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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 6905 | . . . 4 β’ π΅ β V |
9 | 8 | a1i 11 | . . 3 β’ (π β π΅ β V) |
10 | fex2 7923 | . . 3 β’ ((πΉ:π΄βΆπ΅ β§ π΄ β π β§ π΅ β V) β πΉ β V) | |
11 | 6, 7, 9, 10 | syl3anc 1371 | . 2 β’ (π β πΉ β V) |
12 | 6 | fdmd 6728 | . 2 β’ (π β dom πΉ = π΄) |
13 | 1, 2, 3, 4, 5, 11, 12 | tsmsval2 23633 | 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 1541 β wcel 2106 {crab 3432 Vcvv 3474 β© cin 3947 β wss 3948 π« cpw 4602 β¦ cmpt 5231 ran crn 5677 βΎ cres 5678 βΆwf 6539 βcfv 6543 (class class class)co 7408 Fincfn 8938 Basecbs 17143 TopOpenctopn 17366 Ξ£g cgsu 17385 filGencfg 20932 fLimf cflf 23438 tsums ctsu 23629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-tsms 23630 |
This theorem is referenced by: eltsms 23636 haustsms 23639 tsmscls 23641 tsmsmhm 23649 tsmsadd 23650 |
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