ) /\ ( z + T ) <_ ( B + T ) ) )
31	12, 16	readdcld 10069	. . . . . . . . . . . . 13 |- ( ph -> ( A + T ) e. RR )
32	31	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( A + T ) e. RR )
33	13, 16	readdcld 10069	. . . . . . . . . . . . 13 |- ( ph -> ( B + T ) e. RR )
34	33	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( B + T ) e. RR )
35		elicc2 12238	. . . . . . . . . . . 12 |- ( ( ( A + T ) e. RR /\ ( B + T ) e. RR ) -> ( ( z + T ) e. ( ( A + T ) [,] ( B + T ) ) <-> ( ( z + T ) e. RR /\ ( A + T ) <_ ( z + T ) /\ ( z + T ) <_ ( B + T ) ) ) )
36	32, 34, 35	syl2anc 693	. . . . . . . . . . 11 |- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( ( z + T ) e. ( ( A + T ) [,] ( B + T ) ) <-> ( ( z + T ) e. RR /\ ( A + T ) <_ ( z + T ) /\ ( z + T ) <_ ( B + T ) ) ) )
37	30, 36	mpbird 247	. . . . . . . . . 10 |- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( z + T ) e. ( ( A + T ) [,] ( B + T ) ) )
38	11, 37	eqeltrd 2701	. . . . . . . . 9 |- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> x e. ( ( A + T ) [,] ( B + T ) ) )
39	38	3exp 1264	. . . . . . . 8 |- ( ph -> ( z e. ( A [,] B ) -> ( x = ( z + T ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) ) )
40	39	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) -> ( z e. ( A [,] B ) -> ( x = ( z + T ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) ) )
41	9, 10, 40	rexlimd 3026	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) -> ( E. z e. ( A [,] B ) x = ( z + T ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) )
42	4, 41	mpd 15	. . . . 5 |- ( ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) -> x e. ( ( A + T ) [,] ( B + T ) ) )
43	3, 42	sylan2b 492	. . . 4 |- ( ( ph /\ x e. { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } ) -> x e. ( ( A + T ) [,] ( B + T ) ) )
44	31	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( A + T ) e. RR )
45	33	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( B + T ) e. RR )
46		simpr 477	. . . . . . 7 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. ( ( A + T ) [,] ( B + T ) ) )
47		eliccre 39728	. . . . . . 7 |- ( ( ( A + T ) e. RR /\ ( B + T ) e. RR /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. RR )
48	44, 45, 46, 47	syl3anc 1326	. . . . . 6 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. RR )
49	48	recnd 10068	. . . . 5 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. CC )
50	12	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> A e. RR )
51	13	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> B e. RR )
52	16	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> T e. RR )
53	48, 52	resubcld 10458	. . . . . . 7 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) e. RR )
54	12	recnd 10068	. . . . . . . . . . 11 |- ( ph -> A e. CC )
55	16	recnd 10068	. . . . . . . . . . 11 |- ( ph -> T e. CC )
56	54, 55	pncand 10393	. . . . . . . . . 10 |- ( ph -> ( ( A + T ) - T ) = A )
57	56	eqcomd 2628	. . . . . . . . 9 |- ( ph -> A = ( ( A + T ) - T ) )
58	57	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> A = ( ( A + T ) - T ) )
59		elicc2 12238	. . . . . . . . . . . 12 |- ( ( ( A + T ) e. RR /\ ( B + T ) e. RR ) -> ( x e. ( ( A + T ) [,] ( B + T ) ) <-> ( x e. RR /\ ( A + T ) <_ x /\ x <_ ( B + T ) ) ) )
60	44, 45, 59	syl2anc 693	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x e. ( ( A + T ) [,] ( B + T ) ) <-> ( x e. RR /\ ( A + T ) <_ x /\ x <_ ( B + T ) ) ) )
61	46, 60	mpbid 222	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x e. RR /\ ( A + T ) <_ x /\ x <_ ( B + T ) ) )
62	61	simp2d 1074	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( A + T ) <_ x )
63	44, 48, 52, 62	lesub1dd 10643	. . . . . . . 8 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( ( A + T ) - T ) <_ ( x - T ) )
64	58, 63	eqbrtrd 4675	. . . . . . 7 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> A <_ ( x - T ) )
65	61	simp3d 1075	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x <_ ( B + T ) )
66	48, 45, 52, 65	lesub1dd 10643	. . . . . . . 8 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) <_ ( ( B + T ) - T ) )
67	13	recnd 10068	. . . . . . . . . 10 |- ( ph -> B e. CC )
68	67, 55	pncand 10393	. . . . . . . . 9 |- ( ph -> ( ( B + T ) - T ) = B )
69	68	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( ( B + T ) - T ) = B )
70	66, 69	breqtrd 4679	. . . . . . 7 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) <_ B )
71	50, 51, 53, 64, 70	eliccd 39726	. . . . . 6 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) e. ( A [,] B ) )
72	55	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> T e. CC )
73	49, 72	npcand 10396	. . . . . . 7 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( ( x - T ) + T ) = x )
74	73	eqcomd 2628	. . . . . 6 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x = ( ( x - T ) + T ) )
75		oveq1 6657	. . . . . . . 8 |- ( z = ( x - T ) -> ( z + T ) = ( ( x - T ) + T ) )
76	75	eqeq2d 2632	. . . . . . 7 |- ( z = ( x - T ) -> ( x = ( z + T ) <-> x = ( ( x - T ) + T ) ) )
77	76	rspcev 3309	. . . . . 6 |- ( ( ( x - T ) e. ( A [,] B ) /\ x = ( ( x - T ) + T ) ) -> E. z e. ( A [,] B ) x = ( z + T ) )
78	71, 74, 77	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> E. z e. ( A [,] B ) x = ( z + T ) )
79	49, 78, 3	sylanbrc 698	. . . 4 |- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } )
80	43, 79	impbida 877	. . 3 |- ( ph -> ( x e. { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } <-> x e. ( ( A + T ) [,] ( B + T ) ) ) )
81	80	eqrdv 2620	. 2 |- ( ph -> { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } = ( ( A + T ) [,] ( B + T ) ) )
82	81	eqcomd 2628	1 |- ( ph -> ( ( A + T ) [,] ( B + T ) ) = { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   + caddc 9939   <_ cle 10075   - cmin 10266   [,]cicc 12178

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

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-icc 12182

This theorem is referenced by:  itgiccshift  40196  itgperiod  40197