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Mirrors > Home > ILE Home > Th. List > climcncf | Unicode version |
Description: Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.) |
Ref | Expression |
---|---|
climcncf.1 |
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climcncf.2 |
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climcncf.4 |
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climcncf.5 |
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climcncf.6 |
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climcncf.7 |
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Ref | Expression |
---|---|
climcncf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climcncf.1 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | climcncf.2 |
. 2
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3 | climcncf.7 |
. 2
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4 | climcncf.4 |
. . . . 5
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5 | cncff 11899 |
. . . . 5
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6 | 4, 5 | syl 14 |
. . . 4
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7 | 6 | ffvelrnda 5448 |
. . 3
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8 | cncfrss2 11898 |
. . . . 5
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9 | 4, 8 | syl 14 |
. . . 4
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10 | 9 | sselda 3026 |
. . 3
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11 | 7, 10 | syldan 277 |
. 2
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12 | climcncf.6 |
. 2
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13 | climcncf.5 |
. . . 4
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14 | zex 8813 |
. . . . . 6
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15 | uzssz 9092 |
. . . . . 6
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16 | 14, 15 | ssexi 3983 |
. . . . 5
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17 | 1, 16 | eqeltri 2161 |
. . . 4
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18 | fex 5538 |
. . . 4
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19 | 13, 17, 18 | sylancl 405 |
. . 3
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20 | coexg 4988 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 4, 19, 20 | syl2anc 404 |
. 2
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22 | cncfi 11900 |
. . . . 5
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23 | 22 | 3expia 1146 |
. . . 4
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24 | 4, 3, 23 | syl2anc 404 |
. . 3
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25 | 24 | imp 123 |
. 2
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26 | 13 | ffvelrnda 5448 |
. 2
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27 | fvco3 5388 |
. . 3
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28 | 13, 27 | sylan 278 |
. 2
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29 | 1, 2, 3, 11, 12, 21, 25, 26, 28 | climcn1 10751 |
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-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 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-mulrcl 7498 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-mulass 7502 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-1rid 7506 ax-0id 7507 ax-rnegex 7508 ax-precex 7509 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-apti 7514 ax-pre-ltadd 7515 ax-pre-mulgt0 7516 ax-pre-mulext 7517 |
This theorem depends on definitions: df-bi 116 df-dc 782 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-rmo 2368 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-if 3398 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-id 4129 df-po 4132 df-iso 4133 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-map 6421 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-reap 8106 df-ap 8113 df-div 8194 df-inn 8477 df-2 8535 df-n0 8728 df-z 8805 df-uz 9074 df-cj 10330 df-re 10331 df-im 10332 df-rsqrt 10485 df-abs 10486 df-clim 10721 df-cncf 11893 |
This theorem is referenced by: (None) |
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