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Mirrors > Home > ILE Home > Th. List > icoshftf1o | Unicode version |
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.) |
Ref | Expression |
---|---|
icoshftf1o.1 |
Ref | Expression |
---|---|
icoshftf1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icoshft 9895 | . . 3 | |
2 | 1 | ralrimiv 2529 | . 2 |
3 | readdcl 7859 | . . . . . . . . 9 | |
4 | 3 | 3adant2 1001 | . . . . . . . 8 |
5 | readdcl 7859 | . . . . . . . . 9 | |
6 | 5 | 3adant1 1000 | . . . . . . . 8 |
7 | renegcl 8137 | . . . . . . . . 9 | |
8 | 7 | 3ad2ant3 1005 | . . . . . . . 8 |
9 | icoshft 9895 | . . . . . . . 8 | |
10 | 4, 6, 8, 9 | syl3anc 1220 | . . . . . . 7 |
11 | 10 | imp 123 | . . . . . 6 |
12 | 6 | rexrd 7928 | . . . . . . . . . 10 |
13 | icossre 9859 | . . . . . . . . . 10 | |
14 | 4, 12, 13 | syl2anc 409 | . . . . . . . . 9 |
15 | 14 | sselda 3128 | . . . . . . . 8 |
16 | 15 | recnd 7907 | . . . . . . 7 |
17 | simpl3 987 | . . . . . . . 8 | |
18 | 17 | recnd 7907 | . . . . . . 7 |
19 | 16, 18 | negsubd 8193 | . . . . . 6 |
20 | 4 | recnd 7907 | . . . . . . . . . 10 |
21 | simp3 984 | . . . . . . . . . . 11 | |
22 | 21 | recnd 7907 | . . . . . . . . . 10 |
23 | 20, 22 | negsubd 8193 | . . . . . . . . 9 |
24 | simp1 982 | . . . . . . . . . . 11 | |
25 | 24 | recnd 7907 | . . . . . . . . . 10 |
26 | 25, 22 | pncand 8188 | . . . . . . . . 9 |
27 | 23, 26 | eqtrd 2190 | . . . . . . . 8 |
28 | 6 | recnd 7907 | . . . . . . . . . 10 |
29 | 28, 22 | negsubd 8193 | . . . . . . . . 9 |
30 | simp2 983 | . . . . . . . . . . 11 | |
31 | 30 | recnd 7907 | . . . . . . . . . 10 |
32 | 31, 22 | pncand 8188 | . . . . . . . . 9 |
33 | 29, 32 | eqtrd 2190 | . . . . . . . 8 |
34 | 27, 33 | oveq12d 5843 | . . . . . . 7 |
35 | 34 | adantr 274 | . . . . . 6 |
36 | 11, 19, 35 | 3eltr3d 2240 | . . . . 5 |
37 | reueq 2911 | . . . . 5 | |
38 | 36, 37 | sylib 121 | . . . 4 |
39 | 15 | adantr 274 | . . . . . . . 8 |
40 | 39 | recnd 7907 | . . . . . . 7 |
41 | simpll3 1023 | . . . . . . . 8 | |
42 | 41 | recnd 7907 | . . . . . . 7 |
43 | simpl1 985 | . . . . . . . . . 10 | |
44 | simpl2 986 | . . . . . . . . . . 11 | |
45 | 44 | rexrd 7928 | . . . . . . . . . 10 |
46 | icossre 9859 | . . . . . . . . . 10 | |
47 | 43, 45, 46 | syl2anc 409 | . . . . . . . . 9 |
48 | 47 | sselda 3128 | . . . . . . . 8 |
49 | 48 | recnd 7907 | . . . . . . 7 |
50 | 40, 42, 49 | subadd2d 8206 | . . . . . 6 |
51 | eqcom 2159 | . . . . . 6 | |
52 | eqcom 2159 | . . . . . 6 | |
53 | 50, 51, 52 | 3bitr4g 222 | . . . . 5 |
54 | 53 | reubidva 2639 | . . . 4 |
55 | 38, 54 | mpbid 146 | . . 3 |
56 | 55 | ralrimiva 2530 | . 2 |
57 | icoshftf1o.1 | . . 3 | |
58 | 57 | f1ompt 5619 | . 2 |
59 | 2, 56, 58 | sylanbrc 414 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wceq 1335 wcel 2128 wral 2435 wreu 2437 wss 3102 cmpt 4026 wf1o 5170 (class class class)co 5825 cr 7732 caddc 7736 cxr 7912 cmin 8047 cneg 8048 cico 9795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-addass 7835 ax-distr 7837 ax-i2m1 7838 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-ltadd 7849 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-po 4257 df-iso 4258 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-ico 9799 |
This theorem is referenced by: (None) |
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