Theorem icoshftf1o 9013

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 9012	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( x e. ( A [,) B ) -> ( x + C ) e. ( ( A + C ) [,) ( B + C ) ) ) )
2	1	ralrimiv 2433	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A. x e. ( A [,) B ) ( x + C ) e. ( ( A + C ) [,) ( B + C ) ) )
3		readdcl 7099	. . . . . . . . 9 |- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR )
4	3	3adant2 957	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. RR )
5		readdcl 7099	. . . . . . . . 9 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
6	5	3adant1 956	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
7		renegcl 7369	. . . . . . . . 9 |- ( C e. RR -> -u C e. RR )
8	7	3ad2ant3 961	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u C e. RR )
9		icoshft 9012	. . . . . . . 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 1169	. . . . . . 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 122	. . . . . 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 7168	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR* )
13		icossre 8977	. . . . . . . . . 10 |- ( ( ( A + C ) e. RR /\ ( B + C ) e. RR* ) -> ( ( A + C ) [,) ( B + C ) ) C_ RR )
14	4, 12, 13	syl2anc 403	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) [,) ( B + C ) ) C_ RR )
15	14	sselda 2999	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> y e. RR )
16	15	recnd 7147	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> y e. CC )
17		simpl3 943	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> C e. RR )
18	17	recnd 7147	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> C e. CC )
19	16, 18	negsubd 7425	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( y + -u C ) = ( y - C ) )
20	4	recnd 7147	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. CC )
21		simp3 940	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
22	21	recnd 7147	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
23	20, 22	negsubd 7425	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) + -u C ) = ( ( A + C ) - C ) )
24		simp1 938	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
25	24	recnd 7147	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
26	25, 22	pncand 7420	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) - C ) = A )
27	23, 26	eqtrd 2113	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) + -u C ) = A )
28	6	recnd 7147	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. CC )
29	28, 22	negsubd 7425	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) + -u C ) = ( ( B + C ) - C ) )
30		simp2 939	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
31	30	recnd 7147	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
32	31, 22	pncand 7420	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) - C ) = B )
33	29, 32	eqtrd 2113	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) + -u C ) = B )
34	27, 33	oveq12d 5550	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( A + C ) + -u C ) [,) ( ( B + C ) + -u C ) ) = ( A [,) B ) )
35	34	adantr 270	. . . . . 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 2161	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( y - C ) e. ( A [,) B ) )
37		reueq 2789	. . . . 5 |- ( ( y - C ) e. ( A [,) B ) <-> E! x e. ( A [,) B ) x = ( y - C ) )
38	36, 37	sylib 120	. . . 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 270	. . . . . . . 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 7147	. . . . . . 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 979	. . . . . . . 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 7147	. . . . . . 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 941	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> A e. RR )
44		simpl2 942	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> B e. RR )
45	44	rexrd 7168	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> B e. RR* )
46		icossre 8977	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) C_ RR )
47	43, 45, 46	syl2anc 403	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( A [,) B ) C_ RR )
48	47	sselda 2999	. . . . . . . 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 7147	. . . . . . 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 7438	. . . . . 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 2083	. . . . . 6 |- ( x = ( y - C ) <-> ( y - C ) = x )
52		eqcom 2083	. . . . . 6 |- ( y = ( x + C ) <-> ( x + C ) = y )
53	50, 51, 52	3bitr4g 221	. . . . 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 2536	. . . 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 145	. . 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 2434	. 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 5341	. 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 408	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 102   /\ w3a 919   = wceq 1284   e. wcel 1433   A.wral 2348   E!wreu 2350   C_ wss 2973   |-> cmpt 3839   -1-1-onto->wf1o 4921  (class class class)co 5532   RRcr 6980   + caddc 6984   RR*cxr 7152   - cmin 7279   -ucneg 7280   [,)cico 8913

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-ico 8917

This theorem is referenced by: (None)