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Mirrors > Home > MPE Home > Th. List > eltsms | Structured version Visualization version Unicode version |
Description: The property of being a sum of the sequence in the topological commutative monoid . (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
eltsms.b | |
eltsms.j | |
eltsms.s | |
eltsms.1 | CMnd |
eltsms.2 | |
eltsms.a | |
eltsms.f |
Ref | Expression |
---|---|
eltsms | tsums g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eltsms.b | . . . 4 | |
2 | eltsms.j | . . . 4 | |
3 | eltsms.s | . . . 4 | |
4 | eqid 2622 | . . . 4 | |
5 | eltsms.1 | . . . 4 CMnd | |
6 | eltsms.a | . . . 4 | |
7 | eltsms.f | . . . 4 | |
8 | 1, 2, 3, 4, 5, 6, 7 | tsmsval 21934 | . . 3 tsums g |
9 | 8 | eleq2d 2687 | . 2 tsums g |
10 | eltsms.2 | . . . 4 | |
11 | 1, 2 | istps 20738 | . . . 4 TopOn |
12 | 10, 11 | sylib 208 | . . 3 TopOn |
13 | eqid 2622 | . . . 4 | |
14 | 3, 13, 4, 6 | tsmsfbas 21931 | . . 3 |
15 | 1, 3, 5, 6, 7 | tsmslem1 21932 | . . . 4 g |
16 | eqid 2622 | . . . 4 g g | |
17 | 15, 16 | fmptd 6385 | . . 3 g |
18 | eqid 2622 | . . . 4 | |
19 | 18 | flffbas 21799 | . . 3 TopOn g g g |
20 | 12, 14, 17, 19 | syl3anc 1326 | . 2 g g |
21 | pwexg 4850 | . . . . . . . . . . . 12 | |
22 | inex1g 4801 | . . . . . . . . . . . 12 | |
23 | 6, 21, 22 | 3syl 18 | . . . . . . . . . . 11 |
24 | 3, 23 | syl5eqel 2705 | . . . . . . . . . 10 |
25 | 24 | adantr 481 | . . . . . . . . 9 |
26 | rabexg 4812 | . . . . . . . . 9 | |
27 | 25, 26 | syl 17 | . . . . . . . 8 |
28 | 27 | ralrimivw 2967 | . . . . . . 7 |
29 | imaeq2 5462 | . . . . . . . . 9 g g | |
30 | 29 | sseq1d 3632 | . . . . . . . 8 g g |
31 | 13, 30 | rexrnmpt 6369 | . . . . . . 7 g g |
32 | 28, 31 | syl 17 | . . . . . 6 g g |
33 | funmpt 5926 | . . . . . . . . 9 g | |
34 | ssrab2 3687 | . . . . . . . . . 10 | |
35 | ovex 6678 | . . . . . . . . . . 11 g | |
36 | 35, 16 | dmmpti 6023 | . . . . . . . . . 10 g |
37 | 34, 36 | sseqtr4i 3638 | . . . . . . . . 9 g |
38 | funimass3 6333 | . . . . . . . . 9 g g g g | |
39 | 33, 37, 38 | mp2an 708 | . . . . . . . 8 g g |
40 | 16 | mptpreima 5628 | . . . . . . . . 9 g g |
41 | 40 | sseq2i 3630 | . . . . . . . 8 g g |
42 | ss2rab 3678 | . . . . . . . 8 g g | |
43 | 39, 41, 42 | 3bitri 286 | . . . . . . 7 g g |
44 | 43 | rexbii 3041 | . . . . . 6 g g |
45 | 32, 44 | syl6bb 276 | . . . . 5 g g |
46 | 45 | imbi2d 330 | . . . 4 g g |
47 | 46 | ralbidva 2985 | . . 3 g g |
48 | 47 | anbi2d 740 | . 2 g g |
49 | 9, 20, 48 | 3bitrd 294 | 1 tsums g |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 wrex 2913 crab 2916 cvv 3200 cin 3573 wss 3574 cpw 4158 cmpt 4729 ccnv 5113 cdm 5114 crn 5115 cres 5116 cima 5117 wfun 5882 wf 5884 cfv 5888 (class class class)co 6650 cfn 7955 cbs 15857 ctopn 16082 g cgsu 16101 CMndccmn 18193 cfbas 19734 cfg 19735 TopOnctopon 20715 ctps 20736 cflf 21739 tsums ctsu 21929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-0g 16102 df-gsum 16103 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-cntz 17750 df-cmn 18195 df-fbas 19743 df-fg 19744 df-top 20699 df-topon 20716 df-topsp 20737 df-ntr 20824 df-nei 20902 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tsms 21930 |
This theorem is referenced by: tsmsi 21937 tsmscl 21938 tsmsgsum 21942 tsmssubm 21946 tsmsres 21947 tsmsf1o 21948 tsmsxp 21958 xrge0tsms 22637 xrge0tsmsd 29785 |
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