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Theorem tsmscls 24419

Description: One half of tgptsmscls 24431, 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 2760	. . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin)
5		eqid 2760	. . . . . 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 24412	. . . . 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 23214	. . . . . . 7 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
11	6, 10	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵))
12		eqid 2760	. . . . . . . 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 24409	. . . . . . 7 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦}) ∈ (fBas‘(𝒫 𝐴 ∩ Fin)))
14		fgcl 24159	. . . . . . 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 18	. . . . . 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 24410	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐺 Σ_g (𝐹 ↾ 𝑦)) ∈ 𝐵)
18	17	fmpttd 7103	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦))):(𝒫 𝐴 ∩ Fin)⟶𝐵)
19		flfval 24271	. . . . . 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 1398	. . . . 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 2795	. . . 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 2862	. . 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 24267	. . 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 18	. 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 3967	1 ⊢ (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 {crab 3412 ∩ cin 3897 ⊆ wss 3898 𝒫 cpw 4556 {csn 4583 ↦ cmpt 5185 ran crn 5648 ↾ cres 5649 ⟶wf 6523 ‘cfv 6527 (class class class)co 7408 Fincfn 8951 Basecbs 17349 TopOpenctopn 17554 Σ_g cgsu 17573 CMndccmn 19956 fBascfbas 21628 filGencfg 21629 TopOnctopon 23190 TopSpctps 23212 clsccl 23298 Filcfil 24126 FilMap cfm 24214 fLim cflim 24215 fLimf cflf 24216 tsums ctsu 24407

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11228 ax-resscn 11229 ax-1cn 11230 ax-icn 11231 ax-addcl 11232 ax-addrcl 11233 ax-mulcl 11234 ax-mulrcl 11235 ax-mulcom 11236 ax-addass 11237 ax-mulass 11238 ax-distr 11239 ax-i2m1 11240 ax-1ne0 11241 ax-1rid 11242 ax-rnegex 11243 ax-rrecex 11244 ax-cnre 11245 ax-pre-lttri 11246 ax-pre-lttrn 11247 ax-pre-ltadd 11248 ax-pre-mulgt0 11249

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-isom 6536 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-er 8695 df-map 8827 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-fsupp 9332 df-oi 9482 df-card 9992 df-pnf 11317 df-mnf 11318 df-xr 11319 df-ltxr 11320 df-le 11321 df-sub 11515 df-neg 11516 df-nn 12306 df-n0 12577 df-z 12664 df-uz 12936 df-fz 13610 df-fzo 13758 df-seq 14114 df-hash 14443 df-0g 17574 df-gsum 17575 df-mgm 18778 df-sgrp 18870 df-mnd 18886 df-cntz 19493 df-cmn 19958 df-fbas 21637 df-fg 21638 df-top 23174 df-topon 23191 df-topsp 23213 df-cld 23299 df-ntr 23300 df-cls 23301 df-nei 23378 df-fil 24127 df-flim 24220 df-flf 24221 df-tsms 24408

This theorem is used by: tgptsmscls 24431

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