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Mirrors > Home > ILE Home > Th. List > iser3shft | Unicode version |
Description: Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Jim Kingdon, 17-Oct-2022.) |
Ref | Expression |
---|---|
iser3shft.ex |
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iser3shft.m |
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iser3shft.n |
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iser3shft.fm |
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iser3shft.pl |
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Ref | Expression |
---|---|
iser3shft |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iser3shft.ex |
. . . . 5
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2 | iser3shft.m |
. . . . . 6
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3 | iser3shft.n |
. . . . . 6
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4 | 2, 3 | zaddcld 9398 |
. . . . 5
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5 | 2 | zcnd 9395 |
. . . . . . . . . 10
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6 | 3 | zcnd 9395 |
. . . . . . . . . 10
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7 | 5, 6 | pncand 8288 |
. . . . . . . . 9
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8 | 7 | fveq2d 5534 |
. . . . . . . 8
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9 | 8 | eleq2d 2259 |
. . . . . . 7
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10 | 9 | pm5.32i 454 |
. . . . . 6
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11 | iser3shft.fm |
. . . . . 6
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12 | 10, 11 | sylbi 121 |
. . . . 5
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13 | iser3shft.pl |
. . . . 5
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14 | 1, 4, 3, 12, 13 | seq3shft 10866 |
. . . 4
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15 | 7 | seqeq1d 10470 |
. . . . 5
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16 | 15 | oveq1d 5906 |
. . . 4
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17 | 14, 16 | eqtrd 2222 |
. . 3
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18 | 17 | breq1d 4028 |
. 2
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19 | seqex 10466 |
. . 3
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20 | climshft 11331 |
. . 3
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21 | 3, 19, 20 | sylancl 413 |
. 2
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22 | 18, 21 | bitr2d 189 |
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 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-addcom 7930 ax-addass 7932 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-0id 7938 ax-rnegex 7939 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 |
This theorem depends on definitions: df-bi 117 df-dc 836 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-if 3550 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 4308 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-frec 6410 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-inn 8939 df-n0 9196 df-z 9273 df-uz 9548 df-fz 10028 df-seqfrec 10465 df-shft 10843 df-clim 11306 |
This theorem is referenced by: isumshft 11517 |
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