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Mirrors > Home > ILE Home > Th. List > climsub | Unicode version |
Description: Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.) |
Ref | Expression |
---|---|
climadd.1 |
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climadd.2 |
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climadd.4 |
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climadd.6 |
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climadd.7 |
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climadd.8 |
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climadd.9 |
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climsub.h |
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Ref | Expression |
---|---|
climsub |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climadd.1 |
. 2
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2 | climadd.2 |
. 2
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3 | climadd.4 |
. . 3
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4 | climcl 10936 |
. . 3
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5 | 3, 4 | syl 14 |
. 2
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6 | climadd.7 |
. . 3
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7 | climcl 10936 |
. . 3
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8 | 6, 7 | syl 14 |
. 2
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9 | subcl 7877 |
. . 3
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10 | 9 | adantl 273 |
. 2
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11 | climadd.6 |
. 2
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12 | simpr 109 |
. . 3
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13 | 5 | adantr 272 |
. . 3
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14 | 8 | adantr 272 |
. . 3
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15 | subcn2 10965 |
. . 3
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16 | 12, 13, 14, 15 | syl3anc 1197 |
. 2
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17 | climadd.8 |
. 2
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18 | climadd.9 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | climsub.h |
. 2
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20 | 1, 2, 5, 8, 10, 3, 6, 11, 16, 17, 18, 19 | climcn2 10963 |
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 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-iinf 4460 ax-cnex 7629 ax-resscn 7630 ax-1cn 7631 ax-1re 7632 ax-icn 7633 ax-addcl 7634 ax-addrcl 7635 ax-mulcl 7636 ax-mulrcl 7637 ax-addcom 7638 ax-mulcom 7639 ax-addass 7640 ax-mulass 7641 ax-distr 7642 ax-i2m1 7643 ax-0lt1 7644 ax-1rid 7645 ax-0id 7646 ax-rnegex 7647 ax-precex 7648 ax-cnre 7649 ax-pre-ltirr 7650 ax-pre-ltwlin 7651 ax-pre-lttrn 7652 ax-pre-apti 7653 ax-pre-ltadd 7654 ax-pre-mulgt0 7655 ax-pre-mulext 7656 ax-arch 7657 ax-caucvg 7658 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rmo 2396 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-if 3439 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-id 4173 df-po 4176 df-iso 4177 df-iord 4246 df-on 4248 df-ilim 4249 df-suc 4251 df-iom 4463 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5989 df-2nd 5990 df-recs 6153 df-frec 6239 df-pnf 7719 df-mnf 7720 df-xr 7721 df-ltxr 7722 df-le 7723 df-sub 7851 df-neg 7852 df-reap 8248 df-ap 8255 df-div 8339 df-inn 8624 df-2 8682 df-3 8683 df-4 8684 df-n0 8875 df-z 8952 df-uz 9222 df-rp 9337 df-seqfrec 10105 df-exp 10179 df-cj 10500 df-re 10501 df-im 10502 df-rsqrt 10655 df-abs 10656 df-clim 10933 |
This theorem is referenced by: climsubc1 10986 climsubc2 10987 climle 10988 |
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