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Theorem tsmsxp 24178

Description: Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 20008 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 24176 for the main proof; this part mostly sets up the local assumptions. (Contributed by Mario Carneiro, 21-Sep-2015.)

**Hypotheses**
Ref	Expression
tsmsxp.b	⊢ 𝐵 = (Base‘𝐺)
tsmsxp.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
tsmsxp.2	⊢ (𝜑 → 𝐺 ∈ TopGrp)
tsmsxp.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
tsmsxp.c	⊢ (𝜑 → 𝐶 ∈ 𝑊)
tsmsxp.f	⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)
tsmsxp.h	⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
tsmsxp.1	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))

**Assertion**
Ref	Expression
tsmsxp	⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ (𝐺 tsums 𝐻))

Distinct variable groups: 𝑗,𝑘,𝐺 𝐵,𝑘 𝐴,𝑗,𝑘 𝑗,𝐻,𝑘 𝐶,𝑗,𝑘 𝑗,𝐹,𝑘 𝜑,𝑗,𝑘 Allowed substitution hints: 𝐵(𝑗) 𝑉(𝑗,𝑘) 𝑊(𝑗,𝑘)

Proof of Theorem tsmsxp Dummy variables 𝑔 𝑦 𝑧 𝑎 𝑏 𝑐 𝑑 ℎ 𝑛 𝑠 𝑡 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tsmsxp.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ TopGrp)
2		tgptmd 24102	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ TopMnd)
4	3	3ad2ant1 1132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝐺 ∈ TopMnd)
5		simp2 1136	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑢 ∈ (TopOpen‘𝐺))
6		eqid 2734	. . . . . . . . . . . . 13 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
7		tsmsxp.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐺)
8	6, 7	tmdtopon 24104	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ TopMnd → (TopOpen‘𝐺) ∈ (TopOn‘𝐵))
9	4, 8	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (TopOpen‘𝐺) ∈ (TopOn‘𝐵))
10		toponss 22948	. . . . . . . . . . 11 ⊢ (((TopOpen‘𝐺) ∈ (TopOn‘𝐵) ∧ 𝑢 ∈ (TopOpen‘𝐺)) → 𝑢 ⊆ 𝐵)
11	9, 5, 10	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑢 ⊆ 𝐵)
12		simp3 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑥 ∈ 𝑢)
13	11, 12	sseldd 3995	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑥 ∈ 𝐵)
14		tmdmnd 24098	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
15	4, 14	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝐺 ∈ Mnd)
16		eqid 2734	. . . . . . . . . . 11 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
17	7, 16	mndidcl 18774	. . . . . . . . . 10 ⊢ (𝐺 ∈ Mnd → (0_g‘𝐺) ∈ 𝐵)
18	15, 17	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (0_g‘𝐺) ∈ 𝐵)
19		eqid 2734	. . . . . . . . . . . 12 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
20	7, 19, 16	mndrid 18780	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) = 𝑥)
21	15, 13, 20	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) = 𝑥)
22	21, 12	eqeltrd 2838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) ∈ 𝑢)
23	7, 6, 19	tmdcn2 24112	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopMnd ∧ 𝑢 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝐵 ∧ (0_g‘𝐺) ∈ 𝐵 ∧ (𝑥(+_g‘𝐺)(0_g‘𝐺)) ∈ 𝑢)) → ∃𝑣 ∈ (TopOpen‘𝐺)∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢))
24	4, 5, 13, 18, 22, 23	syl23anc 1376	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → ∃𝑣 ∈ (TopOpen‘𝐺)∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢))
25		r19.29 3111	. . . . . . . . 9 ⊢ ((∀𝑣 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ ∃𝑣 ∈ (TopOpen‘𝐺)∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → ∃𝑣 ∈ (TopOpen‘𝐺)((𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ ∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)))
26		simp31 1208	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → 𝑥 ∈ 𝑣)
27		elfpw 9391	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ (𝑦 ⊆ (𝐴 × 𝐶) ∧ 𝑦 ∈ Fin))
28	27	simplbi 497	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ⊆ (𝐴 × 𝐶))
29	28	ad2antrl 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → 𝑦 ⊆ (𝐴 × 𝐶))
30		dmss 5915	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ⊆ (𝐴 × 𝐶) → dom 𝑦 ⊆ dom (𝐴 × 𝐶))
31	29, 30	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → dom 𝑦 ⊆ dom (𝐴 × 𝐶))
32		dmxpss 6192	. . . . . . . . . . . . . . . . . . 19 ⊢ dom (𝐴 × 𝐶) ⊆ 𝐴
33	31, 32	sstrdi 4007	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → dom 𝑦 ⊆ 𝐴)
34		elinel2 4211	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ∈ Fin)
35	34	ad2antrl 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → 𝑦 ∈ Fin)
36		dmfi 9372	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ Fin → dom 𝑦 ∈ Fin)
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → dom 𝑦 ∈ Fin)
38		elfpw 9391	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↔ (dom 𝑦 ⊆ 𝐴 ∧ dom 𝑦 ∈ Fin))
39	33, 37, 38	sylanbrc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → dom 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
40		eqid 2734	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (._g‘𝐺) = (._g‘𝐺)
41		simpl11 1247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝜑)
42		tsmsxp.g	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐺 ∈ CMnd)
43	41, 42	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝐺 ∈ CMnd)
44	41, 3	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝐺 ∈ TopMnd)
45		simprrl 781	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
46	45	elin2d 4214	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝑏 ∈ Fin)
47		simpl2r 1226	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝑡 ∈ (TopOpen‘𝐺))
48	44, 14	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝐺 ∈ Mnd)
49	48, 17	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → (0_g‘𝐺) ∈ 𝐵)
50		hashcl 14391	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 ∈ Fin → (♯‘𝑏) ∈ ℕ₀)
51	46, 50	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → (♯‘𝑏) ∈ ℕ₀)
52	7, 40, 16	mulgnn0z 19131	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Mnd ∧ (♯‘𝑏) ∈ ℕ₀) → ((♯‘𝑏)(._g‘𝐺)(0_g‘𝐺)) = (0_g‘𝐺))
53	48, 51, 52	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → ((♯‘𝑏)(._g‘𝐺)(0_g‘𝐺)) = (0_g‘𝐺))
54		simpl32 1254	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → (0_g‘𝐺) ∈ 𝑡)
55	53, 54	eqeltrd 2838	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → ((♯‘𝑏)(._g‘𝐺)(0_g‘𝐺)) ∈ 𝑡)
56	6, 7, 40, 43, 44, 46, 47, 49, 55	tmdgsum2 24119	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → ∃𝑠 ∈ (TopOpen‘𝐺)((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))
57		simp111 1301	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝜑)
58	57, 42	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝐺 ∈ CMnd)
59	57, 1	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝐺 ∈ TopGrp)
60		tsmsxp.a	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
61	57, 60	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝐴 ∈ 𝑉)
62		tsmsxp.c	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
63	57, 62	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝐶 ∈ 𝑊)
64		tsmsxp.f	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)
65	57, 64	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
66		tsmsxp.h	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
67	57, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝐻:𝐴⟶𝐵)
68		tsmsxp.1	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))
69	57, 68	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))
70		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (-_g‘𝐺) = (-_g‘𝐺)
71		simp3l 1200	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝑠 ∈ (TopOpen‘𝐺))
72		simp3rl 1245	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → (0_g‘𝐺) ∈ 𝑠)
73		simp2rl 1241	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
74		simp2rr 1242	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → dom 𝑦 ⊆ 𝑏)
75		simp2ll 1239	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
76	7, 58, 59, 61, 63, 65, 67, 69, 6, 16, 19, 70, 71, 72, 73, 74, 75	tsmsxplem1 24176	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠))
77	43	3adant3 1131	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐺 ∈ CMnd)
78	59	3adant3r 1180	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐺 ∈ TopGrp)
79	61	3adant3r 1180	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐴 ∈ 𝑉)
80	63	3adant3r 1180	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐶 ∈ 𝑊)
81	65	3adant3r 1180	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
82	67	3adant3r 1180	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐻:𝐴⟶𝐵)
83	41	3adant3 1131	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝜑)
84	83, 68	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))
85		simp3ll 1243	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑠 ∈ (TopOpen‘𝐺))
86	72	3adant3r 1180	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (0_g‘𝐺) ∈ 𝑠)
87		simp2rl 1241	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
88		simp133 1309	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)
89		simp3rl 1245	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑛 ∈ (𝒫 𝐶 ∩ Fin))
90		simp2ll 1239	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
91		simp2rr 1242	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → dom 𝑦 ⊆ 𝑏)
92		simp3rr 1246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠))
93	92	simpld 494	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ran 𝑦 ⊆ 𝑛)
94		relxp 5706	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ Rel (𝐴 × 𝐶)
95		relss 5793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ⊆ (𝐴 × 𝐶) → (Rel (𝐴 × 𝐶) → Rel 𝑦))
96	28, 94, 95	mpisyl 21	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → Rel 𝑦)
97		relssdmrn 6289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Rel 𝑦 → 𝑦 ⊆ (dom 𝑦 × ran 𝑦))
98	96, 97	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ⊆ (dom 𝑦 × ran 𝑦))
99		xpss12 5703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((dom 𝑦 ⊆ 𝑏 ∧ ran 𝑦 ⊆ 𝑛) → (dom 𝑦 × ran 𝑦) ⊆ (𝑏 × 𝑛))
100	98, 99	sylan9ss 4008	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ (dom 𝑦 ⊆ 𝑏 ∧ ran 𝑦 ⊆ 𝑛)) → 𝑦 ⊆ (𝑏 × 𝑛))
101	90, 91, 93, 100	syl12anc 837	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑦 ⊆ (𝑏 × 𝑛))
102	92	simprd 495	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)
103		sseq2 4021	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ (𝑏 × 𝑛)))
104		reseq2 5994	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝐹 ↾ 𝑧) = (𝐹 ↾ (𝑏 × 𝑛)))
105	104	oveq2d 7446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝐺 Σ_g (𝐹 ↾ 𝑧)) = (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))))
106	105	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑏 × 𝑛) → ((𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣 ↔ (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣))
107	103, 106	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑏 × 𝑛) → ((𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣) ↔ (𝑦 ⊆ (𝑏 × 𝑛) → (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣)))
108		simp2lr 1240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))
109		elfpw 9391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑏 ⊆ 𝐴 ∧ 𝑏 ∈ Fin))
110		elfpw 9391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑛 ⊆ 𝐶 ∧ 𝑛 ∈ Fin))
111		xpss12 5703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑏 ⊆ 𝐴 ∧ 𝑛 ⊆ 𝐶) → (𝑏 × 𝑛) ⊆ (𝐴 × 𝐶))
112		xpfi 9355	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑏 ∈ Fin ∧ 𝑛 ∈ Fin) → (𝑏 × 𝑛) ∈ Fin)
113	111, 112	anim12i 613	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑏 ⊆ 𝐴 ∧ 𝑛 ⊆ 𝐶) ∧ (𝑏 ∈ Fin ∧ 𝑛 ∈ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
114	113	an4s 660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑏 ⊆ 𝐴 ∧ 𝑏 ∈ Fin) ∧ (𝑛 ⊆ 𝐶 ∧ 𝑛 ∈ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
115	109, 110, 114	syl2anb 598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑛 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
116		elfpw 9391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
117	115, 116	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑛 ∈ (𝒫 𝐶 ∩ Fin)) → (𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
118	87, 89, 117	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
119	107, 108, 118	rspcdva 3622	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (𝑦 ⊆ (𝑏 × 𝑛) → (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣))
120	101, 119	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣)
121		simp3lr 1244	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))
122	121	simprd 495	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)
123		oveq2 7438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑔 = ℎ → (𝐺 Σ_g 𝑔) = (𝐺 Σ_g ℎ))
124	123	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑔 = ℎ → ((𝐺 Σ_g 𝑔) ∈ 𝑡 ↔ (𝐺 Σ_g ℎ) ∈ 𝑡))
125	124	cbvralvw 3234	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡 ↔ ∀ℎ ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g ℎ) ∈ 𝑡)
126	122, 125	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀ℎ ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g ℎ) ∈ 𝑡)
127	7, 77, 78, 79, 80, 81, 82, 84, 6, 16, 19, 70, 85, 86, 87, 88, 89, 101, 102, 120, 126	tsmsxplem2 24177	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)
128	127	3exp 1118	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → (((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) → (((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠))) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
129	128	exp4a 431	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → (((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) → ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) → ((𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))))
130	129	3imp1 1346	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠))) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)
131	76, 130	rexlimddv 3158	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)
132	131	3expa 1117	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)
133	56, 132	rexlimddv 3158	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)
134	133	anassrs 467	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)
135	134	expr 456	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → (dom 𝑦 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))
136	135	ralrimiva 3143	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → ∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(dom 𝑦 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))
137		sseq1 4020	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = dom 𝑦 → (𝑎 ⊆ 𝑏 ↔ dom 𝑦 ⊆ 𝑏))
138	137	rspceaimv 3627	. . . . . . . . . . . . . . . . 17 ⊢ ((dom 𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∧ ∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(dom 𝑦 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))
139	39, 136, 138	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))
140	139	rexlimdvaa 3153	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → (∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
141	26, 140	embantd 59	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → ((𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
142	141	3expia 1120	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺))) → ((𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢) → ((𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))))
143	142	anassrs 467	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘𝐺)) ∧ 𝑡 ∈ (TopOpen‘𝐺)) → ((𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢) → ((𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))))
144	143	rexlimdva 3152	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘𝐺)) → (∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢) → ((𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))))
145	144	impcomd 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘𝐺)) → (((𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ ∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
146	145	rexlimdva 3152	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (∃𝑣 ∈ (TopOpen‘𝐺)((𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ ∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
147	25, 146	syl5 34	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → ((∀𝑣 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ ∃𝑣 ∈ (TopOpen‘𝐺)∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
148	24, 147	mpan2d 694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (∀𝑣 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
149	148	3expia 1120	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺)) → (𝑥 ∈ 𝑢 → (∀𝑣 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))))
150	149	com23 86	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺)) → (∀𝑣 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) → (𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))))
151	150	ralrimdva 3151	. . . 4 ⊢ (𝜑 → (∀𝑣 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) → ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))))
152	151	anim2d 612	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ ∀𝑣 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → (𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))))
153		eqid 2734	. . . 4 ⊢ (𝒫 (𝐴 × 𝐶) ∩ Fin) = (𝒫 (𝐴 × 𝐶) ∩ Fin)
154		tgptps 24103	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
155	1, 154	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TopSp)
156	60, 62	xpexd 7769	. . . 4 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V)
157	7, 6, 153, 42, 155, 156, 64	eltsms 24156	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑣 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)))))
158		eqid 2734	. . . 4 ⊢ (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin)
159	7, 6, 158, 42, 155, 60, 66	eltsms 24156	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐻) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))))
160	152, 157, 159	3imtr4d 294	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ (𝐺 tsums 𝐻)))
161	160	ssrdv 4000	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ (𝐺 tsums 𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 Vcvv 3477 ∩ cin 3961 ⊆ wss 3962 𝒫 cpw 4604 {csn 4630 ↦ cmpt 5230 × cxp 5686 dom cdm 5688 ran crn 5689 ↾ cres 5690 Rel wrel 5693 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ↑_m cmap 8864 Fincfn 8983 ℕ₀cn0 12523 ♯chash 14365 Basecbs 17244 +_gcplusg 17297 TopOpenctopn 17467 0_gc0g 17485 Σ_g cgsu 17486 Mndcmnd 18759 -_gcsg 18965 ._gcmg 19097 CMndccmn 19812 TopOnctopon 22931 TopSpctps 22953 TopMndctmd 24093 TopGrpctgp 24094 tsums ctsu 24149

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-rest 17468 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-mre 17630 df-mrc 17631 df-acs 17633 df-plusf 18664 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-ghm 19243 df-cntz 19347 df-cmn 19814 df-abl 19815 df-fbas 21378 df-fg 21379 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-ntr 23043 df-nei 23121 df-cn 23250 df-cnp 23251 df-cmp 23410 df-tx 23585 df-xko 23586 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-tmd 24095 df-tgp 24096 df-tsms 24150

This theorem is referenced by: (None)

W3C validator