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Mirrors > Home > MPE Home > Th. List > seqcl2 | Structured version Visualization version Unicode version |
Description: Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.) |
Ref | Expression |
---|---|
seqcl2.1 | |
seqcl2.2 | |
seqcl2.3 | |
seqcl2.4 |
Ref | Expression |
---|---|
seqcl2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqcl2.3 | . . 3 | |
2 | eluzfz2 12349 | . . 3 | |
3 | 1, 2 | syl 17 | . 2 |
4 | eleq1 2689 | . . . . . 6 | |
5 | fveq2 6191 | . . . . . . 7 | |
6 | 5 | eleq1d 2686 | . . . . . 6 |
7 | 4, 6 | imbi12d 334 | . . . . 5 |
8 | 7 | imbi2d 330 | . . . 4 |
9 | eleq1 2689 | . . . . . 6 | |
10 | fveq2 6191 | . . . . . . 7 | |
11 | 10 | eleq1d 2686 | . . . . . 6 |
12 | 9, 11 | imbi12d 334 | . . . . 5 |
13 | 12 | imbi2d 330 | . . . 4 |
14 | eleq1 2689 | . . . . . 6 | |
15 | fveq2 6191 | . . . . . . 7 | |
16 | 15 | eleq1d 2686 | . . . . . 6 |
17 | 14, 16 | imbi12d 334 | . . . . 5 |
18 | 17 | imbi2d 330 | . . . 4 |
19 | eleq1 2689 | . . . . . 6 | |
20 | fveq2 6191 | . . . . . . 7 | |
21 | 20 | eleq1d 2686 | . . . . . 6 |
22 | 19, 21 | imbi12d 334 | . . . . 5 |
23 | 22 | imbi2d 330 | . . . 4 |
24 | seqcl2.1 | . . . . . 6 | |
25 | seq1 12814 | . . . . . . 7 | |
26 | 25 | eleq1d 2686 | . . . . . 6 |
27 | 24, 26 | syl5ibr 236 | . . . . 5 |
28 | 27 | a1dd 50 | . . . 4 |
29 | peano2fzr 12354 | . . . . . . . . . 10 | |
30 | 29 | adantl 482 | . . . . . . . . 9 |
31 | 30 | expr 643 | . . . . . . . 8 |
32 | 31 | imim1d 82 | . . . . . . 7 |
33 | eluzp1p1 11713 | . . . . . . . . . . . . . 14 | |
34 | 33 | ad2antrl 764 | . . . . . . . . . . . . 13 |
35 | elfzuz3 12339 | . . . . . . . . . . . . . 14 | |
36 | 35 | ad2antll 765 | . . . . . . . . . . . . 13 |
37 | elfzuzb 12336 | . . . . . . . . . . . . 13 | |
38 | 34, 36, 37 | sylanbrc 698 | . . . . . . . . . . . 12 |
39 | seqcl2.4 | . . . . . . . . . . . . . 14 | |
40 | 39 | ralrimiva 2966 | . . . . . . . . . . . . 13 |
41 | 40 | adantr 481 | . . . . . . . . . . . 12 |
42 | fveq2 6191 | . . . . . . . . . . . . . 14 | |
43 | 42 | eleq1d 2686 | . . . . . . . . . . . . 13 |
44 | 43 | rspcv 3305 | . . . . . . . . . . . 12 |
45 | 38, 41, 44 | sylc 65 | . . . . . . . . . . 11 |
46 | seqcl2.2 | . . . . . . . . . . . . . 14 | |
47 | 46 | caovclg 6826 | . . . . . . . . . . . . 13 |
48 | 47 | ex 450 | . . . . . . . . . . . 12 |
49 | 48 | adantr 481 | . . . . . . . . . . 11 |
50 | 45, 49 | mpan2d 710 | . . . . . . . . . 10 |
51 | seqp1 12816 | . . . . . . . . . . . 12 | |
52 | 51 | ad2antrl 764 | . . . . . . . . . . 11 |
53 | 52 | eleq1d 2686 | . . . . . . . . . 10 |
54 | 50, 53 | sylibrd 249 | . . . . . . . . 9 |
55 | 54 | expr 643 | . . . . . . . 8 |
56 | 55 | a2d 29 | . . . . . . 7 |
57 | 32, 56 | syld 47 | . . . . . 6 |
58 | 57 | expcom 451 | . . . . 5 |
59 | 58 | a2d 29 | . . . 4 |
60 | 8, 13, 18, 23, 28, 59 | uzind4 11746 | . . 3 |
61 | 1, 60 | mpcom 38 | . 2 |
62 | 3, 61 | mpd 15 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 cfv 5888 (class class class)co 6650 c1 9937 caddc 9939 cz 11377 cuz 11687 cfz 12326 cseq 12801 |
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-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-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-seq 12802 |
This theorem is referenced by: seqf2 12820 seqcl 12821 seqz 12849 |
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