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Mirrors > Home > MPE Home > Th. List > divsqrsumlem | Unicode version |
Description: Lemma for divsqrsum 20781 and divsqrsum2 20782. (Contributed by Mario Carneiro, 18-May-2016.) |
Ref | Expression |
---|---|
divsqrsum.2 |
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Ref | Expression |
---|---|
divsqrsumlem |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioorp 10952 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 1 | eqcomi 2416 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | nnuz 10485 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 1z 10275 |
. . . . . 6
![]() ![]() ![]() ![]() | |
5 | 4 | a1i 11 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 0re 9055 |
. . . . . 6
![]() ![]() ![]() ![]() | |
7 | 6 | a1i 11 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 1re 9054 |
. . . . . . 7
![]() ![]() ![]() ![]() | |
9 | 0nn0 10200 |
. . . . . . 7
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10 | 8, 9 | nn0addge2i 10233 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 10 | a1i 11 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 2re 10033 |
. . . . . 6
![]() ![]() ![]() ![]() | |
13 | rpsqrcl 12033 |
. . . . . . . 8
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14 | 13 | adantl 453 |
. . . . . . 7
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15 | 14 | rpred 10612 |
. . . . . 6
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16 | remulcl 9039 |
. . . . . 6
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17 | 12, 15, 16 | sylancr 645 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 14 | rprecred 10623 |
. . . . 5
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19 | nnrp 10585 |
. . . . . 6
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20 | 19, 18 | sylan2 461 |
. . . . 5
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21 | reex 9045 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
22 | 21 | prid1 3880 |
. . . . . . . 8
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23 | 22 | a1i 11 |
. . . . . . 7
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24 | 14 | rpcnd 10614 |
. . . . . . 7
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25 | 2rp 10581 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
26 | rpmulcl 10597 |
. . . . . . . . 9
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27 | 25, 14, 26 | sylancr 645 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 27 | rpreccld 10622 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | dvsqr 20589 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | 29 | a1i 11 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 2cn 10034 |
. . . . . . . 8
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32 | 31 | a1i 11 |
. . . . . . 7
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33 | 23, 24, 28, 30, 32 | dvmptcmul 19811 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 31 | a1i 11 |
. . . . . . . . 9
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35 | ax-1cn 9012 |
. . . . . . . . . 10
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36 | 35 | a1i 11 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 27 | rpcnne0d 10621 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | divass 9660 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
39 | 34, 36, 37, 38 | syl3anc 1184 |
. . . . . . . 8
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40 | 14 | rpcnne0d 10621 |
. . . . . . . . 9
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41 | rpcnne0 10593 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
42 | 25, 41 | mp1i 12 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | divcan5 9680 |
. . . . . . . . 9
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44 | 36, 40, 42, 43 | syl3anc 1184 |
. . . . . . . 8
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45 | 39, 44 | eqtr3d 2446 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | 45 | mpteq2dva 4263 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | 33, 46 | eqtrd 2444 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | fveq2 5695 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
49 | 48 | oveq2d 6064 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | simp3r 986 |
. . . . . . 7
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51 | simp2l 983 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
52 | 51 | rprege0d 10619 |
. . . . . . . 8
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53 | simp2r 984 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
54 | 53 | rprege0d 10619 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
55 | sqrle 12029 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
56 | 52, 54, 55 | syl2anc 643 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
57 | 50, 56 | mpbid 202 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
58 | 51 | rpsqrcld 12177 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
59 | 53 | rpsqrcld 12177 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
60 | 58, 59 | lerecd 10631 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
61 | 57, 60 | mpbid 202 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
62 | divsqrsum.2 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
63 | sqrlim 20772 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
64 | 63 | a1i 11 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
65 | fveq2 5695 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
66 | 65 | oveq2d 6064 |
. . . . 5
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67 | 2, 3, 5, 7, 11, 7, 17, 18, 20, 47, 49, 61, 62, 64, 66 | dvfsumrlim3 19878 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
68 | 67 | simp1d 969 |
. . 3
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69 | 68 | trud 1329 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() |
70 | 67 | simp2d 970 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
71 | 70 | trud 1329 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() |
72 | rpge0 10588 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
73 | 72 | adantl 453 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
74 | 67 | simp3d 971 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
75 | 74 | trud 1329 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
76 | 73, 75 | mpd3an3 1280 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
77 | 69, 71, 76 | 3pm3.2i 1132 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: divsqrsumf 20780 divsqrsum 20781 divsqrsum2 20782 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2393 ax-rep 4288 ax-sep 4298 ax-nul 4306 ax-pow 4345 ax-pr 4371 ax-un 4668 ax-inf2 7560 ax-cnex 9010 ax-resscn 9011 ax-1cn 9012 ax-icn 9013 ax-addcl 9014 ax-addrcl 9015 ax-mulcl 9016 ax-mulrcl 9017 ax-mulcom 9018 ax-addass 9019 ax-mulass 9020 ax-distr 9021 ax-i2m1 9022 ax-1ne0 9023 ax-1rid 9024 ax-rnegex 9025 ax-rrecex 9026 ax-cnre 9027 ax-pre-lttri 9028 ax-pre-lttrn 9029 ax-pre-ltadd 9030 ax-pre-mulgt0 9031 ax-pre-sup 9032 ax-addf 9033 ax-mulf 9034 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2266 df-mo 2267 df-clab 2399 df-cleq 2405 df-clel 2408 df-nfc 2537 df-ne 2577 df-nel 2578 df-ral 2679 df-rex 2680 df-reu 2681 df-rmo 2682 df-rab 2683 df-v 2926 df-sbc 3130 df-csb 3220 df-dif 3291 df-un 3293 df-in 3295 df-ss 3302 df-pss 3304 df-nul 3597 df-if 3708 df-pw 3769 df-sn 3788 df-pr 3789 df-tp 3790 df-op 3791 df-uni 3984 df-int 4019 df-iun 4063 df-iin 4064 df-br 4181 df-opab 4235 df-mpt 4236 df-tr 4271 df-eprel 4462 df-id 4466 df-po 4471 df-so 4472 df-fr 4509 df-se 4510 df-we 4511 df-ord 4552 df-on 4553 df-lim 4554 df-suc 4555 df-om 4813 df-xp 4851 df-rel 4852 df-cnv 4853 df-co 4854 df-dm 4855 df-rn 4856 df-res 4857 df-ima 4858 df-iota 5385 df-fun 5423 df-fn 5424 df-f 5425 df-f1 5426 df-fo 5427 df-f1o 5428 df-fv 5429 df-isom 5430 df-ov 6051 df-oprab 6052 df-mpt2 6053 df-of 6272 df-1st 6316 df-2nd 6317 df-riota 6516 df-recs 6600 df-rdg 6635 df-1o 6691 df-2o 6692 df-oadd 6695 df-er 6872 df-map 6987 df-pm 6988 df-ixp 7031 df-en 7077 df-dom 7078 df-sdom 7079 df-fin 7080 df-fi 7382 df-sup 7412 df-oi 7443 df-card 7790 df-cda 8012 df-pnf 9086 df-mnf 9087 df-xr 9088 df-ltxr 9089 df-le 9090 df-sub 9257 df-neg 9258 df-div 9642 df-nn 9965 df-2 10022 df-3 10023 df-4 10024 df-5 10025 df-6 10026 df-7 10027 df-8 10028 df-9 10029 df-10 10030 df-n0 10186 df-z 10247 df-dec 10347 df-uz 10453 df-q 10539 df-rp 10577 df-xneg 10674 df-xadd 10675 df-xmul 10676 df-ioo 10884 df-ioc 10885 df-ico 10886 df-icc 10887 df-fz 11008 df-fzo 11099 df-fl 11165 df-mod 11214 df-seq 11287 df-exp 11346 df-fac 11530 df-bc 11557 df-hash 11582 df-shft 11845 df-cj 11867 df-re 11868 df-im 11869 df-sqr 12003 df-abs 12004 df-limsup 12228 df-clim 12245 df-rlim 12246 df-sum 12443 df-ef 12633 df-sin 12635 df-cos 12636 df-pi 12638 df-struct 13434 df-ndx 13435 df-slot 13436 df-base 13437 df-sets 13438 df-ress 13439 df-plusg 13505 df-mulr 13506 df-starv 13507 df-sca 13508 df-vsca 13509 df-tset 13511 df-ple 13512 df-ds 13514 df-unif 13515 df-hom 13516 df-cco 13517 df-rest 13613 df-topn 13614 df-topgen 13630 df-pt 13631 df-prds 13634 df-xrs 13689 df-0g 13690 df-gsum 13691 df-qtop 13696 df-imas 13697 df-xps 13699 df-mre 13774 df-mrc 13775 df-acs 13777 df-mnd 14653 df-submnd 14702 df-mulg 14778 df-cntz 15079 df-cmn 15377 df-psmet 16657 df-xmet 16658 df-met 16659 df-bl 16660 df-mopn 16661 df-fbas 16662 df-fg 16663 df-cnfld 16667 df-top 16926 df-bases 16928 df-topon 16929 df-topsp 16930 df-cld 17046 df-ntr 17047 df-cls 17048 df-nei 17125 df-lp 17163 df-perf 17164 df-cn 17253 df-cnp 17254 df-haus 17341 df-cmp 17412 df-tx 17555 df-hmeo 17748 df-fil 17839 df-fm 17931 df-flim 17932 df-flf 17933 df-xms 18311 df-ms 18312 df-tms 18313 df-cncf 18869 df-limc 19714 df-dv 19715 df-log 20415 df-cxp 20416 |
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