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Mirrors > Home > ILE Home > Th. List > iseqcl | Unicode version |
Description: Closure property of the
recursive sequence builder.
New proofs should use seqf 9880 or seq3clss 9887 instead (together with iseqsst 9886 or iseqseq3 9902 if need be). (Contributed by Jim Kingdon, 1-Jun-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
iseqcl.1 |
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iseqcl.2 |
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iseqcl.3 |
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Ref | Expression |
---|---|
iseqcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqcl.1 |
. . . 4
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2 | eluzel2 9024 |
. . . 4
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3 | 1, 2 | syl 14 |
. . 3
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4 | eqid 2088 |
. . 3
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5 | fveq2 5305 |
. . . . 5
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6 | 5 | eleq1d 2156 |
. . . 4
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7 | iseqcl.2 |
. . . . 5
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8 | 7 | ralrimiva 2446 |
. . . 4
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9 | uzid 9033 |
. . . . 5
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10 | 3, 9 | syl 14 |
. . . 4
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11 | 6, 8, 10 | rspcdva 2727 |
. . 3
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12 | iseqcl.3 |
. . . 4
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13 | 7, 12 | iseqovex 9870 |
. . 3
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14 | eqid 2088 |
. . 3
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15 | 14, 7, 12 | iseqval 9871 |
. . 3
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16 | 3, 4, 11, 13, 14, 15 | frecuzrdgtcl 9819 |
. 2
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17 | 16, 1 | ffvelrnd 5435 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 ax-cnex 7436 ax-resscn 7437 ax-1cn 7438 ax-1re 7439 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-addcom 7445 ax-addass 7447 ax-distr 7449 ax-i2m1 7450 ax-0lt1 7451 ax-0id 7453 ax-rnegex 7454 ax-cnre 7456 ax-pre-ltirr 7457 ax-pre-ltwlin 7458 ax-pre-lttrn 7459 ax-pre-ltadd 7461 |
This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-id 4120 df-iord 4193 df-on 4195 df-ilim 4196 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-frec 6156 df-pnf 7524 df-mnf 7525 df-xr 7526 df-ltxr 7527 df-le 7528 df-sub 7655 df-neg 7656 df-inn 8423 df-n0 8674 df-z 8751 df-uz 9020 df-iseq 9853 |
This theorem is referenced by: iseqp1 9882 iseqoveq 9885 isermono 9906 iseqsplit 9908 iseqcaopr2 9911 iseqid3 9937 iseqhomo 9942 iseqz 9943 iseqdistr 9945 ibcval5 10171 iseqcoll 10247 fisum 10778 ialgrp1 11306 |
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