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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 9803 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | fveqeq2 5430 | . . . 4 | |
5 | 4 | imbi2d 229 | . . 3 |
6 | fveqeq2 5430 | . . . 4 | |
7 | 6 | imbi2d 229 | . . 3 |
8 | fveqeq2 5430 | . . . 4 | |
9 | 8 | imbi2d 229 | . . 3 |
10 | fveqeq2 5430 | . . . 4 | |
11 | 10 | imbi2d 229 | . . 3 |
12 | elfzuz 9802 | . . . . . . . . . 10 | |
13 | 1, 12 | syl 14 | . . . . . . . . 9 |
14 | eluzelz 9335 | . . . . . . . . 9 | |
15 | 13, 14 | syl 14 | . . . . . . . 8 |
16 | simpr 109 | . . . . . . . . . 10 | |
17 | 13 | adantr 274 | . . . . . . . . . 10 |
18 | uztrn 9342 | . . . . . . . . . 10 | |
19 | 16, 17, 18 | syl2anc 408 | . . . . . . . . 9 |
20 | seq3homo.2 | . . . . . . . . 9 | |
21 | 19, 20 | syldan 280 | . . . . . . . 8 |
22 | seq3homo.1 | . . . . . . . 8 | |
23 | 15, 21, 22 | seq3-1 10233 | . . . . . . 7 |
24 | seqz.7 | . . . . . . 7 | |
25 | 23, 24 | eqtrd 2172 | . . . . . 6 |
26 | seqeq1 10221 | . . . . . . . 8 | |
27 | 26 | fveq1d 5423 | . . . . . . 7 |
28 | 27 | eqeq1d 2148 | . . . . . 6 |
29 | 25, 28 | syl5ibcom 154 | . . . . 5 |
30 | eluzel2 9331 | . . . . . . . . . 10 | |
31 | 13, 30 | syl 14 | . . . . . . . . 9 |
32 | 31 | adantr 274 | . . . . . . . 8 |
33 | simpr 109 | . . . . . . . 8 | |
34 | 20 | adantlr 468 | . . . . . . . 8 |
35 | 22 | adantlr 468 | . . . . . . . 8 |
36 | 32, 33, 34, 35 | seq3m1 10241 | . . . . . . 7 |
37 | 24 | adantr 274 | . . . . . . . 8 |
38 | 37 | oveq2d 5790 | . . . . . . 7 |
39 | oveq1 5781 | . . . . . . . . 9 | |
40 | 39 | eqeq1d 2148 | . . . . . . . 8 |
41 | seqz.4 | . . . . . . . . . 10 | |
42 | 41 | ralrimiva 2505 | . . . . . . . . 9 |
43 | 42 | adantr 274 | . . . . . . . 8 |
44 | eqid 2139 | . . . . . . . . . 10 | |
45 | 44, 32, 34, 35 | seqf 10234 | . . . . . . . . 9 |
46 | eluzp1m1 9349 | . . . . . . . . . 10 | |
47 | 31, 46 | sylan 281 | . . . . . . . . 9 |
48 | 45, 47 | ffvelrnd 5556 | . . . . . . . 8 |
49 | 40, 43, 48 | rspcdva 2794 | . . . . . . 7 |
50 | 36, 38, 49 | 3eqtrd 2176 | . . . . . 6 |
51 | 50 | ex 114 | . . . . 5 |
52 | uzp1 9359 | . . . . . 6 | |
53 | 13, 52 | syl 14 | . . . . 5 |
54 | 29, 51, 53 | mpjaod 707 | . . . 4 |
55 | 54 | a1i 9 | . . 3 |
56 | simpr 109 | . . . . . . . . . 10 | |
57 | 13 | adantr 274 | . . . . . . . . . 10 |
58 | uztrn 9342 | . . . . . . . . . 10 | |
59 | 56, 57, 58 | syl2anc 408 | . . . . . . . . 9 |
60 | 20 | adantlr 468 | . . . . . . . . 9 |
61 | 22 | adantlr 468 | . . . . . . . . 9 |
62 | 59, 60, 61 | seq3p1 10235 | . . . . . . . 8 |
63 | 62 | adantr 274 | . . . . . . 7 |
64 | simpr 109 | . . . . . . . 8 | |
65 | 64 | oveq1d 5789 | . . . . . . 7 |
66 | oveq2 5782 | . . . . . . . . . 10 | |
67 | 66 | eqeq1d 2148 | . . . . . . . . 9 |
68 | seqz.3 | . . . . . . . . . . 11 | |
69 | 68 | ralrimiva 2505 | . . . . . . . . . 10 |
70 | 69 | adantr 274 | . . . . . . . . 9 |
71 | fveq2 5421 | . . . . . . . . . . 11 | |
72 | 71 | eleq1d 2208 | . . . . . . . . . 10 |
73 | 20 | ralrimiva 2505 | . . . . . . . . . . 11 |
74 | 73 | adantr 274 | . . . . . . . . . 10 |
75 | peano2uz 9378 | . . . . . . . . . . 11 | |
76 | 59, 75 | syl 14 | . . . . . . . . . 10 |
77 | 72, 74, 76 | rspcdva 2794 | . . . . . . . . 9 |
78 | 67, 70, 77 | rspcdva 2794 | . . . . . . . 8 |
79 | 78 | adantr 274 | . . . . . . 7 |
80 | 63, 65, 79 | 3eqtrd 2176 | . . . . . 6 |
81 | 80 | ex 114 | . . . . 5 |
82 | 81 | expcom 115 | . . . 4 |
83 | 82 | a2d 26 | . . 3 |
84 | 5, 7, 9, 11, 55, 83 | uzind4 9383 | . 2 |
85 | 3, 84 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 wceq 1331 wcel 1480 wral 2416 cfv 5123 (class class class)co 5774 c1 7621 caddc 7623 cmin 7933 cz 9054 cuz 9326 cfz 9790 cseq 10218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 df-seqfrec 10219 |
This theorem is referenced by: bcval5 10509 |
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