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Mirrors > Home > ILE Home > Th. List > seq3caopr3 | Unicode version |
Description: Lemma for seq3caopr2 10452. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.) |
Ref | Expression |
---|---|
seqcaopr3.1 | |
seqcaopr3.2 | |
seqcaopr3.3 | |
seq3caopr3.4 | |
seq3caopr3.5 | |
seq3caopr3.6 | |
seqcaopr3.7 | ..^ |
Ref | Expression |
---|---|
seq3caopr3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqcaopr3.3 | . . 3 | |
2 | eluzfz2 10002 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | fveq2 5507 | . . . . 5 | |
5 | fveq2 5507 | . . . . . 6 | |
6 | fveq2 5507 | . . . . . 6 | |
7 | 5, 6 | oveq12d 5883 | . . . . 5 |
8 | 4, 7 | eqeq12d 2190 | . . . 4 |
9 | 8 | imbi2d 230 | . . 3 |
10 | fveq2 5507 | . . . . 5 | |
11 | fveq2 5507 | . . . . . 6 | |
12 | fveq2 5507 | . . . . . 6 | |
13 | 11, 12 | oveq12d 5883 | . . . . 5 |
14 | 10, 13 | eqeq12d 2190 | . . . 4 |
15 | 14 | imbi2d 230 | . . 3 |
16 | fveq2 5507 | . . . . 5 | |
17 | fveq2 5507 | . . . . . 6 | |
18 | fveq2 5507 | . . . . . 6 | |
19 | 17, 18 | oveq12d 5883 | . . . . 5 |
20 | 16, 19 | eqeq12d 2190 | . . . 4 |
21 | 20 | imbi2d 230 | . . 3 |
22 | fveq2 5507 | . . . . 5 | |
23 | fveq2 5507 | . . . . . 6 | |
24 | fveq2 5507 | . . . . . 6 | |
25 | 23, 24 | oveq12d 5883 | . . . . 5 |
26 | 22, 25 | eqeq12d 2190 | . . . 4 |
27 | 26 | imbi2d 230 | . . 3 |
28 | fveq2 5507 | . . . . . . 7 | |
29 | fveq2 5507 | . . . . . . . 8 | |
30 | fveq2 5507 | . . . . . . . 8 | |
31 | 29, 30 | oveq12d 5883 | . . . . . . 7 |
32 | 28, 31 | eqeq12d 2190 | . . . . . 6 |
33 | seq3caopr3.6 | . . . . . . 7 | |
34 | 33 | ralrimiva 2548 | . . . . . 6 |
35 | eluzel2 9506 | . . . . . . . 8 | |
36 | 1, 35 | syl 14 | . . . . . . 7 |
37 | uzid 9515 | . . . . . . 7 | |
38 | 36, 37 | syl 14 | . . . . . 6 |
39 | 32, 34, 38 | rspcdva 2844 | . . . . 5 |
40 | seqcaopr3.2 | . . . . . . . . . . . 12 | |
41 | 40 | ralrimivva 2557 | . . . . . . . . . . 11 |
42 | 41 | adantr 276 | . . . . . . . . . 10 |
43 | seq3caopr3.4 | . . . . . . . . . . 11 | |
44 | seq3caopr3.5 | . . . . . . . . . . 11 | |
45 | oveq1 5872 | . . . . . . . . . . . . 13 | |
46 | 45 | eleq1d 2244 | . . . . . . . . . . . 12 |
47 | oveq2 5873 | . . . . . . . . . . . . 13 | |
48 | 47 | eleq1d 2244 | . . . . . . . . . . . 12 |
49 | 46, 48 | rspc2v 2852 | . . . . . . . . . . 11 |
50 | 43, 44, 49 | syl2anc 411 | . . . . . . . . . 10 |
51 | 42, 50 | mpd 13 | . . . . . . . . 9 |
52 | 33, 51 | eqeltrd 2252 | . . . . . . . 8 |
53 | 52 | ralrimiva 2548 | . . . . . . 7 |
54 | fveq2 5507 | . . . . . . . . 9 | |
55 | 54 | eleq1d 2244 | . . . . . . . 8 |
56 | 55 | rspcv 2835 | . . . . . . 7 |
57 | 53, 56 | mpan9 281 | . . . . . 6 |
58 | seqcaopr3.1 | . . . . . 6 | |
59 | 36, 57, 58 | seq3-1 10430 | . . . . 5 |
60 | 43 | ralrimiva 2548 | . . . . . . . 8 |
61 | fveq2 5507 | . . . . . . . . . 10 | |
62 | 61 | eleq1d 2244 | . . . . . . . . 9 |
63 | 62 | rspcv 2835 | . . . . . . . 8 |
64 | 60, 63 | mpan9 281 | . . . . . . 7 |
65 | 36, 64, 58 | seq3-1 10430 | . . . . . 6 |
66 | 44 | ralrimiva 2548 | . . . . . . . 8 |
67 | fveq2 5507 | . . . . . . . . . 10 | |
68 | 67 | eleq1d 2244 | . . . . . . . . 9 |
69 | 68 | rspcv 2835 | . . . . . . . 8 |
70 | 66, 69 | mpan9 281 | . . . . . . 7 |
71 | 36, 70, 58 | seq3-1 10430 | . . . . . 6 |
72 | 65, 71 | oveq12d 5883 | . . . . 5 |
73 | 39, 59, 72 | 3eqtr4d 2218 | . . . 4 |
74 | 73 | a1i 9 | . . 3 |
75 | oveq1 5872 | . . . . . 6 | |
76 | elfzouz 10121 | . . . . . . . . 9 ..^ | |
77 | 76 | adantl 277 | . . . . . . . 8 ..^ |
78 | 57 | adantlr 477 | . . . . . . . 8 ..^ |
79 | 58 | adantlr 477 | . . . . . . . 8 ..^ |
80 | 77, 78, 79 | seq3p1 10432 | . . . . . . 7 ..^ |
81 | seqcaopr3.7 | . . . . . . . 8 ..^ | |
82 | fveq2 5507 | . . . . . . . . . . 11 | |
83 | fveq2 5507 | . . . . . . . . . . . 12 | |
84 | fveq2 5507 | . . . . . . . . . . . 12 | |
85 | 83, 84 | oveq12d 5883 | . . . . . . . . . . 11 |
86 | 82, 85 | eqeq12d 2190 | . . . . . . . . . 10 |
87 | 34 | adantr 276 | . . . . . . . . . 10 ..^ |
88 | fzofzp1 10197 | . . . . . . . . . . . 12 ..^ | |
89 | elfzuz 9991 | . . . . . . . . . . . 12 | |
90 | 88, 89 | syl 14 | . . . . . . . . . . 11 ..^ |
91 | 90 | adantl 277 | . . . . . . . . . 10 ..^ |
92 | 86, 87, 91 | rspcdva 2844 | . . . . . . . . 9 ..^ |
93 | 92 | oveq2d 5881 | . . . . . . . 8 ..^ |
94 | 64 | adantlr 477 | . . . . . . . . . 10 ..^ |
95 | 77, 94, 79 | seq3p1 10432 | . . . . . . . . 9 ..^ |
96 | 70 | adantlr 477 | . . . . . . . . . 10 ..^ |
97 | 77, 96, 79 | seq3p1 10432 | . . . . . . . . 9 ..^ |
98 | 95, 97 | oveq12d 5883 | . . . . . . . 8 ..^ |
99 | 81, 93, 98 | 3eqtr4rd 2219 | . . . . . . 7 ..^ |
100 | 80, 99 | eqeq12d 2190 | . . . . . 6 ..^ |
101 | 75, 100 | syl5ibr 156 | . . . . 5 ..^ |
102 | 101 | expcom 116 | . . . 4 ..^ |
103 | 102 | a2d 26 | . . 3 ..^ |
104 | 9, 15, 21, 27, 74, 103 | fzind2 10209 | . 2 |
105 | 3, 104 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wceq 1353 wcel 2146 wral 2453 cfv 5208 (class class class)co 5865 c1 7787 caddc 7789 cz 9226 cuz 9501 cfz 9979 ..^cfzo 10112 cseq 10415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-addcom 7886 ax-addass 7888 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-0id 7894 ax-rnegex 7895 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-ltadd 7902 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-inn 8893 df-n0 9150 df-z 9227 df-uz 9502 df-fz 9980 df-fzo 10113 df-seqfrec 10416 |
This theorem is referenced by: seq3caopr2 10452 |
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