Theorem icoshftf1o 10393

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 10392	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( x e. ( A [,) B ) -> ( x + C ) e. ( ( A + C ) [,) ( B + C ) ) ) )
2	1	ralrimiv 2622	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A. x e. ( A [,) B ) ( x + C ) e. ( ( A + C ) [,) ( B + C ) ) )
3		readdcl 8305	. . . . . . . . 9 |- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR )
4	3	3adant2 1047	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. RR )
5		readdcl 8305	. . . . . . . . 9 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
6	5	3adant1 1046	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
7		renegcl 8587	. . . . . . . . 9 |- ( C e. RR -> -u C e. RR )
8	7	3ad2ant3 1051	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u C e. RR )
9		icoshft 10392	. . . . . . . 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 1278	. . . . . . 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 8375	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR* )
13		icossre 10356	. . . . . . . . . 10 |- ( ( ( A + C ) e. RR /\ ( B + C ) e. RR* ) -> ( ( A + C ) [,) ( B + C ) ) C_ RR )
14	4, 12, 13	syl2anc 415	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) [,) ( B + C ) ) C_ RR )
15	14	sselda 3248	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> y e. RR )
16	15	recnd 8354	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> y e. CC )
17		simpl3 1033	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> C e. RR )
18	17	recnd 8354	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> C e. CC )
19	16, 18	negsubd 8643	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( y + -u C ) = ( y - C ) )
20	4	recnd 8354	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. CC )
21		simp3 1030	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
22	21	recnd 8354	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
23	20, 22	negsubd 8643	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) + -u C ) = ( ( A + C ) - C ) )
24		simp1 1028	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
25	24	recnd 8354	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
26	25, 22	pncand 8638	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) - C ) = A )
27	23, 26	eqtrd 2271	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) + -u C ) = A )
28	6	recnd 8354	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. CC )
29	28, 22	negsubd 8643	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) + -u C ) = ( ( B + C ) - C ) )
30		simp2 1029	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
31	30	recnd 8354	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
32	31, 22	pncand 8638	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) - C ) = B )
33	29, 32	eqtrd 2271	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) + -u C ) = B )
34	27, 33	oveq12d 6103	. . . . . . 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 2321	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( y - C ) e. ( A [,) B ) )
37		reueq 3025	. . . . 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 8354	. . . . . . 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 1069	. . . . . . . 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 8354	. . . . . . 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 1031	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> A e. RR )
44		simpl2 1032	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> B e. RR )
45	44	rexrd 8375	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> B e. RR* )
46		icossre 10356	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) C_ RR )
47	43, 45, 46	syl2anc 415	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( A [,) B ) C_ RR )
48	47	sselda 3248	. . . . . . . 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 8354	. . . . . . 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 8656	. . . . . 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 2240	. . . . . 6 |- ( x = ( y - C ) <-> ( y - C ) = x )
52		eqcom 2240	. . . . . 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 2736	. . . 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 2623	. 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 5859	. 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 421	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> F : ( A [,) B ) -1-1-onto-> ( ( A + C ) [,) ( B + C ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E!wreu 2530   C_ wss 3220   |-> cmpt 4192   -1-1-onto->wf1o 5376  (class class class)co 6085   RRcr 8178   + caddc 8182   RR*cxr 8359   - cmin 8497   -ucneg 8498   [,)cico 10292

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-ico 10296

This theorem is used by: (None)