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Mirrors > Home > ILE Home > Th. List > iccf1o | Unicode version |
Description: Describe a bijection from to an arbitrary nontrivial closed interval . (Contributed by Mario Carneiro, 8-Sep-2015.) |
Ref | Expression |
---|---|
iccf1o.1 |
Ref | Expression |
---|---|
iccf1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccf1o.1 | . 2 | |
2 | 0re 7759 | . . . . . . . . 9 | |
3 | 1re 7758 | . . . . . . . . 9 | |
4 | 2, 3 | elicc2i 9715 | . . . . . . . 8 |
5 | 4 | simp1bi 996 | . . . . . . 7 |
6 | 5 | adantl 275 | . . . . . 6 |
7 | 6 | recnd 7787 | . . . . 5 |
8 | simpl2 985 | . . . . . 6 | |
9 | 8 | recnd 7787 | . . . . 5 |
10 | 7, 9 | mulcld 7779 | . . . 4 |
11 | ax-1cn 7706 | . . . . . 6 | |
12 | subcl 7954 | . . . . . 6 | |
13 | 11, 7, 12 | sylancr 410 | . . . . 5 |
14 | simpl1 984 | . . . . . 6 | |
15 | 14 | recnd 7787 | . . . . 5 |
16 | 13, 15 | mulcld 7779 | . . . 4 |
17 | 10, 16 | addcomd 7906 | . . 3 |
18 | lincmb01cmp 9779 | . . 3 | |
19 | 17, 18 | eqeltrd 2214 | . 2 |
20 | simpr 109 | . . . . 5 | |
21 | simpl1 984 | . . . . . 6 | |
22 | simpl2 985 | . . . . . 6 | |
23 | elicc2 9714 | . . . . . . . . 9 | |
24 | 23 | 3adant3 1001 | . . . . . . . 8 |
25 | 24 | biimpa 294 | . . . . . . 7 |
26 | 25 | simp1d 993 | . . . . . 6 |
27 | eqid 2137 | . . . . . . 7 | |
28 | eqid 2137 | . . . . . . 7 | |
29 | 27, 28 | iccshftl 9772 | . . . . . 6 |
30 | 21, 22, 26, 21, 29 | syl22anc 1217 | . . . . 5 |
31 | 20, 30 | mpbid 146 | . . . 4 |
32 | 26, 21 | resubcld 8136 | . . . . . 6 |
33 | 32 | recnd 7787 | . . . . 5 |
34 | difrp 9473 | . . . . . . . 8 | |
35 | 34 | biimp3a 1323 | . . . . . . 7 |
36 | 35 | adantr 274 | . . . . . 6 |
37 | 36 | rpcnd 9478 | . . . . 5 |
38 | rpap0 9451 | . . . . . 6 # | |
39 | 36, 38 | syl 14 | . . . . 5 # |
40 | 33, 37, 39 | divcanap1d 8544 | . . . 4 |
41 | 37 | mul02d 8147 | . . . . . 6 |
42 | 21 | recnd 7787 | . . . . . . 7 |
43 | 42 | subidd 8054 | . . . . . 6 |
44 | 41, 43 | eqtr4d 2173 | . . . . 5 |
45 | 37 | mulid2d 7777 | . . . . 5 |
46 | 44, 45 | oveq12d 5785 | . . . 4 |
47 | 31, 40, 46 | 3eltr4d 2221 | . . 3 |
48 | 0red 7760 | . . . 4 | |
49 | 1red 7774 | . . . 4 | |
50 | 32, 36 | rerpdivcld 9508 | . . . 4 |
51 | eqid 2137 | . . . . 5 | |
52 | eqid 2137 | . . . . 5 | |
53 | 51, 52 | iccdil 9774 | . . . 4 |
54 | 48, 49, 50, 36, 53 | syl22anc 1217 | . . 3 |
55 | 47, 54 | mpbird 166 | . 2 |
56 | eqcom 2139 | . . . 4 | |
57 | 33 | adantrl 469 | . . . . 5 |
58 | 7 | adantrr 470 | . . . . 5 |
59 | 37 | adantrl 469 | . . . . 5 |
60 | 39 | adantrl 469 | . . . . 5 # |
61 | 57, 58, 59, 60 | divmulap3d 8578 | . . . 4 |
62 | 56, 61 | syl5bb 191 | . . 3 |
63 | 26 | adantrl 469 | . . . . . 6 |
64 | 63 | recnd 7787 | . . . . 5 |
65 | 42 | adantrl 469 | . . . . 5 |
66 | 8, 14 | resubcld 8136 | . . . . . . . 8 |
67 | 6, 66 | remulcld 7789 | . . . . . . 7 |
68 | 67 | adantrr 470 | . . . . . 6 |
69 | 68 | recnd 7787 | . . . . 5 |
70 | 64, 65, 69 | subadd2d 8085 | . . . 4 |
71 | eqcom 2139 | . . . 4 | |
72 | 70, 71 | syl6bb 195 | . . 3 |
73 | 7, 15 | mulcld 7779 | . . . . . . 7 |
74 | 10, 73, 15 | subadd23d 8088 | . . . . . 6 |
75 | 7, 9, 15 | subdid 8169 | . . . . . . 7 |
76 | 75 | oveq1d 5782 | . . . . . 6 |
77 | 1cnd 7775 | . . . . . . . . 9 | |
78 | 77, 7, 15 | subdird 8170 | . . . . . . . 8 |
79 | 15 | mulid2d 7777 | . . . . . . . . 9 |
80 | 79 | oveq1d 5782 | . . . . . . . 8 |
81 | 78, 80 | eqtrd 2170 | . . . . . . 7 |
82 | 81 | oveq2d 5783 | . . . . . 6 |
83 | 74, 76, 82 | 3eqtr4d 2180 | . . . . 5 |
84 | 83 | adantrr 470 | . . . 4 |
85 | 84 | eqeq2d 2149 | . . 3 |
86 | 62, 72, 85 | 3bitrd 213 | . 2 |
87 | 1, 19, 55, 86 | f1ocnv2d 5967 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 class class class wbr 3924 cmpt 3984 ccnv 4533 wf1o 5117 (class class class)co 5767 cc 7611 cr 7612 cc0 7613 c1 7614 caddc 7616 cmul 7618 clt 7793 cle 7794 cmin 7926 # cap 8336 cdiv 8425 crp 9434 cicc 9667 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-rp 9435 df-icc 9671 |
This theorem is referenced by: (None) |
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