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Mirrors > Home > MPE Home > Th. List > tsmsval | Structured version Visualization version GIF version |
Description: Definition of the topological group sum(s) of a collection πΉ(π₯) of values in the group with index set π΄. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tsmsval.b | β’ π΅ = (BaseβπΊ) |
tsmsval.j | β’ π½ = (TopOpenβπΊ) |
tsmsval.s | β’ π = (π« π΄ β© Fin) |
tsmsval.l | β’ πΏ = ran (π§ β π β¦ {π¦ β π β£ π§ β π¦}) |
tsmsval.g | β’ (π β πΊ β π) |
tsmsval.a | β’ (π β π΄ β π) |
tsmsval.f | β’ (π β πΉ:π΄βΆπ΅) |
Ref | Expression |
---|---|
tsmsval | β’ (π β (πΊ tsums πΉ) = ((π½ fLimf (πfilGenπΏ))β(π¦ β π β¦ (πΊ Ξ£g (πΉ βΎ π¦))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsmsval.b | . 2 β’ π΅ = (BaseβπΊ) | |
2 | tsmsval.j | . 2 β’ π½ = (TopOpenβπΊ) | |
3 | tsmsval.s | . 2 β’ π = (π« π΄ β© Fin) | |
4 | tsmsval.l | . 2 β’ πΏ = ran (π§ β π β¦ {π¦ β π β£ π§ β π¦}) | |
5 | tsmsval.g | . 2 β’ (π β πΊ β π) | |
6 | tsmsval.f | . . 3 β’ (π β πΉ:π΄βΆπ΅) | |
7 | tsmsval.a | . . 3 β’ (π β π΄ β π) | |
8 | 1 | fvexi 6904 | . . . 4 β’ π΅ β V |
9 | 8 | a1i 11 | . . 3 β’ (π β π΅ β V) |
10 | fex2 7926 | . . 3 β’ ((πΉ:π΄βΆπ΅ β§ π΄ β π β§ π΅ β V) β πΉ β V) | |
11 | 6, 7, 9, 10 | syl3anc 1369 | . 2 β’ (π β πΉ β V) |
12 | 6 | fdmd 6727 | . 2 β’ (π β dom πΉ = π΄) |
13 | 1, 2, 3, 4, 5, 11, 12 | tsmsval2 23854 | 1 β’ (π β (πΊ tsums πΉ) = ((π½ fLimf (πfilGenπΏ))β(π¦ β π β¦ (πΊ Ξ£g (πΉ βΎ π¦))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 {crab 3430 Vcvv 3472 β© cin 3946 β wss 3947 π« cpw 4601 β¦ cmpt 5230 ran crn 5676 βΎ cres 5677 βΆwf 6538 βcfv 6542 (class class class)co 7411 Fincfn 8941 Basecbs 17148 TopOpenctopn 17371 Ξ£g cgsu 17390 filGencfg 21133 fLimf cflf 23659 tsums ctsu 23850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-tsms 23851 |
This theorem is referenced by: eltsms 23857 haustsms 23860 tsmscls 23862 tsmsmhm 23870 tsmsadd 23871 |
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