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Theorem tsmsxp 23651

Description: Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 19839 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 23649 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 23575	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ TopMnd)
4	3	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝐺 ∈ TopMnd)
5		simp2 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑢 ∈ (TopOpen‘𝐺))
6		eqid 2733	. . . . . . . . . . . . 13 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
7		tsmsxp.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐺)
8	6, 7	tmdtopon 23577	. . . . . . . . . . . 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 22421	. . . . . . . . . . 11 ⊢ (((TopOpen‘𝐺) ∈ (TopOn‘𝐵) ∧ 𝑢 ∈ (TopOpen‘𝐺)) → 𝑢 ⊆ 𝐵)
11	9, 5, 10	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑢 ⊆ 𝐵)
12		simp3 1139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑥 ∈ 𝑢)
13	11, 12	sseldd 3983	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑥 ∈ 𝐵)
14		tmdmnd 23571	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
15	4, 14	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝐺 ∈ Mnd)
16		eqid 2733	. . . . . . . . . . 11 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
17	7, 16	mndidcl 18637	. . . . . . . . . 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 2733	. . . . . . . . . . . 12 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
20	7, 19, 16	mndrid 18643	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) = 𝑥)
21	15, 13, 20	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) = 𝑥)
22	21, 12	eqeltrd 2834	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) ∈ 𝑢)
23	7, 6, 19	tmdcn2 23585	. . . . . . . . 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 1378	. . . . . . . 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 3115	. . . . . . . . 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 1210	. . . . . . . . . . . . . . 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 9351	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ (𝑦 ⊆ (𝐴 × 𝐶) ∧ 𝑦 ∈ Fin))
28	27	simplbi 499	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ⊆ (𝐴 × 𝐶))
29	28	ad2antrl 727	. . . . . . . . . . . . . . . . . . . 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 5901	. . . . . . . . . . . . . . . . . . . 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 6168	. . . . . . . . . . . . . . . . . . 19 ⊢ dom (𝐴 × 𝐶) ⊆ 𝐴
33	31, 32	sstrdi 3994	. . . . . . . . . . . . . . . . . 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 4196	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ∈ Fin)
35	34	ad2antrl 727	. . . . . . . . . . . . . . . . . . 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 9327	. . . . . . . . . . . . . . . . . . 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 9351	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↔ (dom 𝑦 ⊆ 𝐴 ∧ dom 𝑦 ∈ Fin))
39	33, 37, 38	sylanbrc 584	. . . . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (._g‘𝐺) = (._g‘𝐺)
41		simpl11 1249	. . . . . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . . . 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 4199	. . . . . . . . . . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . . . . . . . . . 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 14313	. . . . . . . . . . . . . . . . . . . . . . . . 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 18976	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Mnd ∧ (♯‘𝑏) ∈ ℕ₀) → ((♯‘𝑏)(._g‘𝐺)(0_g‘𝐺)) = (0_g‘𝐺))
53	48, 51, 52	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . 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 1256	. . . . . . . . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . . . . . . . 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 23592	. . . . . . . . . . . . . . . . . . . . 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 1303	. . . . . . . . . . . . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (-_g‘𝐺) = (-_g‘𝐺)
71		simp3l 1202	. . . . . . . . . . . . . . . . . . . . . . . 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 1247	. . . . . . . . . . . . . . . . . . . . . . . 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 1243	. . . . . . . . . . . . . . . . . . . . . . . 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 1244	. . . . . . . . . . . . . . . . . . . . . . . 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 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))) → 𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
76	7, 58, 59, 61, 63, 65, 67, 69, 6, 16, 19, 70, 71, 72, 73, 74, 75	tsmsxplem1 23649	. . . . . . . . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1182	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1182	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1182	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1182	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1182	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 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)))) → 𝑠 ∈ (TopOpen‘𝐺))
86	72	3adant3r 1182	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 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)))) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
88		simp133 1311	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1247	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1241	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1244	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ Rel (𝐴 × 𝐶)
95		relss 5780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6265	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((dom 𝑦 ⊆ 𝑏 ∧ ran 𝑦 ⊆ 𝑛) → (dom 𝑦 × ran 𝑦) ⊆ (𝑏 × 𝑛))
100	98, 99	sylan9ss 3995	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ (dom 𝑦 ⊆ 𝑏 ∧ ran 𝑦 ⊆ 𝑛)) → 𝑦 ⊆ (𝑏 × 𝑛))
101	90, 91, 93, 100	syl12anc 836	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4008	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ (𝑏 × 𝑛)))
104		reseq2 5975	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝐹 ↾ 𝑧) = (𝐹 ↾ (𝑏 × 𝑛)))
105	104	oveq2d 7422	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝐺 Σ_g (𝐹 ↾ 𝑧)) = (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))))
106	105	eleq1d 2819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑏 × 𝑛) → ((𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣 ↔ (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣))
107	103, 106	imbi12d 345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑏 × 𝑛) → ((𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣) ↔ (𝑦 ⊆ (𝑏 × 𝑛) → (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣)))
108		simp2lr 1242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9351	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑏 ⊆ 𝐴 ∧ 𝑏 ∈ Fin))
110		elfpw 9351	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑛 ⊆ 𝐶 ∧ 𝑛 ∈ Fin))
111		xpss12 5691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑏 ⊆ 𝐴 ∧ 𝑛 ⊆ 𝐶) → (𝑏 × 𝑛) ⊆ (𝐴 × 𝐶))
112		xpfi 9314	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑏 ∈ Fin ∧ 𝑛 ∈ Fin) → (𝑏 × 𝑛) ∈ Fin)
113	111, 112	anim12i 614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑏 ⊆ 𝐴 ∧ 𝑛 ⊆ 𝐶) ∧ (𝑏 ∈ Fin ∧ 𝑛 ∈ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
114	113	an4s 659	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑏 ⊆ 𝐴 ∧ 𝑏 ∈ Fin) ∧ (𝑛 ⊆ 𝐶 ∧ 𝑛 ∈ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
115	109, 110, 114	syl2anb 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑛 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
116		elfpw 9351	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
117	115, 116	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑛 ∈ (𝒫 𝐶 ∩ Fin)) → (𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
118	87, 89, 117	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3614	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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)))) → ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))
122	121	simprd 497	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7414	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑔 = ℎ → (𝐺 Σ_g 𝑔) = (𝐺 Σ_g ℎ))
124	123	eleq1d 2819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑔 = ℎ → ((𝐺 Σ_g 𝑔) ∈ 𝑡 ↔ (𝐺 Σ_g ℎ) ∈ 𝑡))
125	124	cbvralvw 3235	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡 ↔ ∀ℎ ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g ℎ) ∈ 𝑡)
126	122, 125	sylib 217	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 23650	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1120	. . . . . . . . . . . . . . . . . . . . . . . . 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 433	. . . . . . . . . . . . . . . . . . . . . . . 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 1348	. . . . . . . . . . . . . . . . . . . . . . 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 3162	. . . . . . . . . . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . . . . . . . . . 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 3162	. . . . . . . . . . . . . . . . . . . 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 469	. . . . . . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . . . . . 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 4007	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = dom 𝑦 → (𝑎 ⊆ 𝑏 ↔ dom 𝑦 ⊆ 𝑏))
138	137	rspceaimv 3617	. . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . . . 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 1122	. . . . . . . . . . . . 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 469	. . . . . . . . . . . 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 3156	. . . . . . . . . . 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 413	. . . . . . . . . 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 3156	. . . . . . . . 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 693	. . . . . . 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 1122	. . . . . 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 3155	. . . 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 613	. . 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 2733	. . . 4 ⊢ (𝒫 (𝐴 × 𝐶) ∩ Fin) = (𝒫 (𝐴 × 𝐶) ∩ Fin)
154		tgptps 23576	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
155	1, 154	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TopSp)
156	60, 62	xpexd 7735	. . . 4 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V)
157	7, 6, 153, 42, 155, 156, 64	eltsms 23629	. . 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 2733	. . . 4 ⊢ (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin)
159	7, 6, 158, 42, 155, 60, 66	eltsms 23629	. . 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 3988	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ (𝐺 tsums 𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∩ cin 3947 ⊆ wss 3948 𝒫 cpw 4602 {csn 4628 ↦ cmpt 5231 × cxp 5674 dom cdm 5676 ran crn 5677 ↾ cres 5678 Rel wrel 5681 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ↑_m cmap 8817 Fincfn 8936 ℕ₀cn0 12469 ♯chash 14287 Basecbs 17141 +_gcplusg 17194 TopOpenctopn 17364 0_gc0g 17382 Σ_g cgsu 17383 Mndcmnd 18622 -_gcsg 18818 ._gcmg 18945 CMndccmn 19643 TopOnctopon 22404 TopSpctps 22426 TopMndctmd 23566 TopGrpctgp 23567 tsums ctsu 23622

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-rest 17365 df-0g 17384 df-gsum 17385 df-topgen 17386 df-pt 17387 df-mre 17527 df-mrc 17528 df-acs 17530 df-plusf 18557 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-mhm 18668 df-submnd 18669 df-grp 18819 df-minusg 18820 df-sbg 18821 df-mulg 18946 df-ghm 19085 df-cntz 19176 df-cmn 19645 df-abl 19646 df-fbas 20934 df-fg 20935 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-ntr 22516 df-nei 22594 df-cn 22723 df-cnp 22724 df-cmp 22883 df-tx 23058 df-xko 23059 df-hmeo 23251 df-fil 23342 df-fm 23434 df-flim 23435 df-flf 23436 df-tmd 23568 df-tgp 23569 df-tsms 23623

This theorem is referenced by: (None)

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