Theorem icoshft 7577

Description: A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)

Assertion

Ref	Expression
icoshft	|- ((A e. RR /\ B e. RR /\ C e. RR) -> (X e. (A[,)B) -> (X + C) e. ((A + C)[,)(B + C))))

Proof of Theorem icoshft

Step	Hyp	Ref	Expression
1		elico2 7559	. . . . 5 |- ((A e. RR /\ B e. RR) -> (X e. (A[,)B) <-> (X e. RR /\ A <_ X /\ X < B)))
2	1	biimpd 170	. . . 4 |- ((A e. RR /\ B e. RR) -> (X e. (A[,)B) -> (X e. RR /\ A <_ X /\ X < B)))
3	2	3adant3 896	. . 3 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (X e. (A[,)B) -> (X e. RR /\ A <_ X /\ X < B)))
4		3anass 862	. . 3 |- ((X e. RR /\ A <_ X /\ X < B) <-> (X e. RR /\ (A <_ X /\ X < B)))
5	3, 4	syl6ib 229	. 2 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (X e. (A[,)B) -> (X e. RR /\ (A <_ X /\ X < B))))
6		leadd1 6808	. . . . . . . . . 10 |- ((A e. RR /\ X e. RR /\ C e. RR) -> (A <_ X <-> (A + C) <_ (X + C)))
7	6	3com12 1071	. . . . . . . . 9 |- ((X e. RR /\ A e. RR /\ C e. RR) -> (A <_ X <-> (A + C) <_ (X + C)))
8	7	3expib 1070	. . . . . . . 8 |- (X e. RR -> ((A e. RR /\ C e. RR) -> (A <_ X <-> (A + C) <_ (X + C))))
9	8	com12 14	. . . . . . 7 |- ((A e. RR /\ C e. RR) -> (X e. RR -> (A <_ X <-> (A + C) <_ (X + C))))
10	9	3adant2 895	. . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (X e. RR -> (A <_ X <-> (A + C) <_ (X + C))))
11	10	imp 377	. . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ X e. RR) -> (A <_ X <-> (A + C) <_ (X + C)))
12		ltadd1 6806	. . . . . . . . 9 |- ((X e. RR /\ B e. RR /\ C e. RR) -> (X < B <-> (X + C) < (B + C)))
13	12	3expib 1070	. . . . . . . 8 |- (X e. RR -> ((B e. RR /\ C e. RR) -> (X < B <-> (X + C) < (B + C))))
14	13	com12 14	. . . . . . 7 |- ((B e. RR /\ C e. RR) -> (X e. RR -> (X < B <-> (X + C) < (B + C))))
15	14	3adant1 894	. . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (X e. RR -> (X < B <-> (X + C) < (B + C))))
16	15	imp 377	. . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ X e. RR) -> (X < B <-> (X + C) < (B + C)))
17	11, 16	anbi12d 690	. . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ X e. RR) -> ((A <_ X /\ X < B) <-> ((A + C) <_ (X + C) /\ (X + C) < (B + C))))
18	17	pm5.32da 711	. . 3 |- ((A e. RR /\ B e. RR /\ C e. RR) -> ((X e. RR /\ (A <_ X /\ X < B)) <-> (X e. RR /\ ((A + C) <_ (X + C) /\ (X + C) < (B + C)))))
19		readdcl 6455	. . . . . . . 8 |- ((X e. RR /\ C e. RR) -> (X + C) e. RR)
20	19	expcom 403	. . . . . . 7 |- (C e. RR -> (X e. RR -> (X + C) e. RR))
21	20	anim1d 619	. . . . . 6 |- (C e. RR -> ((X e. RR /\ ((A + C) <_ (X + C) /\ (X + C) < (B + C))) -> ((X + C) e. RR /\ ((A + C) <_ (X + C) /\ (X + C) < (B + C)))))
22		3anass 862	. . . . . 6 |- (((X + C) e. RR /\ (A + C) <_ (X + C) /\ (X + C) < (B + C)) <-> ((X + C) e. RR /\ ((A + C) <_ (X + C) /\ (X + C) < (B + C))))
23	21, 22	syl6ibr 230	. . . . 5 |- (C e. RR -> ((X e. RR /\ ((A + C) <_ (X + C) /\ (X + C) < (B + C))) -> ((X + C) e. RR /\ (A + C) <_ (X + C) /\ (X + C) < (B + C))))
24	23	3ad2ant3 899	. . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR) -> ((X e. RR /\ ((A + C) <_ (X + C) /\ (X + C) < (B + C))) -> ((X + C) e. RR /\ (A + C) <_ (X + C) /\ (X + C) < (B + C))))
25		readdcl 6455	. . . . . 6 |- ((A e. RR /\ C e. RR) -> (A + C) e. RR)
26	25	3adant2 895	. . . . 5 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (A + C) e. RR)
27		readdcl 6455	. . . . . 6 |- ((B e. RR /\ C e. RR) -> (B + C) e. RR)
28	27	3adant1 894	. . . . 5 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (B + C) e. RR)
29		elico2 7559	. . . . . 6 |- (((A + C) e. RR /\ (B + C) e. RR) -> ((X + C) e. ((A + C)[,)(B + C)) <-> ((X + C) e. RR /\ (A + C) <_ (X + C) /\ (X + C) < (B + C))))
30	29	biimprd 171	. . . . 5 |- (((A + C) e. RR /\ (B + C) e. RR) -> (((X + C) e. RR /\ (A + C) <_ (X + C) /\ (X + C) < (B + C)) -> (X + C) e. ((A + C)[,)(B + C))))
31	26, 28, 30	syl11anc 524	. . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (((X + C) e. RR /\ (A + C) <_ (X + C) /\ (X + C) < (B + C)) -> (X + C) e. ((A + C)[,)(B + C))))
32	24, 31	syld 30	. . 3 |- ((A e. RR /\ B e. RR /\ C e. RR) -> ((X e. RR /\ ((A + C) <_ (X + C) /\ (X + C) < (B + C))) -> (X + C) e. ((A + C)[,)(B + C))))
33	18, 32	sylbid 220	. 2 |- ((A e. RR /\ B e. RR /\ C e. RR) -> ((X e. RR /\ (A <_ X /\ X < B)) -> (X + C) e. ((A + C)[,)(B + C))))
34	5, 33	syld 30	1 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (X e. (A[,)B) -> (X + C) e. ((A + C)[,)(B + C))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   e. wcel 1300   class class class wbr 3338  (class class class)co 4884  RRcr 6385   + caddc 6389   <_ cle 6448   < clt 6653  [,)cico 7526

This theorem is referenced by:  icoshftf1oii 7578  shftefif1olem 10095

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  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-inf2 5731

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-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-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  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-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-ico 7530

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