Theorem icoshftf1o 8859

Description: Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Hypothesis

Ref	Expression
icoshftf1o.1	|- F = ( x e. ( A [,) B ) |-> ( x + C ) )

Assertion

Ref	Expression
icoshftf1o	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> F : ( A [,) B ) -1-1-onto-> ( ( A + C ) [,) ( B + C ) ) )

Distinct variable groups:   x, A   x, B   x, C

Allowed substitution hint:   F( x)

Proof of Theorem icoshftf1o

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		icoshft 8858	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( x e. ( A [,) B ) -> ( x + C ) e. ( ( A + C ) [,) ( B + C ) ) ) )
2	1	ralrimiv 2391	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A. x e. ( A [,) B ) ( x + C ) e. ( ( A + C ) [,) ( B + C ) ) )
3		readdcl 7007	. . . . . . . . 9 |- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR )
4	3	3adant2 923	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. RR )
5		readdcl 7007	. . . . . . . . 9 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
6	5	3adant1 922	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
7		renegcl 7272	. . . . . . . . 9 |- ( C e. RR -> -u C e. RR )
8	7	3ad2ant3 927	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u C e. RR )
9		icoshft 8858	. . . . . . . 8 |- ( ( ( A + C ) e. RR /\ ( B + C ) e. RR /\ -u C e. RR ) -> ( y e. ( ( A + C ) [,) ( B + C ) ) -> ( y + -u C ) e. ( ( ( A + C ) + -u C ) [,) ( ( B + C ) + -u C ) ) ) )
10	4, 6, 8, 9	syl3anc 1135	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( y e. ( ( A + C ) [,) ( B + C ) ) -> ( y + -u C ) e. ( ( ( A + C ) + -u C ) [,) ( ( B + C ) + -u C ) ) ) )
11	10	imp 115	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( y + -u C ) e. ( ( ( A + C ) + -u C ) [,) ( ( B + C ) + -u C ) ) )
12	6	rexrd 7075	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR* )
13		icossre 8823	. . . . . . . . . 10 |- ( ( ( A + C ) e. RR /\ ( B + C ) e. RR* ) -> ( ( A + C ) [,) ( B + C ) ) C_ RR )
14	4, 12, 13	syl2anc 391	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) [,) ( B + C ) ) C_ RR )
15	14	sselda 2945	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> y e. RR )
16	15	recnd 7054	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> y e. CC )
17		simpl3 909	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> C e. RR )
18	17	recnd 7054	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> C e. CC )
19	16, 18	negsubd 7328	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( y + -u C ) = ( y - C ) )
20	4	recnd 7054	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. CC )
21		simp3 906	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
22	21	recnd 7054	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
23	20, 22	negsubd 7328	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) + -u C ) = ( ( A + C ) - C ) )
24		simp1 904	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
25	24	recnd 7054	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
26	25, 22	pncand 7323	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) - C ) = A )
27	23, 26	eqtrd 2072	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) + -u C ) = A )
28	6	recnd 7054	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. CC )
29	28, 22	negsubd 7328	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) + -u C ) = ( ( B + C ) - C ) )
30		simp2 905	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
31	30	recnd 7054	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
32	31, 22	pncand 7323	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) - C ) = B )
33	29, 32	eqtrd 2072	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) + -u C ) = B )
34	27, 33	oveq12d 5530	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( A + C ) + -u C ) [,) ( ( B + C ) + -u C ) ) = ( A [,) B ) )
35	34	adantr 261	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( ( ( A + C ) + -u C ) [,) ( ( B + C ) + -u C ) ) = ( A [,) B ) )
36	11, 19, 35	3eltr3d 2120	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( y - C ) e. ( A [,) B ) )
37		reueq 2738	. . . . 5 |- ( ( y - C ) e. ( A [,) B ) <-> E! x e. ( A [,) B ) x = ( y - C ) )
38	36, 37	sylib 127	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> E! x e. ( A [,) B ) x = ( y - C ) )
39	15	adantr 261	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> y e. RR )
40	39	recnd 7054	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> y e. CC )
41		simpll3 945	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> C e. RR )
42	41	recnd 7054	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> C e. CC )
43		simpl1 907	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> A e. RR )
44		simpl2 908	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> B e. RR )
45	44	rexrd 7075	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> B e. RR* )
46		icossre 8823	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) C_ RR )
47	43, 45, 46	syl2anc 391	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( A [,) B ) C_ RR )
48	47	sselda 2945	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> x e. RR )
49	48	recnd 7054	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> x e. CC )
50	40, 42, 49	subadd2d 7341	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> ( ( y - C ) = x <-> ( x + C ) = y ) )
51		eqcom 2042	. . . . . 6 |- ( x = ( y - C ) <-> ( y - C ) = x )
52		eqcom 2042	. . . . . 6 |- ( y = ( x + C ) <-> ( x + C ) = y )
53	50, 51, 52	3bitr4g 212	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> ( x = ( y - C ) <-> y = ( x + C ) ) )
54	53	reubidva 2492	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( E! x e. ( A [,) B ) x = ( y - C ) <-> E! x e. ( A [,) B ) y = ( x + C ) ) )
55	38, 54	mpbid 135	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> E! x e. ( A [,) B ) y = ( x + C ) )
56	55	ralrimiva 2392	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A. y e. ( ( A + C ) [,) ( B + C ) ) E! x e. ( A [,) B ) y = ( x + C ) )
57		icoshftf1o.1	. . 3 |- F = ( x e. ( A [,) B ) |-> ( x + C ) )
58	57	f1ompt 5320	. 2 |- ( F : ( A [,) B ) -1-1-onto-> ( ( A + C ) [,) ( B + C ) ) <-> ( A. x e. ( A [,) B ) ( x + C ) e. ( ( A + C ) [,) ( B + C ) ) /\ A. y e. ( ( A + C ) [,) ( B + C ) ) E! x e. ( A [,) B ) y = ( x + C ) ) )
59	2, 56, 58	sylanbrc 394	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> F : ( A [,) B ) -1-1-onto-> ( ( A + C ) [,) ( B + C ) ) )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   e. wcel 1393   A.wral 2306   E!wreu 2308   C_ wss 2917   |-> cmpt 3818   -1-1-onto->wf1o 4901  (class class class)co 5512   RRcr 6888   + caddc 6892   RR*cxr 7059   - cmin 7182   -ucneg 7183   [,)cico 8759

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-po 4033  df-iso 4034  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-ico 8763

This theorem is referenced by: (None)