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Mirrors > Home > ILE Home > Th. List > seq3homo | Unicode version |
Description: Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 10-Oct-2022.) |
Ref | Expression |
---|---|
seq3homo.1 | |
seq3homo.2 | |
seq3homo.3 | |
seq3homo.4 | |
seq3homo.5 | |
seq3homo.g | |
seq3homo.qcl |
Ref | Expression |
---|---|
seq3homo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seq3homo.3 | . 2 | |
2 | 2fveq3 5426 | . . . . 5 | |
3 | fveq2 5421 | . . . . 5 | |
4 | 2, 3 | eqeq12d 2154 | . . . 4 |
5 | 4 | imbi2d 229 | . . 3 |
6 | 2fveq3 5426 | . . . . 5 | |
7 | fveq2 5421 | . . . . 5 | |
8 | 6, 7 | eqeq12d 2154 | . . . 4 |
9 | 8 | imbi2d 229 | . . 3 |
10 | 2fveq3 5426 | . . . . 5 | |
11 | fveq2 5421 | . . . . 5 | |
12 | 10, 11 | eqeq12d 2154 | . . . 4 |
13 | 12 | imbi2d 229 | . . 3 |
14 | 2fveq3 5426 | . . . . 5 | |
15 | fveq2 5421 | . . . . 5 | |
16 | 14, 15 | eqeq12d 2154 | . . . 4 |
17 | 16 | imbi2d 229 | . . 3 |
18 | 2fveq3 5426 | . . . . . . 7 | |
19 | fveq2 5421 | . . . . . . 7 | |
20 | 18, 19 | eqeq12d 2154 | . . . . . 6 |
21 | seq3homo.5 | . . . . . . 7 | |
22 | 21 | ralrimiva 2505 | . . . . . 6 |
23 | eluzel2 9331 | . . . . . . . 8 | |
24 | 1, 23 | syl 14 | . . . . . . 7 |
25 | uzid 9340 | . . . . . . 7 | |
26 | 24, 25 | syl 14 | . . . . . 6 |
27 | 20, 22, 26 | rspcdva 2794 | . . . . 5 |
28 | seq3homo.2 | . . . . . . 7 | |
29 | seq3homo.1 | . . . . . . 7 | |
30 | 24, 28, 29 | seq3-1 10233 | . . . . . 6 |
31 | 30 | fveq2d 5425 | . . . . 5 |
32 | seq3homo.g | . . . . . 6 | |
33 | seq3homo.qcl | . . . . . 6 | |
34 | 24, 32, 33 | seq3-1 10233 | . . . . 5 |
35 | 27, 31, 34 | 3eqtr4d 2182 | . . . 4 |
36 | 35 | a1i 9 | . . 3 |
37 | oveq1 5781 | . . . . . 6 | |
38 | simpr 109 | . . . . . . . . . 10 | |
39 | 28 | adantlr 468 | . . . . . . . . . 10 |
40 | 29 | adantlr 468 | . . . . . . . . . 10 |
41 | 38, 39, 40 | seq3p1 10235 | . . . . . . . . 9 |
42 | 41 | fveq2d 5425 | . . . . . . . 8 |
43 | seq3homo.4 | . . . . . . . . . . 11 | |
44 | 43 | ralrimivva 2514 | . . . . . . . . . 10 |
45 | 44 | adantr 274 | . . . . . . . . 9 |
46 | eqid 2139 | . . . . . . . . . . . 12 | |
47 | 46, 24, 28, 29 | seqf 10234 | . . . . . . . . . . 11 |
48 | 47 | ffvelrnda 5555 | . . . . . . . . . 10 |
49 | fveq2 5421 | . . . . . . . . . . . 12 | |
50 | 49 | eleq1d 2208 | . . . . . . . . . . 11 |
51 | 28 | ralrimiva 2505 | . . . . . . . . . . . 12 |
52 | 51 | adantr 274 | . . . . . . . . . . 11 |
53 | peano2uz 9378 | . . . . . . . . . . . 12 | |
54 | 38, 53 | syl 14 | . . . . . . . . . . 11 |
55 | 50, 52, 54 | rspcdva 2794 | . . . . . . . . . 10 |
56 | oveq1 5781 | . . . . . . . . . . . . 13 | |
57 | 56 | fveq2d 5425 | . . . . . . . . . . . 12 |
58 | fveq2 5421 | . . . . . . . . . . . . 13 | |
59 | 58 | oveq1d 5789 | . . . . . . . . . . . 12 |
60 | 57, 59 | eqeq12d 2154 | . . . . . . . . . . 11 |
61 | oveq2 5782 | . . . . . . . . . . . . 13 | |
62 | 61 | fveq2d 5425 | . . . . . . . . . . . 12 |
63 | fveq2 5421 | . . . . . . . . . . . . 13 | |
64 | 63 | oveq2d 5790 | . . . . . . . . . . . 12 |
65 | 62, 64 | eqeq12d 2154 | . . . . . . . . . . 11 |
66 | 60, 65 | rspc2v 2802 | . . . . . . . . . 10 |
67 | 48, 55, 66 | syl2anc 408 | . . . . . . . . 9 |
68 | 45, 67 | mpd 13 | . . . . . . . 8 |
69 | 2fveq3 5426 | . . . . . . . . . . 11 | |
70 | fveq2 5421 | . . . . . . . . . . 11 | |
71 | 69, 70 | eqeq12d 2154 | . . . . . . . . . 10 |
72 | 22 | adantr 274 | . . . . . . . . . 10 |
73 | 71, 72, 54 | rspcdva 2794 | . . . . . . . . 9 |
74 | 73 | oveq2d 5790 | . . . . . . . 8 |
75 | 42, 68, 74 | 3eqtrd 2176 | . . . . . . 7 |
76 | 32 | adantlr 468 | . . . . . . . 8 |
77 | 33 | adantlr 468 | . . . . . . . 8 |
78 | 38, 76, 77 | seq3p1 10235 | . . . . . . 7 |
79 | 75, 78 | eqeq12d 2154 | . . . . . 6 |
80 | 37, 79 | syl5ibr 155 | . . . . 5 |
81 | 80 | expcom 115 | . . . 4 |
82 | 81 | a2d 26 | . . 3 |
83 | 5, 9, 13, 17, 36, 82 | uzind4 9383 | . 2 |
84 | 1, 83 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2416 cfv 5123 (class class class)co 5774 c1 7621 caddc 7623 cz 9054 cuz 9326 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-seqfrec 10219 |
This theorem is referenced by: seqfeq3 10285 seq3distr 10286 efcj 11379 |
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