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Mirrors > Home > MPE Home > Th. List > tsmsid | Unicode version |
Description: If a sum is finite, the usual sum is always a limit point of the topological sum (although it may not be the only limit point). (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tsmsid.b |
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tsmsid.z |
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tsmsid.1 |
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tsmsid.2 |
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tsmsid.a |
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tsmsid.f |
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tsmsid.w |
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Ref | Expression |
---|---|
tsmsid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsmsid.2 |
. . . . . 6
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2 | tsmsid.b |
. . . . . . 7
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3 | eqid 2412 |
. . . . . . 7
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4 | 2, 3 | istps 16964 |
. . . . . 6
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5 | 1, 4 | sylib 189 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | topontop 16954 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | syl 16 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | tsmsid.z |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | tsmsid.1 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | tsmsid.a |
. . . . . . 7
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11 | tsmsid.f |
. . . . . . 7
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12 | tsmsid.w |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 2, 8, 9, 10, 11, 12 | gsumcl 15484 |
. . . . . 6
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14 | 13 | snssd 3911 |
. . . . 5
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15 | toponuni 16955 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 5, 15 | syl 16 |
. . . . 5
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17 | 14, 16 | sseqtrd 3352 |
. . . 4
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18 | eqid 2412 |
. . . . 5
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19 | 18 | sscls 17083 |
. . . 4
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20 | 7, 17, 19 | syl2anc 643 |
. . 3
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21 | ovex 6073 |
. . . 4
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22 | 21 | snss 3894 |
. . 3
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23 | 20, 22 | sylibr 204 |
. 2
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24 | 2, 8, 9, 1, 10, 11, 12, 3 | tsmsgsum 18129 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 23, 24 | eleqtrrd 2489 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: haustsmsid 18131 tsms0 18132 tayl0 20239 |
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-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 |
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-1st 6316 df-2nd 6317 df-riota 6516 df-recs 6600 df-rdg 6635 df-1o 6691 df-oadd 6695 df-er 6872 df-map 6987 df-en 7077 df-dom 7078 df-sdom 7079 df-fin 7080 df-oi 7443 df-card 7790 df-pnf 9086 df-mnf 9087 df-xr 9088 df-ltxr 9089 df-le 9090 df-sub 9257 df-neg 9258 df-nn 9965 df-n0 10186 df-z 10247 df-uz 10453 df-fz 11008 df-fzo 11099 df-seq 11287 df-hash 11582 df-0g 13690 df-gsum 13691 df-mnd 14653 df-cntz 15079 df-cmn 15377 df-fbas 16662 df-fg 16663 df-top 16926 df-topon 16929 df-topsp 16930 df-cld 17046 df-ntr 17047 df-cls 17048 df-nei 17125 df-fil 17839 df-fm 17931 df-flim 17932 df-flf 17933 df-tsms 18117 |
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