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Mirrors > Home > ILE Home > Th. List > seq3z | Unicode version |
Description: If the operation has an absorbing element (a.k.a. zero element), then any sequence containing a evaluates to . (Contributed by Mario Carneiro, 27-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.) |
Ref | Expression |
---|---|
seq3homo.1 | |
seq3homo.2 | |
seqz.3 | |
seqz.4 | |
seqz.5 | |
seqz.7 |
Ref | Expression |
---|---|
seq3z |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqz.5 | . . 3 | |
2 | elfzuz3 9992 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | fveqeq2 5516 | . . . 4 | |
5 | 4 | imbi2d 230 | . . 3 |
6 | fveqeq2 5516 | . . . 4 | |
7 | 6 | imbi2d 230 | . . 3 |
8 | fveqeq2 5516 | . . . 4 | |
9 | 8 | imbi2d 230 | . . 3 |
10 | fveqeq2 5516 | . . . 4 | |
11 | 10 | imbi2d 230 | . . 3 |
12 | elfzuz 9991 | . . . . . . . . . 10 | |
13 | 1, 12 | syl 14 | . . . . . . . . 9 |
14 | eluzelz 9510 | . . . . . . . . 9 | |
15 | 13, 14 | syl 14 | . . . . . . . 8 |
16 | simpr 110 | . . . . . . . . . 10 | |
17 | 13 | adantr 276 | . . . . . . . . . 10 |
18 | uztrn 9517 | . . . . . . . . . 10 | |
19 | 16, 17, 18 | syl2anc 411 | . . . . . . . . 9 |
20 | seq3homo.2 | . . . . . . . . 9 | |
21 | 19, 20 | syldan 282 | . . . . . . . 8 |
22 | seq3homo.1 | . . . . . . . 8 | |
23 | 15, 21, 22 | seq3-1 10430 | . . . . . . 7 |
24 | seqz.7 | . . . . . . 7 | |
25 | 23, 24 | eqtrd 2208 | . . . . . 6 |
26 | seqeq1 10418 | . . . . . . . 8 | |
27 | 26 | fveq1d 5509 | . . . . . . 7 |
28 | 27 | eqeq1d 2184 | . . . . . 6 |
29 | 25, 28 | syl5ibcom 155 | . . . . 5 |
30 | eluzel2 9506 | . . . . . . . . . 10 | |
31 | 13, 30 | syl 14 | . . . . . . . . 9 |
32 | 31 | adantr 276 | . . . . . . . 8 |
33 | simpr 110 | . . . . . . . 8 | |
34 | 20 | adantlr 477 | . . . . . . . 8 |
35 | 22 | adantlr 477 | . . . . . . . 8 |
36 | 32, 33, 34, 35 | seq3m1 10438 | . . . . . . 7 |
37 | 24 | adantr 276 | . . . . . . . 8 |
38 | 37 | oveq2d 5881 | . . . . . . 7 |
39 | oveq1 5872 | . . . . . . . . 9 | |
40 | 39 | eqeq1d 2184 | . . . . . . . 8 |
41 | seqz.4 | . . . . . . . . . 10 | |
42 | 41 | ralrimiva 2548 | . . . . . . . . 9 |
43 | 42 | adantr 276 | . . . . . . . 8 |
44 | eqid 2175 | . . . . . . . . . 10 | |
45 | 44, 32, 34, 35 | seqf 10431 | . . . . . . . . 9 |
46 | eluzp1m1 9524 | . . . . . . . . . 10 | |
47 | 31, 46 | sylan 283 | . . . . . . . . 9 |
48 | 45, 47 | ffvelcdmd 5644 | . . . . . . . 8 |
49 | 40, 43, 48 | rspcdva 2844 | . . . . . . 7 |
50 | 36, 38, 49 | 3eqtrd 2212 | . . . . . 6 |
51 | 50 | ex 115 | . . . . 5 |
52 | uzp1 9534 | . . . . . 6 | |
53 | 13, 52 | syl 14 | . . . . 5 |
54 | 29, 51, 53 | mpjaod 718 | . . . 4 |
55 | 54 | a1i 9 | . . 3 |
56 | simpr 110 | . . . . . . . . . 10 | |
57 | 13 | adantr 276 | . . . . . . . . . 10 |
58 | uztrn 9517 | . . . . . . . . . 10 | |
59 | 56, 57, 58 | syl2anc 411 | . . . . . . . . 9 |
60 | 20 | adantlr 477 | . . . . . . . . 9 |
61 | 22 | adantlr 477 | . . . . . . . . 9 |
62 | 59, 60, 61 | seq3p1 10432 | . . . . . . . 8 |
63 | 62 | adantr 276 | . . . . . . 7 |
64 | simpr 110 | . . . . . . . 8 | |
65 | 64 | oveq1d 5880 | . . . . . . 7 |
66 | oveq2 5873 | . . . . . . . . . 10 | |
67 | 66 | eqeq1d 2184 | . . . . . . . . 9 |
68 | seqz.3 | . . . . . . . . . . 11 | |
69 | 68 | ralrimiva 2548 | . . . . . . . . . 10 |
70 | 69 | adantr 276 | . . . . . . . . 9 |
71 | fveq2 5507 | . . . . . . . . . . 11 | |
72 | 71 | eleq1d 2244 | . . . . . . . . . 10 |
73 | 20 | ralrimiva 2548 | . . . . . . . . . . 11 |
74 | 73 | adantr 276 | . . . . . . . . . 10 |
75 | peano2uz 9556 | . . . . . . . . . . 11 | |
76 | 59, 75 | syl 14 | . . . . . . . . . 10 |
77 | 72, 74, 76 | rspcdva 2844 | . . . . . . . . 9 |
78 | 67, 70, 77 | rspcdva 2844 | . . . . . . . 8 |
79 | 78 | adantr 276 | . . . . . . 7 |
80 | 63, 65, 79 | 3eqtrd 2212 | . . . . . 6 |
81 | 80 | ex 115 | . . . . 5 |
82 | 81 | expcom 116 | . . . 4 |
83 | 82 | a2d 26 | . . 3 |
84 | 5, 7, 9, 11, 55, 83 | uzind4 9561 | . 2 |
85 | 3, 84 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wo 708 wceq 1353 wcel 2146 wral 2453 cfv 5208 (class class class)co 5865 c1 7787 caddc 7789 cmin 8102 cz 9226 cuz 9501 cfz 9979 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-seqfrec 10416 |
This theorem is referenced by: bcval5 10711 lgsne0 14019 |
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