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Theorem tsmscls 24094

Description: One half of tgptsmscls 24106, 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 24087	. . . . 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 22890	. . . . . . 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 24084	. . . . . . 7 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ {𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∣ 𝑥 ⊆ 𝑦}) ∈ (fBas‘(𝒫 𝐴 ∩ Fin)))
14		fgcl 23834	. . . . . . 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 24085	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐺 Σ_g (𝐹 ↾ 𝑦)) ∈ 𝐵)
18	17	fmpttd 7069	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σ_g (𝐹 ↾ 𝑦))):(𝒫 𝐴 ∩ Fin)⟶𝐵)
19		flfval 23946	. . . . . 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 23942	. . 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 3973	1 ⊢ (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 {crab 3401 ∩ cin 3902 ⊆ wss 3903 𝒫 cpw 4556 {csn 4582 ↦ cmpt 5181 ran crn 5633 ↾ cres 5634 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 Fincfn 8895 Basecbs 17148 TopOpenctopn 17353 Σ_g cgsu 17372 CMndccmn 19721 fBascfbas 21309 filGencfg 21310 TopOnctopon 22866 TopSpctps 22888 clsccl 22974 Filcfil 23801 FilMap cfm 23889 fLim cflim 23890 fLimf cflf 23891 tsums ctsu 24082

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-0g 17373 df-gsum 17374 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-cntz 19258 df-cmn 19723 df-fbas 21318 df-fg 21319 df-top 22850 df-topon 22867 df-topsp 22889 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-fil 23802 df-flim 23895 df-flf 23896 df-tsms 24083

This theorem is referenced by: tgptsmscls 24106

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