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Mirrors > Home > MPE Home > Th. List > seqcaopr3 | Structured version Visualization version Unicode version |
Description: Lemma for seqcaopr2 12837. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
seqcaopr3.1 | |
seqcaopr3.2 | |
seqcaopr3.3 | |
seqcaopr3.4 | |
seqcaopr3.5 | |
seqcaopr3.6 | |
seqcaopr3.7 | ..^ |
Ref | Expression |
---|---|
seqcaopr3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqcaopr3.3 | . . 3 | |
2 | eluzfz2 12349 | . . 3 | |
3 | 1, 2 | syl 17 | . 2 |
4 | fveq2 6191 | . . . . 5 | |
5 | fveq2 6191 | . . . . . 6 | |
6 | fveq2 6191 | . . . . . 6 | |
7 | 5, 6 | oveq12d 6668 | . . . . 5 |
8 | 4, 7 | eqeq12d 2637 | . . . 4 |
9 | 8 | imbi2d 330 | . . 3 |
10 | fveq2 6191 | . . . . 5 | |
11 | fveq2 6191 | . . . . . 6 | |
12 | fveq2 6191 | . . . . . 6 | |
13 | 11, 12 | oveq12d 6668 | . . . . 5 |
14 | 10, 13 | eqeq12d 2637 | . . . 4 |
15 | 14 | imbi2d 330 | . . 3 |
16 | fveq2 6191 | . . . . 5 | |
17 | fveq2 6191 | . . . . . 6 | |
18 | fveq2 6191 | . . . . . 6 | |
19 | 17, 18 | oveq12d 6668 | . . . . 5 |
20 | 16, 19 | eqeq12d 2637 | . . . 4 |
21 | 20 | imbi2d 330 | . . 3 |
22 | fveq2 6191 | . . . . 5 | |
23 | fveq2 6191 | . . . . . 6 | |
24 | fveq2 6191 | . . . . . 6 | |
25 | 23, 24 | oveq12d 6668 | . . . . 5 |
26 | 22, 25 | eqeq12d 2637 | . . . 4 |
27 | 26 | imbi2d 330 | . . 3 |
28 | eluzfz1 12348 | . . . . . . 7 | |
29 | 1, 28 | syl 17 | . . . . . 6 |
30 | seqcaopr3.6 | . . . . . . 7 | |
31 | 30 | ralrimiva 2966 | . . . . . 6 |
32 | fveq2 6191 | . . . . . . . 8 | |
33 | fveq2 6191 | . . . . . . . . 9 | |
34 | fveq2 6191 | . . . . . . . . 9 | |
35 | 33, 34 | oveq12d 6668 | . . . . . . . 8 |
36 | 32, 35 | eqeq12d 2637 | . . . . . . 7 |
37 | 36 | rspcv 3305 | . . . . . 6 |
38 | 29, 31, 37 | sylc 65 | . . . . 5 |
39 | eluzel2 11692 | . . . . . . 7 | |
40 | 1, 39 | syl 17 | . . . . . 6 |
41 | seq1 12814 | . . . . . 6 | |
42 | 40, 41 | syl 17 | . . . . 5 |
43 | seq1 12814 | . . . . . . 7 | |
44 | seq1 12814 | . . . . . . 7 | |
45 | 43, 44 | oveq12d 6668 | . . . . . 6 |
46 | 40, 45 | syl 17 | . . . . 5 |
47 | 38, 42, 46 | 3eqtr4d 2666 | . . . 4 |
48 | 47 | a1i 11 | . . 3 |
49 | oveq1 6657 | . . . . . 6 | |
50 | elfzouz 12474 | . . . . . . . . 9 ..^ | |
51 | 50 | adantl 482 | . . . . . . . 8 ..^ |
52 | seqp1 12816 | . . . . . . . 8 | |
53 | 51, 52 | syl 17 | . . . . . . 7 ..^ |
54 | seqcaopr3.7 | . . . . . . . 8 ..^ | |
55 | fzofzp1 12565 | . . . . . . . . . . 11 ..^ | |
56 | 55 | adantl 482 | . . . . . . . . . 10 ..^ |
57 | 31 | adantr 481 | . . . . . . . . . 10 ..^ |
58 | fveq2 6191 | . . . . . . . . . . . 12 | |
59 | fveq2 6191 | . . . . . . . . . . . . 13 | |
60 | fveq2 6191 | . . . . . . . . . . . . 13 | |
61 | 59, 60 | oveq12d 6668 | . . . . . . . . . . . 12 |
62 | 58, 61 | eqeq12d 2637 | . . . . . . . . . . 11 |
63 | 62 | rspcv 3305 | . . . . . . . . . 10 |
64 | 56, 57, 63 | sylc 65 | . . . . . . . . 9 ..^ |
65 | 64 | oveq2d 6666 | . . . . . . . 8 ..^ |
66 | seqp1 12816 | . . . . . . . . . 10 | |
67 | seqp1 12816 | . . . . . . . . . 10 | |
68 | 66, 67 | oveq12d 6668 | . . . . . . . . 9 |
69 | 51, 68 | syl 17 | . . . . . . . 8 ..^ |
70 | 54, 65, 69 | 3eqtr4rd 2667 | . . . . . . 7 ..^ |
71 | 53, 70 | eqeq12d 2637 | . . . . . 6 ..^ |
72 | 49, 71 | syl5ibr 236 | . . . . 5 ..^ |
73 | 72 | expcom 451 | . . . 4 ..^ |
74 | 73 | a2d 29 | . . 3 ..^ |
75 | 9, 15, 21, 27, 48, 74 | fzind2 12586 | . 2 |
76 | 3, 75 | mpcom 38 | 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 ..^cfzo 12465 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-fzo 12466 df-seq 12802 |
This theorem is referenced by: seqcaopr2 12837 gsumzaddlem 18321 |
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