Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7766 | . . . . . . . . 9 | |
3 | 1re 7765 | . . . . . . . . 9 | |
4 | 2, 3 | elicc2i 9722 | . . . . . . . 8 |
5 | 4 | simp1bi 996 | . . . . . . 7 |
6 | 5 | adantl 275 | . . . . . 6 |
7 | 6 | recnd 7794 | . . . . 5 |
8 | simpl2 985 | . . . . . 6 | |
9 | 8 | recnd 7794 | . . . . 5 |
10 | 7, 9 | mulcld 7786 | . . . 4 |
11 | ax-1cn 7713 | . . . . . 6 | |
12 | subcl 7961 | . . . . . 6 | |
13 | 11, 7, 12 | sylancr 410 | . . . . 5 |
14 | simpl1 984 | . . . . . 6 | |
15 | 14 | recnd 7794 | . . . . 5 |
16 | 13, 15 | mulcld 7786 | . . . 4 |
17 | 10, 16 | addcomd 7913 | . . 3 |
18 | lincmb01cmp 9786 | . . 3 | |
19 | 17, 18 | eqeltrd 2216 | . 2 |
20 | simpr 109 | . . . . 5 | |
21 | simpl1 984 | . . . . . 6 | |
22 | simpl2 985 | . . . . . 6 | |
23 | elicc2 9721 | . . . . . . . . 9 | |
24 | 23 | 3adant3 1001 | . . . . . . . 8 |
25 | 24 | biimpa 294 | . . . . . . 7 |
26 | 25 | simp1d 993 | . . . . . 6 |
27 | eqid 2139 | . . . . . . 7 | |
28 | eqid 2139 | . . . . . . 7 | |
29 | 27, 28 | iccshftl 9779 | . . . . . 6 |
30 | 21, 22, 26, 21, 29 | syl22anc 1217 | . . . . 5 |
31 | 20, 30 | mpbid 146 | . . . 4 |
32 | 26, 21 | resubcld 8143 | . . . . . 6 |
33 | 32 | recnd 7794 | . . . . 5 |
34 | difrp 9480 | . . . . . . . 8 | |
35 | 34 | biimp3a 1323 | . . . . . . 7 |
36 | 35 | adantr 274 | . . . . . 6 |
37 | 36 | rpcnd 9485 | . . . . 5 |
38 | rpap0 9458 | . . . . . 6 # | |
39 | 36, 38 | syl 14 | . . . . 5 # |
40 | 33, 37, 39 | divcanap1d 8551 | . . . 4 |
41 | 37 | mul02d 8154 | . . . . . 6 |
42 | 21 | recnd 7794 | . . . . . . 7 |
43 | 42 | subidd 8061 | . . . . . 6 |
44 | 41, 43 | eqtr4d 2175 | . . . . 5 |
45 | 37 | mulid2d 7784 | . . . . 5 |
46 | 44, 45 | oveq12d 5792 | . . . 4 |
47 | 31, 40, 46 | 3eltr4d 2223 | . . 3 |
48 | 0red 7767 | . . . 4 | |
49 | 1red 7781 | . . . 4 | |
50 | 32, 36 | rerpdivcld 9515 | . . . 4 |
51 | eqid 2139 | . . . . 5 | |
52 | eqid 2139 | . . . . 5 | |
53 | 51, 52 | iccdil 9781 | . . . 4 |
54 | 48, 49, 50, 36, 53 | syl22anc 1217 | . . 3 |
55 | 47, 54 | mpbird 166 | . 2 |
56 | eqcom 2141 | . . . 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 8585 | . . . 4 |
62 | 56, 61 | syl5bb 191 | . . 3 |
63 | 26 | adantrl 469 | . . . . . 6 |
64 | 63 | recnd 7794 | . . . . 5 |
65 | 42 | adantrl 469 | . . . . 5 |
66 | 8, 14 | resubcld 8143 | . . . . . . . 8 |
67 | 6, 66 | remulcld 7796 | . . . . . . 7 |
68 | 67 | adantrr 470 | . . . . . 6 |
69 | 68 | recnd 7794 | . . . . 5 |
70 | 64, 65, 69 | subadd2d 8092 | . . . 4 |
71 | eqcom 2141 | . . . 4 | |
72 | 70, 71 | syl6bb 195 | . . 3 |
73 | 7, 15 | mulcld 7786 | . . . . . . 7 |
74 | 10, 73, 15 | subadd23d 8095 | . . . . . 6 |
75 | 7, 9, 15 | subdid 8176 | . . . . . . 7 |
76 | 75 | oveq1d 5789 | . . . . . 6 |
77 | 1cnd 7782 | . . . . . . . . 9 | |
78 | 77, 7, 15 | subdird 8177 | . . . . . . . 8 |
79 | 15 | mulid2d 7784 | . . . . . . . . 9 |
80 | 79 | oveq1d 5789 | . . . . . . . 8 |
81 | 78, 80 | eqtrd 2172 | . . . . . . 7 |
82 | 81 | oveq2d 5790 | . . . . . 6 |
83 | 74, 76, 82 | 3eqtr4d 2182 | . . . . 5 |
84 | 83 | adantrr 470 | . . . 4 |
85 | 84 | eqeq2d 2151 | . . 3 |
86 | 62, 72, 85 | 3bitrd 213 | . 2 |
87 | 1, 19, 55, 86 | f1ocnv2d 5974 | 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 3929 cmpt 3989 ccnv 4538 wf1o 5122 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 caddc 7623 cmul 7625 clt 7800 cle 7801 cmin 7933 # cap 8343 cdiv 8432 crp 9441 cicc 9674 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-rp 9442 df-icc 9678 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |