Theorem icoshft 6341

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		elico2t 6323	. . . . 5 |- ((A e. RR /\ B e. RR) -> (X e. (A[,)B) <-> (X e. RR /\ A <_ X /\ X < B)))
2	1	biimpd 153	. . . 4 |- ((A e. RR /\ B e. RR) -> (X e. (A[,)B) -> (X e. RR /\ A <_ X /\ X < B)))
3	2	3adant3 797	. . 3 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (X e. (A[,)B) -> (X e. RR /\ A <_ X /\ X < B)))
4		3anass 777	. . 3 |- ((X e. RR /\ A <_ X /\ X < B) <-> (X e. RR /\ (A <_ X /\ X < B)))
5	3, 4	syl6ib 212	. 2 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (X e. (A[,)B) -> (X e. RR /\ (A <_ X /\ X < B))))
6		leadd1t 5599	. . . . . . . . . 10 |- ((A e. RR /\ X e. RR /\ C e. RR) -> (A <_ X <-> (A + C) <_ (X + C)))
7	6	3com12 835	. . . . . . . . 9 |- ((X e. RR /\ A e. RR /\ C e. RR) -> (A <_ X <-> (A + C) <_ (X + C)))
8	7	3expib 834	. . . . . . . 8 |- (X e. RR -> ((A e. RR /\ C e. RR) -> (A <_ X <-> (A + C) <_ (X + C))))
9	8	com12 11	. . . . . . 7 |- ((A e. RR /\ C e. RR) -> (X e. RR -> (A <_ X <-> (A + C) <_ (X + C))))
10	9	3adant2 796	. . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (X e. RR -> (A <_ X <-> (A + C) <_ (X + C))))
11	10	imp 350	. . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ X e. RR) -> (A <_ X <-> (A + C) <_ (X + C)))
12		ltadd1t 5597	. . . . . . . . 9 |- ((X e. RR /\ B e. RR /\ C e. RR) -> (X < B <-> (X + C) < (B + C)))
13	12	3expib 834	. . . . . . . 8 |- (X e. RR -> ((B e. RR /\ C e. RR) -> (X < B <-> (X + C) < (B + C))))
14	13	com12 11	. . . . . . 7 |- ((B e. RR /\ C e. RR) -> (X e. RR -> (X < B <-> (X + C) < (B + C))))
15	14	3adant1 795	. . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (X e. RR -> (X < B <-> (X + C) < (B + C))))
16	15	imp 350	. . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ X e. RR) -> (X < B <-> (X + C) < (B + C)))
17	11, 16	anbi12d 626	. . . 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 647	. . 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		axaddrcl 5244	. . . . . . . 8 |- ((X e. RR /\ C e. RR) -> (X + C) e. RR)
20	19	expcom 374	. . . . . . 7 |- (C e. RR -> (X e. RR -> (X + C) e. RR))
21	20	anim1d 558	. . . . . 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 777	. . . . . 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 213	. . . . 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 800	. . . 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		elico2t 6323	. . . . . 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))))
26	25	biimprd 154	. . . . 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))))
27		axaddrcl 5244	. . . . . 6 |- ((A e. RR /\ C e. RR) -> (A + C) e. RR)
28	27	3adant2 796	. . . . 5 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (A + C) e. RR)
29		axaddrcl 5244	. . . . . 6 |- ((B e. RR /\ C e. RR) -> (B + C) e. RR)
30	29	3adant1 795	. . . . 5 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (B + C) e. RR)
31	26, 28, 30	sylanc 471	. . . 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 27	. . 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 203	. 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 27	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 146   /\ wa 223   /\ w3a 773   e. wcel 955   class class class wbr 2609  (class class class)co 3948  RRcr 5205   + caddc 5209   <_ cle 5267   < clt 5458  [,)cico 6296

This theorem is referenced by:  icoshftf1oi 6342  shftefif1olem 8661  shftefif1olemOLD 8662

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-ico 6300

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