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Mirrors > Home > ILE Home > Th. List > ser0f | Unicode version |
Description: A zero-valued infinite series is equal to the constant zero function. (Contributed by Mario Carneiro, 8-Feb-2014.) |
Ref | Expression |
---|---|
ser0.1 |
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Ref | Expression |
---|---|
ser0f |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 |
. . 3
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2 | ser0.1 |
. . . . . 6
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3 | 2 | eleq2i 2155 |
. . . . 5
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4 | c0ex 7536 |
. . . . . . 7
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5 | 4 | fvconst2 5527 |
. . . . . 6
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6 | 0cnd 7535 |
. . . . . 6
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7 | 5, 6 | eqeltrd 2165 |
. . . . 5
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8 | 3, 7 | sylbir 134 |
. . . 4
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9 | 8 | adantl 272 |
. . 3
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10 | 1, 9 | iseqseq3 9956 |
. 2
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11 | 2 | iser0 10001 |
. . . . 5
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12 | 11, 5 | eqtr4d 2124 |
. . . 4
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13 | 12 | rgen 2429 |
. . 3
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14 | eqid 2089 |
. . . . . . 7
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15 | 14, 1, 9 | iserf 9957 |
. . . . . 6
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16 | 15 | ffnd 5175 |
. . . . 5
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17 | 2 | fneq2i 5122 |
. . . . 5
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18 | 16, 17 | sylibr 133 |
. . . 4
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19 | 4 | fconst 5219 |
. . . . 5
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20 | ffn 5174 |
. . . . 5
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21 | 19, 20 | ax-mp 7 |
. . . 4
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22 | eqfnfv 5411 |
. . . 4
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23 | 18, 21, 22 | sylancl 405 |
. . 3
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24 | 13, 23 | mpbiri 167 |
. 2
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25 | 10, 24 | eqtr3d 2123 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-addcom 7499 ax-addass 7501 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-0id 7507 ax-rnegex 7508 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-ltadd 7515 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-frec 6170 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-inn 8477 df-n0 8728 df-z 8805 df-uz 9074 df-fz 9479 df-fzo 9608 df-iseq 9907 df-seq3 9908 |
This theorem is referenced by: serclim0 10747 |
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