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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 6857 | . . . 4 β’ π΅ β V |
9 | 8 | a1i 11 | . . 3 β’ (π β π΅ β V) |
10 | fex2 7871 | . . 3 β’ ((πΉ:π΄βΆπ΅ β§ π΄ β π β§ π΅ β V) β πΉ β V) | |
11 | 6, 7, 9, 10 | syl3anc 1372 | . 2 β’ (π β πΉ β V) |
12 | 6 | fdmd 6680 | . 2 β’ (π β dom πΉ = π΄) |
13 | 1, 2, 3, 4, 5, 11, 12 | tsmsval2 23497 | 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 1542 β wcel 2107 {crab 3406 Vcvv 3444 β© cin 3910 β wss 3911 π« cpw 4561 β¦ cmpt 5189 ran crn 5635 βΎ cres 5636 βΆwf 6493 βcfv 6497 (class class class)co 7358 Fincfn 8886 Basecbs 17088 TopOpenctopn 17308 Ξ£g cgsu 17327 filGencfg 20801 fLimf cflf 23302 tsums ctsu 23493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-tsms 23494 |
This theorem is referenced by: eltsms 23500 haustsms 23503 tsmscls 23505 tsmsmhm 23513 tsmsadd 23514 |
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