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Theorem tsmscls 24063

Description: One half of tgptsmscls 24075, 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 2733	. . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin)
5		eqid 2733	. . . . . 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 24056	. . . . 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 22859	. . . . . . 7 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
11	6, 10	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵))
12		eqid 2733	. . . . . . . 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 24053	. . . . . . 7 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦}) ∈ (fBas‘(𝒫 𝐴 ∩ Fin)))
14		fgcl 23803	. . . . . . 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 24054	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐺 Σ_g (𝐹 ↾ 𝑦)) ∈ 𝐵)
18	17	fmpttd 7057	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦))):(𝒫 𝐴 ∩ Fin)⟶𝐵)
19		flfval 23915	. . . . . 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 1373	. . . . 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 2768	. . . 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 2835	. . 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 23911	. . 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 3969	1 ⊢ (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 {crab 3397 ∩ cin 3898 ⊆ wss 3899 𝒫 cpw 4551 {csn 4577 ↦ cmpt 5176 ran crn 5622 ↾ cres 5623 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 Fincfn 8878 Basecbs 17130 TopOpenctopn 17335 Σ_g cgsu 17354 CMndccmn 19702 fBascfbas 21289 filGencfg 21290 TopOnctopon 22835 TopSpctps 22857 clsccl 22943 Filcfil 23770 FilMap cfm 23858 fLim cflim 23859 fLimf cflf 23860 tsums ctsu 24051

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11072 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-pre-mulgt0 11093

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-map 8761 df-en 8879 df-dom 8880 df-sdom 8881 df-fin 8882 df-fsupp 9256 df-oi 9406 df-card 9842 df-pnf 11158 df-mnf 11159 df-xr 11160 df-ltxr 11161 df-le 11162 df-sub 11356 df-neg 11357 df-nn 12136 df-n0 12392 df-z 12479 df-uz 12743 df-fz 13418 df-fzo 13565 df-seq 13919 df-hash 14248 df-0g 17355 df-gsum 17356 df-mgm 18558 df-sgrp 18637 df-mnd 18653 df-cntz 19239 df-cmn 19704 df-fbas 21298 df-fg 21299 df-top 22819 df-topon 22836 df-topsp 22858 df-cld 22944 df-ntr 22945 df-cls 22946 df-nei 23023 df-fil 23771 df-flim 23864 df-flf 23865 df-tsms 24052

This theorem is referenced by: tgptsmscls 24075

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