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Mirrors > Home > ILE Home > Th. List > iseqp1 | Unicode version |
Description: Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.) |
Ref | Expression |
---|---|
iseqp1.m |
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iseqp1.f |
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iseqp1.pl |
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Ref | Expression |
---|---|
iseqp1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqp1.m |
. . 3
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2 | eluzel2 8919 |
. . . . 5
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3 | 1, 2 | syl 14 |
. . . 4
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4 | eqid 2083 |
. . . 4
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5 | fveq2 5253 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 5 | eleq1d 2151 |
. . . . 5
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7 | iseqp1.f |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 7 | ralrimiva 2440 |
. . . . 5
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9 | uzid 8928 |
. . . . . 6
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10 | 3, 9 | syl 14 |
. . . . 5
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11 | 6, 8, 10 | rspcdva 2717 |
. . . 4
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12 | iseqp1.pl |
. . . . 5
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13 | 7, 12 | iseqovex 9748 |
. . . 4
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14 | eqid 2083 |
. . . 4
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15 | 14, 7, 12 | iseqval 9749 |
. . . 4
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16 | 3, 4, 11, 13, 14, 15 | frecuzrdgsuc 9710 |
. . 3
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17 | 1, 16 | mpdan 412 |
. 2
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18 | 1, 7, 12 | iseqcl 9756 |
. . 3
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19 | fveq2 5253 |
. . . . . 6
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20 | 19 | eleq1d 2151 |
. . . . 5
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21 | peano2uz 8966 |
. . . . . 6
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22 | 1, 21 | syl 14 |
. . . . 5
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23 | 20, 8, 22 | rspcdva 2717 |
. . . 4
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24 | 12, 18, 23 | caovcld 5733 |
. . 3
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25 | oveq1 5598 |
. . . . . 6
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26 | 25 | fveq2d 5257 |
. . . . 5
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27 | 26 | oveq2d 5607 |
. . . 4
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28 | oveq1 5598 |
. . . 4
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29 | eqid 2083 |
. . . 4
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30 | 27, 28, 29 | ovmpt2g 5714 |
. . 3
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31 | 1, 18, 24, 30 | syl3anc 1170 |
. 2
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32 | 17, 31 | eqtrd 2115 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-addcom 7348 ax-addass 7350 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-0id 7356 ax-rnegex 7357 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-ltadd 7364 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4084 df-iord 4157 df-on 4159 df-ilim 4160 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-1st 5846 df-2nd 5847 df-recs 6002 df-frec 6088 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-inn 8317 df-n0 8566 df-z 8647 df-uz 8915 df-iseq 9741 |
This theorem is referenced by: iseqoveq 9759 iseqss 9760 iseqsst 9761 iseqm1 9762 iseqfveq2 9763 iseqshft2 9767 isermono 9772 iseqsplit 9773 iseqcaopr3 9775 iseqid3s 9781 iseqid2 9783 iseqhomo 9784 iseqz 9785 expivallem 9793 expp1 9799 facp1 9973 resqrexlemfp1 10269 climserile 10557 ialgrp1 10808 |
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