Theorem icoshftf1o 10225

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 10224	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( x e. ( A [,) B ) -> ( x + C ) e. ( ( A + C ) [,) ( B + C ) ) ) )
2	1	ralrimiv 2604	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A. x e. ( A [,) B ) ( x + C ) e. ( ( A + C ) [,) ( B + C ) ) )
3		readdcl 8157	. . . . . . . . 9 |- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR )
4	3	3adant2 1042	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. RR )
5		readdcl 8157	. . . . . . . . 9 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
6	5	3adant1 1041	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
7		renegcl 8439	. . . . . . . . 9 |- ( C e. RR -> -u C e. RR )
8	7	3ad2ant3 1046	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u C e. RR )
9		icoshft 10224	. . . . . . . 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 1273	. . . . . . 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 124	. . . . . 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 8228	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR* )
13		icossre 10188	. . . . . . . . . 10 |- ( ( ( A + C ) e. RR /\ ( B + C ) e. RR* ) -> ( ( A + C ) [,) ( B + C ) ) C_ RR )
14	4, 12, 13	syl2anc 411	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) [,) ( B + C ) ) C_ RR )
15	14	sselda 3227	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> y e. RR )
16	15	recnd 8207	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> y e. CC )
17		simpl3 1028	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> C e. RR )
18	17	recnd 8207	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> C e. CC )
19	16, 18	negsubd 8495	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( y + -u C ) = ( y - C ) )
20	4	recnd 8207	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. CC )
21		simp3 1025	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
22	21	recnd 8207	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
23	20, 22	negsubd 8495	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) + -u C ) = ( ( A + C ) - C ) )
24		simp1 1023	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
25	24	recnd 8207	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
26	25, 22	pncand 8490	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) - C ) = A )
27	23, 26	eqtrd 2264	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) + -u C ) = A )
28	6	recnd 8207	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. CC )
29	28, 22	negsubd 8495	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) + -u C ) = ( ( B + C ) - C ) )
30		simp2 1024	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
31	30	recnd 8207	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
32	31, 22	pncand 8490	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) - C ) = B )
33	29, 32	eqtrd 2264	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) + -u C ) = B )
34	27, 33	oveq12d 6035	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( A + C ) + -u C ) [,) ( ( B + C ) + -u C ) ) = ( A [,) B ) )
35	34	adantr 276	. . . . . 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 2314	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( y - C ) e. ( A [,) B ) )
37		reueq 3005	. . . . 5 |- ( ( y - C ) e. ( A [,) B ) <-> E! x e. ( A [,) B ) x = ( y - C ) )
38	36, 37	sylib 122	. . . 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 276	. . . . . . . 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 8207	. . . . . . 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 1064	. . . . . . . 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 8207	. . . . . . 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 1026	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> A e. RR )
44		simpl2 1027	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> B e. RR )
45	44	rexrd 8228	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> B e. RR* )
46		icossre 10188	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) C_ RR )
47	43, 45, 46	syl2anc 411	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( A [,) B ) C_ RR )
48	47	sselda 3227	. . . . . . . 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 8207	. . . . . . 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 8508	. . . . . 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 2233	. . . . . 6 |- ( x = ( y - C ) <-> ( y - C ) = x )
52		eqcom 2233	. . . . . 6 |- ( y = ( x + C ) <-> ( x + C ) = y )
53	50, 51, 52	3bitr4g 223	. . . . 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 2717	. . . 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 147	. . 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 2605	. 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 5798	. 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 417	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 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   E!wreu 2512   C_ wss 3200   |-> cmpt 4150   -1-1-onto->wf1o 5325  (class class class)co 6017   RRcr 8030   + caddc 8034   RR*cxr 8212   - cmin 8349   -ucneg 8350   [,)cico 10124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-ico 10128

This theorem is referenced by: (None)