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Mirrors > Home > ILE Home > Th. List > seq3p1 | Unicode version |
Description: Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.) |
Ref | Expression |
---|---|
seq3p1.m |
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seq3p1.f |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
seq3p1.pl |
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Ref | Expression |
---|---|
seq3p1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seq3p1.m |
. . 3
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2 | eluzel2 9564 |
. . . . 5
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3 | 1, 2 | syl 14 |
. . . 4
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4 | fveq2 5534 |
. . . . . 6
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5 | 4 | eleq1d 2258 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | seq3p1.f |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 6 | ralrimiva 2563 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | uzid 9573 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 3, 8 | syl 14 |
. . . . 5
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10 | 5, 7, 9 | rspcdva 2861 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | ssv 3192 |
. . . . 5
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12 | 11 | a1i 9 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | seq3p1.pl |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 6, 13 | iseqovex 10489 |
. . . 4
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15 | iseqvalcbv 10490 |
. . . 4
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16 | 3, 15, 6, 13 | seq3val 10491 |
. . . 4
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17 | 3, 10, 12, 14, 15, 16 | frecuzrdgsuct 10457 |
. . 3
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18 | 1, 17 | mpdan 421 |
. 2
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19 | eqid 2189 |
. . . . 5
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20 | 19, 3, 6, 13 | seqf 10494 |
. . . 4
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21 | 20, 1 | ffvelcdmd 5673 |
. . 3
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22 | fveq2 5534 |
. . . . . 6
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23 | 22 | eleq1d 2258 |
. . . . 5
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24 | peano2uz 9615 |
. . . . . 6
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25 | 1, 24 | syl 14 |
. . . . 5
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26 | 23, 7, 25 | rspcdva 2861 |
. . . 4
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27 | 13, 21, 26 | caovcld 6051 |
. . 3
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28 | fvoveq1 5920 |
. . . . 5
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29 | 28 | oveq2d 5913 |
. . . 4
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30 | oveq1 5904 |
. . . 4
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31 | eqid 2189 |
. . . 4
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32 | 29, 30, 31 | ovmpog 6032 |
. . 3
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33 | 1, 21, 27, 32 | syl3anc 1249 |
. 2
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34 | 18, 33 | eqtrd 2222 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-ltadd 7958 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-frec 6417 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-inn 8951 df-n0 9208 df-z 9285 df-uz 9560 df-seqfrec 10479 |
This theorem is referenced by: seq3clss 10500 seq3m1 10501 seq3fveq2 10502 seq3shft2 10506 ser3mono 10511 seq3split 10512 seq3caopr3 10514 seq3id3 10540 seq3id2 10542 seq3homo 10543 seq3z 10544 seqfeq4g 10546 ser3ge0 10551 exp3vallem 10555 expp1 10561 facp1 10745 seq3coll 10857 resqrexlemfp1 11053 climserle 11388 clim2prod 11582 prodfap0 11588 prodfrecap 11589 ege2le3 11714 efgt1p2 11738 efgt1p 11739 algrp1 12081 pcmpt 12378 nninfdclemp1 12504 gsumsplit1r 12876 gsumprval 12877 mulgnnp1 13087 |
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