![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tsmsid | Structured version Visualization version GIF 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.) (Revised by AV, 24-Jul-2019.) |
Ref | Expression |
---|---|
tsmsid.b | ⊢ 𝐵 = (Base‘𝐺) |
tsmsid.z | ⊢ 0 = (0g‘𝐺) |
tsmsid.1 | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
tsmsid.2 | ⊢ (𝜑 → 𝐺 ∈ TopSp) |
tsmsid.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
tsmsid.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
tsmsid.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Ref | Expression |
---|---|
tsmsid | ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 tsums 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsmsid.2 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TopSp) | |
2 | tsmsid.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2727 | . . . . . . 7 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
4 | 2, 3 | istps 22854 | . . . . . 6 ⊢ (𝐺 ∈ TopSp ↔ (TopOpen‘𝐺) ∈ (TopOn‘𝐵)) |
5 | 1, 4 | sylib 217 | . . . . 5 ⊢ (𝜑 → (TopOpen‘𝐺) ∈ (TopOn‘𝐵)) |
6 | topontop 22833 | . . . . 5 ⊢ ((TopOpen‘𝐺) ∈ (TopOn‘𝐵) → (TopOpen‘𝐺) ∈ Top) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → (TopOpen‘𝐺) ∈ Top) |
8 | tsmsid.z | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
9 | tsmsid.1 | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
10 | tsmsid.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
11 | tsmsid.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
12 | tsmsid.w | . . . . . . 7 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
13 | 2, 8, 9, 10, 11, 12 | gsumcl 19875 | . . . . . 6 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) |
14 | 13 | snssd 4815 | . . . . 5 ⊢ (𝜑 → {(𝐺 Σg 𝐹)} ⊆ 𝐵) |
15 | toponuni 22834 | . . . . . 6 ⊢ ((TopOpen‘𝐺) ∈ (TopOn‘𝐵) → 𝐵 = ∪ (TopOpen‘𝐺)) | |
16 | 5, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐵 = ∪ (TopOpen‘𝐺)) |
17 | 14, 16 | sseqtrd 4020 | . . . 4 ⊢ (𝜑 → {(𝐺 Σg 𝐹)} ⊆ ∪ (TopOpen‘𝐺)) |
18 | eqid 2727 | . . . . 5 ⊢ ∪ (TopOpen‘𝐺) = ∪ (TopOpen‘𝐺) | |
19 | 18 | sscls 22978 | . . . 4 ⊢ (((TopOpen‘𝐺) ∈ Top ∧ {(𝐺 Σg 𝐹)} ⊆ ∪ (TopOpen‘𝐺)) → {(𝐺 Σg 𝐹)} ⊆ ((cls‘(TopOpen‘𝐺))‘{(𝐺 Σg 𝐹)})) |
20 | 7, 17, 19 | syl2anc 582 | . . 3 ⊢ (𝜑 → {(𝐺 Σg 𝐹)} ⊆ ((cls‘(TopOpen‘𝐺))‘{(𝐺 Σg 𝐹)})) |
21 | ovex 7457 | . . . 4 ⊢ (𝐺 Σg 𝐹) ∈ V | |
22 | 21 | snss 4792 | . . 3 ⊢ ((𝐺 Σg 𝐹) ∈ ((cls‘(TopOpen‘𝐺))‘{(𝐺 Σg 𝐹)}) ↔ {(𝐺 Σg 𝐹)} ⊆ ((cls‘(TopOpen‘𝐺))‘{(𝐺 Σg 𝐹)})) |
23 | 20, 22 | sylibr 233 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ ((cls‘(TopOpen‘𝐺))‘{(𝐺 Σg 𝐹)})) |
24 | 2, 8, 9, 1, 10, 11, 12, 3 | tsmsgsum 24061 | . 2 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘(TopOpen‘𝐺))‘{(𝐺 Σg 𝐹)})) |
25 | 23, 24 | eleqtrrd 2831 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 tsums 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3947 {csn 4630 ∪ cuni 4910 class class class wbr 5150 ⟶wf 6547 ‘cfv 6551 (class class class)co 7424 finSupp cfsupp 9391 Basecbs 17185 TopOpenctopn 17408 0gc0g 17426 Σg cgsu 17427 CMndccmn 19740 Topctop 22813 TopOnctopon 22830 TopSpctps 22852 clsccl 22940 tsums ctsu 24048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-n0 12509 df-z 12595 df-uz 12859 df-fz 13523 df-fzo 13666 df-seq 14005 df-hash 14328 df-0g 17428 df-gsum 17429 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-cntz 19273 df-cmn 19742 df-fbas 21281 df-fg 21282 df-top 22814 df-topon 22831 df-topsp 22853 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-tsms 24049 |
This theorem is referenced by: haustsmsid 24063 tsms0 24064 tayl0 26314 esumgsum 33669 |
Copyright terms: Public domain | W3C validator |