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Mirrors > Home > MPE Home > Th. List > lmres | Structured version Visualization version Unicode version |
Description: A function converges iff its restriction to an upper integers set converges. (Contributed by Mario Carneiro, 31-Dec-2013.) |
Ref | Expression |
---|---|
lmres.2 |
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lmres.4 |
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lmres.5 |
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Ref | Expression |
---|---|
lmres |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmres.2 |
. . . . . . 7
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2 | toponmax 19998 |
. . . . . . 7
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3 | 1, 2 | syl 17 |
. . . . . 6
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4 | cnex 9651 |
. . . . . 6
![]() ![]() ![]() ![]() | |
5 | ssid 3463 |
. . . . . . 7
![]() ![]() ![]() ![]() | |
6 | uzssz 11212 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | zsscn 10979 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
8 | 6, 7 | sstri 3453 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | pmss12g 7529 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 5, 8, 9 | mpanl12 693 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 3, 4, 10 | sylancl 673 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | fvex 5902 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | lmres.4 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | pmresg 7530 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 12, 13, 14 | sylancr 674 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 11, 15 | sseldd 3445 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16, 13 | 2thd 248 |
. . 3
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18 | eqid 2462 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 18 | uztrn2 11210 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | dmres 5147 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 20 | elin2 3633 |
. . . . . . . . . . 11
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22 | 21 | baib 919 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | fvres 5906 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 23 | eleq1d 2524 |
. . . . . . . . . 10
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25 | 22, 24 | anbi12d 722 |
. . . . . . . . 9
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26 | 19, 25 | syl 17 |
. . . . . . . 8
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27 | 26 | ralbidva 2836 |
. . . . . . 7
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28 | 27 | rexbiia 2900 |
. . . . . 6
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29 | 28 | imbi2i 318 |
. . . . 5
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30 | 29 | ralbii 2831 |
. . . 4
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31 | 30 | a1i 11 |
. . 3
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32 | 17, 31 | 3anbi13d 1350 |
. 2
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33 | lmres.5 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
34 | 1, 18, 33 | lmbr2 20330 |
. 2
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35 | 1, 18, 33 | lmbr2 20330 |
. 2
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36 | 32, 34, 35 | 3bitr4rd 294 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4541 ax-nul 4550 ax-pow 4598 ax-pr 4656 ax-un 6615 ax-cnex 9626 ax-resscn 9627 ax-pre-lttri 9644 ax-pre-lttrn 9645 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-nel 2636 df-ral 2754 df-rex 2755 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-iun 4294 df-br 4419 df-opab 4478 df-mpt 4479 df-id 4771 df-po 4777 df-so 4778 df-xp 4862 df-rel 4863 df-cnv 4864 df-co 4865 df-dm 4866 df-rn 4867 df-res 4868 df-ima 4869 df-iota 5569 df-fun 5607 df-fn 5608 df-f 5609 df-f1 5610 df-fo 5611 df-f1o 5612 df-fv 5613 df-ov 6323 df-oprab 6324 df-mpt2 6325 df-1st 6825 df-2nd 6826 df-er 7394 df-pm 7506 df-en 7601 df-dom 7602 df-sdom 7603 df-pnf 9708 df-mnf 9709 df-xr 9710 df-ltxr 9711 df-le 9712 df-neg 9894 df-z 10972 df-uz 11194 df-top 19976 df-topon 19978 df-lm 20300 |
This theorem is referenced by: esumcvg 28958 |
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