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Mirrors > Home > ILE Home > Th. List > seq3feq2 | Unicode version |
Description: Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) |
Ref | Expression |
---|---|
seq3fveq2.1 |
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seq3fveq2.2 |
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seq3fveq2.f |
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seq3fveq2.g |
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seq3fveq2.pl |
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seq3feq2.4 |
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Ref | Expression |
---|---|
seq3feq2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2113 |
. . . . 5
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2 | seq3fveq2.1 |
. . . . . 6
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3 | eluzel2 9227 |
. . . . . 6
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4 | 2, 3 | syl 14 |
. . . . 5
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5 | seq3fveq2.f |
. . . . 5
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6 | seq3fveq2.pl |
. . . . 5
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7 | 1, 4, 5, 6 | seqf 10121 |
. . . 4
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8 | 7 | ffnd 5229 |
. . 3
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9 | uzss 9242 |
. . . 4
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10 | 2, 9 | syl 14 |
. . 3
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11 | fnssres 5192 |
. . 3
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12 | 8, 10, 11 | syl2anc 406 |
. 2
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13 | eqid 2113 |
. . . 4
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14 | eluzelz 9231 |
. . . . 5
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15 | 2, 14 | syl 14 |
. . . 4
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16 | seq3fveq2.g |
. . . 4
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17 | 13, 15, 16, 6 | seqf 10121 |
. . 3
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18 | 17 | ffnd 5229 |
. 2
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19 | fvres 5397 |
. . . 4
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20 | 19 | adantl 273 |
. . 3
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21 | 2 | adantr 272 |
. . . 4
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22 | seq3fveq2.2 |
. . . . 5
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23 | 22 | adantr 272 |
. . . 4
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24 | 5 | adantlr 466 |
. . . 4
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25 | 16 | adantlr 466 |
. . . 4
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26 | 6 | adantlr 466 |
. . . 4
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27 | simpr 109 |
. . . 4
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28 | elfzuz 9689 |
. . . . . 6
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29 | seq3feq2.4 |
. . . . . 6
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30 | 28, 29 | sylan2 282 |
. . . . 5
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31 | 30 | adantlr 466 |
. . . 4
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32 | 21, 23, 24, 25, 26, 27, 31 | seq3fveq2 10129 |
. . 3
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33 | 20, 32 | eqtrd 2145 |
. 2
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34 | 12, 18, 33 | eqfnfvd 5473 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-iinf 4460 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-addcom 7639 ax-addass 7641 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-0id 7647 ax-rnegex 7648 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-ltadd 7655 |
This theorem depends on definitions: df-bi 116 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-id 4173 df-iord 4246 df-on 4248 df-ilim 4249 df-suc 4251 df-iom 4463 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-recs 6154 df-frec 6240 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-inn 8625 df-n0 8876 df-z 8953 df-uz 9223 df-fz 9678 df-seqfrec 10106 |
This theorem is referenced by: seq3id 10168 |
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