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Mirrors > Home > MPE Home > Th. List > axlowdimlem15 | Structured version Visualization version Unicode version |
Description: Lemma for axlowdim 25841. Set up a one-to-one function of points. (Contributed by Scott Fenton, 21-Apr-2013.) |
Ref | Expression |
---|---|
axlowdimlem15.1 |
Ref | Expression |
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axlowdimlem15 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . . 6 | |
2 | 1 | axlowdimlem7 25828 | . . . . 5 |
3 | 2 | adantr 481 | . . . 4 |
4 | eluzge3nn 11730 | . . . . 5 | |
5 | eqid 2622 | . . . . . 6 | |
6 | 5 | axlowdimlem10 25831 | . . . . 5 |
7 | 4, 6 | sylan 488 | . . . 4 |
8 | 3, 7 | ifcld 4131 | . . 3 |
9 | axlowdimlem15.1 | . . 3 | |
10 | 8, 9 | fmptd 6385 | . 2 |
11 | eqeq1 2626 | . . . . . . . 8 | |
12 | oveq1 6657 | . . . . . . . . . . 11 | |
13 | 12 | opeq1d 4408 | . . . . . . . . . 10 |
14 | 13 | sneqd 4189 | . . . . . . . . 9 |
15 | 12 | sneqd 4189 | . . . . . . . . . . 11 |
16 | 15 | difeq2d 3728 | . . . . . . . . . 10 |
17 | 16 | xpeq1d 5138 | . . . . . . . . 9 |
18 | 14, 17 | uneq12d 3768 | . . . . . . . 8 |
19 | 11, 18 | ifbieq2d 4111 | . . . . . . 7 |
20 | snex 4908 | . . . . . . . . 9 | |
21 | ovex 6678 | . . . . . . . . . . 11 | |
22 | difexg 4808 | . . . . . . . . . . 11 | |
23 | 21, 22 | ax-mp 5 | . . . . . . . . . 10 |
24 | snex 4908 | . . . . . . . . . 10 | |
25 | 23, 24 | xpex 6962 | . . . . . . . . 9 |
26 | 20, 25 | unex 6956 | . . . . . . . 8 |
27 | snex 4908 | . . . . . . . . 9 | |
28 | difexg 4808 | . . . . . . . . . . 11 | |
29 | 21, 28 | ax-mp 5 | . . . . . . . . . 10 |
30 | 29, 24 | xpex 6962 | . . . . . . . . 9 |
31 | 27, 30 | unex 6956 | . . . . . . . 8 |
32 | 26, 31 | ifex 4156 | . . . . . . 7 |
33 | 19, 9, 32 | fvmpt 6282 | . . . . . 6 |
34 | eqeq1 2626 | . . . . . . . 8 | |
35 | oveq1 6657 | . . . . . . . . . . 11 | |
36 | 35 | opeq1d 4408 | . . . . . . . . . 10 |
37 | 36 | sneqd 4189 | . . . . . . . . 9 |
38 | 35 | sneqd 4189 | . . . . . . . . . . 11 |
39 | 38 | difeq2d 3728 | . . . . . . . . . 10 |
40 | 39 | xpeq1d 5138 | . . . . . . . . 9 |
41 | 37, 40 | uneq12d 3768 | . . . . . . . 8 |
42 | 34, 41 | ifbieq2d 4111 | . . . . . . 7 |
43 | snex 4908 | . . . . . . . . 9 | |
44 | difexg 4808 | . . . . . . . . . . 11 | |
45 | 21, 44 | ax-mp 5 | . . . . . . . . . 10 |
46 | 45, 24 | xpex 6962 | . . . . . . . . 9 |
47 | 43, 46 | unex 6956 | . . . . . . . 8 |
48 | 26, 47 | ifex 4156 | . . . . . . 7 |
49 | 42, 9, 48 | fvmpt 6282 | . . . . . 6 |
50 | 33, 49 | eqeqan12d 2638 | . . . . 5 |
51 | 50 | adantl 482 | . . . 4 |
52 | eqtr3 2643 | . . . . . 6 | |
53 | 52 | 2a1d 26 | . . . . 5 |
54 | eqid 2622 | . . . . . . . . . . 11 | |
55 | 1, 54 | axlowdimlem13 25834 | . . . . . . . . . 10 |
56 | 55 | neneqd 2799 | . . . . . . . . 9 |
57 | 56 | pm2.21d 118 | . . . . . . . 8 |
58 | 57 | adantrl 752 | . . . . . . 7 |
59 | 4, 58 | sylan 488 | . . . . . 6 |
60 | iftrue 4092 | . . . . . . . 8 | |
61 | iffalse 4095 | . . . . . . . 8 | |
62 | 60, 61 | eqeqan12d 2638 | . . . . . . 7 |
63 | 62 | imbi1d 331 | . . . . . 6 |
64 | 59, 63 | syl5ibr 236 | . . . . 5 |
65 | eqid 2622 | . . . . . . . . . . . 12 | |
66 | 1, 65 | axlowdimlem13 25834 | . . . . . . . . . . 11 |
67 | 66 | necomd 2849 | . . . . . . . . . 10 |
68 | 67 | neneqd 2799 | . . . . . . . . 9 |
69 | 68 | pm2.21d 118 | . . . . . . . 8 |
70 | 4, 69 | sylan 488 | . . . . . . 7 |
71 | 70 | adantrr 753 | . . . . . 6 |
72 | iffalse 4095 | . . . . . . . 8 | |
73 | iftrue 4092 | . . . . . . . 8 | |
74 | 72, 73 | eqeqan12d 2638 | . . . . . . 7 |
75 | 74 | imbi1d 331 | . . . . . 6 |
76 | 71, 75 | syl5ibr 236 | . . . . 5 |
77 | 65, 54 | axlowdimlem14 25835 | . . . . . . . 8 |
78 | 77 | 3expb 1266 | . . . . . . 7 |
79 | 4, 78 | sylan 488 | . . . . . 6 |
80 | 72, 61 | eqeqan12d 2638 | . . . . . . 7 |
81 | 80 | imbi1d 331 | . . . . . 6 |
82 | 79, 81 | syl5ibr 236 | . . . . 5 |
83 | 53, 64, 76, 82 | 4cases 990 | . . . 4 |
84 | 51, 83 | sylbid 230 | . . 3 |
85 | 84 | ralrimivva 2971 | . 2 |
86 | dff13 6512 | . 2 | |
87 | 10, 85, 86 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 cvv 3200 cdif 3571 cun 3572 cif 4086 csn 4177 cop 4183 cmpt 4729 cxp 5112 wf 5884 wf1 5885 cfv 5888 (class class class)co 6650 cc0 9936 c1 9937 caddc 9939 cmin 10266 cneg 10267 cn 11020 c3 11071 cuz 11687 cfz 12326 cee 25768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-ee 25771 |
This theorem is referenced by: axlowdim 25841 |
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