tsmscls - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tsmscls		Structured version Visualization version GIF version

Theorem tsmscls 24103

Description: One half of tgptsmscls 24115, true in any commutative monoid topological space. (Contributed by Mario Carneiro, 21-Sep-2015.)

**Hypotheses**
Ref	Expression
tsmscls.b	⊢ 𝐵 = (Base‘𝐺)
tsmscls.j	⊢ 𝐽 = (TopOpen‘𝐺)
tsmscls.1	⊢ (𝜑 → 𝐺 ∈ CMnd)
tsmscls.2	⊢ (𝜑 → 𝐺 ∈ TopSp)
tsmscls.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
tsmscls.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
tsmscls.x	⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))

**Assertion**
Ref	Expression
tsmscls	⊢ (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))

Proof of Theorem tsmscls Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tsmscls.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))
2		tsmscls.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
3		tsmscls.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝐺)
4		eqid 2737	. . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin)
5		eqid 2737	. . . . . 6 ⊢ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦}) = ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})
6		tsmscls.2	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TopSp)
7		tsmscls.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		tsmscls.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
9	2, 3, 4, 5, 6, 7, 8	tsmsval 24096	. . . . 5 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf ((𝒫 𝐴 ∩ Fin)filGenran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})))‘(𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦)))))
10	2, 3	istps 22899	. . . . . . 7 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
11	6, 10	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵))
12		eqid 2737	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦}) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})
13	4, 12, 5, 7	tsmsfbas 24093	. . . . . . 7 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦}) ∈ (fBas‘(𝒫 𝐴 ∩ Fin)))
14		fgcl 23843	. . . . . . 7 ⊢ (ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦}) ∈ (fBas‘(𝒫 𝐴 ∩ Fin)) → ((𝒫 𝐴 ∩ Fin)filGenran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})) ∈ (Fil‘(𝒫 𝐴 ∩ Fin)))
15	13, 14	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝒫 𝐴 ∩ Fin)filGenran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})) ∈ (Fil‘(𝒫 𝐴 ∩ Fin)))
16		tsmscls.1	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ CMnd)
17	2, 4, 16, 7, 8	tsmslem1 24094	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐺 Σ_g (𝐹 ↾ 𝑦)) ∈ 𝐵)
18	17	fmpttd 7068	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦))):(𝒫 𝐴 ∩ Fin)⟶𝐵)
19		flfval 23955	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝐵) ∧ ((𝒫 𝐴 ∩ Fin)filGenran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})) ∈ (Fil‘(𝒫 𝐴 ∩ Fin)) ∧ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦))):(𝒫 𝐴 ∩ Fin)⟶𝐵) → ((𝐽 fLimf ((𝒫 𝐴 ∩ Fin)filGenran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})))‘(𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦)))) = (𝐽 fLim ((𝐵 FilMap (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦))))‘((𝒫 𝐴 ∩ Fin)filGenran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})))))
20	11, 15, 18, 19	syl3anc 1374	. . . . 5 ⊢ (𝜑 → ((𝐽 fLimf ((𝒫 𝐴 ∩ Fin)filGenran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})))‘(𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦)))) = (𝐽 fLim ((𝐵 FilMap (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦))))‘((𝒫 𝐴 ∩ Fin)filGenran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})))))
21	9, 20	eqtrd 2772	. . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐽 fLim ((𝐵 FilMap (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦))))‘((𝒫 𝐴 ∩ Fin)filGenran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})))))
22	1, 21	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐽 fLim ((𝐵 FilMap (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦))))‘((𝒫 𝐴 ∩ Fin)filGenran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})))))
23		flimsncls 23951	. . 3 ⊢ (𝑋 ∈ (𝐽 fLim ((𝐵 FilMap (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦))))‘((𝒫 𝐴 ∩ Fin)filGenran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})))) → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐽 fLim ((𝐵 FilMap (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦))))‘((𝒫 𝐴 ∩ Fin)filGenran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})))))
24	22, 23	syl 17	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐽 fLim ((𝐵 FilMap (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦))))‘((𝒫 𝐴 ∩ Fin)filGenran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦})))))
25	24, 21	sseqtrrd 3960	1 ⊢ (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 {crab 3390 ∩ cin 3889 ⊆ wss 3890 𝒫 cpw 4542 {csn 4568 ↦ cmpt 5167 ran crn 5632 ↾ cres 5633 ⟶wf 6495 ‘cfv 6499 (class class class)co 7367 Fincfn 8893 Basecbs 17179 TopOpenctopn 17384 Σ_g cgsu 17403 CMndccmn 19755 fBascfbas 21340 filGencfg 21341 TopOnctopon 22875 TopSpctps 22897 clsccl 22983 Filcfil 23810 FilMap cfm 23898 fLim cflim 23899 fLimf cflf 23900 tsums ctsu 24091

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-seq 13964 df-hash 14293 df-0g 17404 df-gsum 17405 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-cntz 19292 df-cmn 19757 df-fbas 21349 df-fg 21350 df-top 22859 df-topon 22876 df-topsp 22898 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-fil 23811 df-flim 23904 df-flf 23905 df-tsms 24092

This theorem is referenced by: tgptsmscls 24115

W3C validator