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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldioph2lem1 | Structured version Visualization version Unicode version |
Description: Lemma for eldioph2 37325. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
Ref | Expression |
---|---|
eldioph2lem1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 11301 | . . . . . . . . . 10 | |
2 | 1 | 3ad2ant1 1082 | . . . . . . . . 9 |
3 | 2 | recnd 10068 | . . . . . . . 8 |
4 | ax-1cn 9994 | . . . . . . . 8 | |
5 | addcom 10222 | . . . . . . . 8 | |
6 | 3, 4, 5 | sylancl 694 | . . . . . . 7 |
7 | diffi 8192 | . . . . . . . . . 10 | |
8 | 7 | 3ad2ant2 1083 | . . . . . . . . 9 |
9 | fzfid 12772 | . . . . . . . . 9 | |
10 | incom 3805 | . . . . . . . . . . 11 | |
11 | disjdif 4040 | . . . . . . . . . . 11 | |
12 | 10, 11 | eqtri 2644 | . . . . . . . . . 10 |
13 | 12 | a1i 11 | . . . . . . . . 9 |
14 | hashun 13171 | . . . . . . . . 9 | |
15 | 8, 9, 13, 14 | syl3anc 1326 | . . . . . . . 8 |
16 | uncom 3757 | . . . . . . . . . 10 | |
17 | simp3 1063 | . . . . . . . . . . 11 | |
18 | undif 4049 | . . . . . . . . . . 11 | |
19 | 17, 18 | sylib 208 | . . . . . . . . . 10 |
20 | 16, 19 | syl5eq 2668 | . . . . . . . . 9 |
21 | 20 | fveq2d 6195 | . . . . . . . 8 |
22 | hashfz1 13134 | . . . . . . . . . 10 | |
23 | 22 | 3ad2ant1 1082 | . . . . . . . . 9 |
24 | 23 | oveq2d 6666 | . . . . . . . 8 |
25 | 15, 21, 24 | 3eqtr3d 2664 | . . . . . . 7 |
26 | 6, 25 | oveq12d 6668 | . . . . . 6 |
27 | 26 | fveq2d 6195 | . . . . 5 |
28 | 1zzd 11408 | . . . . . . . 8 | |
29 | hashcl 13147 | . . . . . . . . . 10 | |
30 | 8, 29 | syl 17 | . . . . . . . . 9 |
31 | 30 | nn0zd 11480 | . . . . . . . 8 |
32 | nn0z 11400 | . . . . . . . . 9 | |
33 | 32 | 3ad2ant1 1082 | . . . . . . . 8 |
34 | fzen 12358 | . . . . . . . 8 | |
35 | 28, 31, 33, 34 | syl3anc 1326 | . . . . . . 7 |
36 | 35 | ensymd 8007 | . . . . . 6 |
37 | fzfi 12771 | . . . . . . 7 | |
38 | fzfi 12771 | . . . . . . 7 | |
39 | hashen 13135 | . . . . . . 7 | |
40 | 37, 38, 39 | mp2an 708 | . . . . . 6 |
41 | 36, 40 | sylibr 224 | . . . . 5 |
42 | hashfz1 13134 | . . . . . 6 | |
43 | 30, 42 | syl 17 | . . . . 5 |
44 | 27, 41, 43 | 3eqtrd 2660 | . . . 4 |
45 | fzfi 12771 | . . . . 5 | |
46 | hashen 13135 | . . . . 5 | |
47 | 45, 8, 46 | sylancr 695 | . . . 4 |
48 | 44, 47 | mpbid 222 | . . 3 |
49 | bren 7964 | . . 3 | |
50 | 48, 49 | sylib 208 | . 2 |
51 | simpl1 1064 | . . . . 5 | |
52 | 51 | nn0zd 11480 | . . . 4 |
53 | simpl2 1065 | . . . . . 6 | |
54 | hashcl 13147 | . . . . . 6 | |
55 | 53, 54 | syl 17 | . . . . 5 |
56 | 55 | nn0zd 11480 | . . . 4 |
57 | nn0addge2 11340 | . . . . . . 7 | |
58 | 2, 30, 57 | syl2anc 693 | . . . . . 6 |
59 | 58, 25 | breqtrrd 4681 | . . . . 5 |
60 | 59 | adantr 481 | . . . 4 |
61 | eluz2 11693 | . . . 4 | |
62 | 52, 56, 60, 61 | syl3anbrc 1246 | . . 3 |
63 | vex 3203 | . . . . 5 | |
64 | ovex 6678 | . . . . . 6 | |
65 | resiexg 7102 | . . . . . 6 | |
66 | 64, 65 | ax-mp 5 | . . . . 5 |
67 | 63, 66 | unex 6956 | . . . 4 |
68 | 67 | a1i 11 | . . 3 |
69 | simpr 477 | . . . . 5 | |
70 | f1oi 6174 | . . . . . 6 | |
71 | 70 | a1i 11 | . . . . 5 |
72 | incom 3805 | . . . . . 6 | |
73 | 51 | nn0red 11352 | . . . . . . . 8 |
74 | 73 | ltp1d 10954 | . . . . . . 7 |
75 | fzdisj 12368 | . . . . . . 7 | |
76 | 74, 75 | syl 17 | . . . . . 6 |
77 | 72, 76 | syl5eq 2668 | . . . . 5 |
78 | 12 | a1i 11 | . . . . 5 |
79 | f1oun 6156 | . . . . 5 | |
80 | 69, 71, 77, 78, 79 | syl22anc 1327 | . . . 4 |
81 | fzsplit1nn0 37317 | . . . . . . 7 | |
82 | 51, 55, 60, 81 | syl3anc 1326 | . . . . . 6 |
83 | uncom 3757 | . . . . . 6 | |
84 | 82, 83 | syl6reqr 2675 | . . . . 5 |
85 | simpl3 1066 | . . . . . . 7 | |
86 | 85, 18 | sylib 208 | . . . . . 6 |
87 | 16, 86 | syl5eq 2668 | . . . . 5 |
88 | f1oeq23 6130 | . . . . 5 | |
89 | 84, 87, 88 | syl2anc 693 | . . . 4 |
90 | 80, 89 | mpbid 222 | . . 3 |
91 | resundir 5411 | . . . 4 | |
92 | dmres 5419 | . . . . . . . 8 | |
93 | f1odm 6141 | . . . . . . . . . . 11 | |
94 | 93 | adantl 482 | . . . . . . . . . 10 |
95 | 94 | ineq2d 3814 | . . . . . . . . 9 |
96 | 95, 76 | eqtrd 2656 | . . . . . . . 8 |
97 | 92, 96 | syl5eq 2668 | . . . . . . 7 |
98 | relres 5426 | . . . . . . . 8 | |
99 | reldm0 5343 | . . . . . . . 8 | |
100 | 98, 99 | ax-mp 5 | . . . . . . 7 |
101 | 97, 100 | sylibr 224 | . . . . . 6 |
102 | residm 5430 | . . . . . . 7 | |
103 | 102 | a1i 11 | . . . . . 6 |
104 | 101, 103 | uneq12d 3768 | . . . . 5 |
105 | uncom 3757 | . . . . . 6 | |
106 | un0 3967 | . . . . . 6 | |
107 | 105, 106 | eqtri 2644 | . . . . 5 |
108 | 104, 107 | syl6eq 2672 | . . . 4 |
109 | 91, 108 | syl5eq 2668 | . . 3 |
110 | oveq2 6658 | . . . . . 6 | |
111 | f1oeq2 6128 | . . . . . 6 | |
112 | 110, 111 | syl 17 | . . . . 5 |
113 | 112 | anbi1d 741 | . . . 4 |
114 | f1oeq1 6127 | . . . . 5 | |
115 | reseq1 5390 | . . . . . 6 | |
116 | 115 | eqeq1d 2624 | . . . . 5 |
117 | 114, 116 | anbi12d 747 | . . . 4 |
118 | 113, 117 | rspc2ev 3324 | . . 3 |
119 | 62, 68, 90, 109, 118 | syl112anc 1330 | . 2 |
120 | 50, 119 | exlimddv 1863 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wex 1704 wcel 1990 wrex 2913 cvv 3200 cdif 3571 cun 3572 cin 3573 wss 3574 c0 3915 class class class wbr 4653 cid 5023 cdm 5114 cres 5116 wrel 5119 wf1o 5887 cfv 5888 (class class class)co 6650 cen 7952 cfn 7955 cc 9934 cr 9935 c1 9937 caddc 9939 clt 10074 cle 10075 cn0 11292 cz 11377 cuz 11687 cfz 12326 chash 13117 |
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-rep 4771 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-rmo 2920 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-int 4476 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-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 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-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 |
This theorem is referenced by: eldioph2 37325 |
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