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Theorem tsmsxp 21768

Description: Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 18198 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 21766 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 21693	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ TopMnd)
4	3	3ad2ant1 1075	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝐺 ∈ TopMnd)
5		simp2 1055	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑢 ∈ (TopOpen‘𝐺))
6		eqid 2610	. . . . . . . . . . . . 13 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
7		tsmsxp.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐺)
8	6, 7	tmdtopon 21695	. . . . . . . . . . . 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 20544	. . . . . . . . . . 11 ⊢ (((TopOpen‘𝐺) ∈ (TopOn‘𝐵) ∧ 𝑢 ∈ (TopOpen‘𝐺)) → 𝑢 ⊆ 𝐵)
11	9, 5, 10	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑢 ⊆ 𝐵)
12		simp3 1056	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑥 ∈ 𝑢)
13	11, 12	sseldd 3569	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑥 ∈ 𝐵)
14		tmdmnd 21689	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
15	4, 14	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝐺 ∈ Mnd)
16		eqid 2610	. . . . . . . . . . 11 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
17	7, 16	mndidcl 17131	. . . . . . . . . 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 2610	. . . . . . . . . . . 12 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
20	7, 19, 16	mndrid 17135	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) = 𝑥)
21	15, 13, 20	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) = 𝑥)
22	21, 12	eqeltrd 2688	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) ∈ 𝑢)
23	7, 6, 19	tmdcn2 21703	. . . . . . . . 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 1325	. . . . . . . 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 3054	. . . . . . . . 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 1090	. . . . . . . . . . . . . . . 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)) → 𝑥 ∈ 𝑣)
27		elfpw 8151	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ (𝑦 ⊆ (𝐴 × 𝐶) ∧ 𝑦 ∈ Fin))
28	27	simplbi 475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ⊆ (𝐴 × 𝐶))
29	28	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . 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))) → 𝑦 ⊆ (𝐴 × 𝐶))
30		dmss 5245	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ⊆ (𝐴 × 𝐶) → dom 𝑦 ⊆ dom (𝐴 × 𝐶))
31	29, 30	syl 17	. . . . . . . . . . . . . . . . . . . 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))) → dom 𝑦 ⊆ dom (𝐴 × 𝐶))
32		dmxpss 5484	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom (𝐴 × 𝐶) ⊆ 𝐴
33	31, 32	syl6ss 3580	. . . . . . . . . . . . . . . . . . 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 ⊆ 𝐴)
34	27	simprbi 479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ∈ Fin)
35	34	ad2antrl 760	. . . . . . . . . . . . . . . . . . . 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 ∈ Fin)
36		dmfi 8129	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ Fin → dom 𝑦 ∈ Fin)
37	35, 36	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 ∈ Fin)
38		elfpw 8151	. . . . . . . . . . . . . . . . . . 19 ⊢ (dom 𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↔ (dom 𝑦 ⊆ 𝐴 ∧ dom 𝑦 ∈ Fin))
39	33, 37, 38	sylanbrc 695	. . . . . . . . . . . . . . . . . 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 ∈ (𝒫 𝐴 ∩ Fin))
40		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (._g‘𝐺) = (._g‘𝐺)
41		simpl11 1129	. . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝜑)
42		tsmsxp.g	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐺 ∈ CMnd)
43	41, 42	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 ∈ CMnd)
44	41, 3	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 ∈ TopMnd)
45		simprrl 800	. . . . . . . . . . . . . . . . . . . . . . . 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 ∈ (𝒫 𝐴 ∩ Fin))
46		elfpw 8151	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑏 ⊆ 𝐴 ∧ 𝑏 ∈ Fin))
47	46	simprbi 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) → 𝑏 ∈ Fin)
48	45, 47	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))) → 𝑏 ∈ Fin)
49		simpl2r 1108	. . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑡 ∈ (TopOpen‘𝐺))
50	44, 14	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))) → 𝐺 ∈ Mnd)
51	50, 17	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))) → (0_g‘𝐺) ∈ 𝐵)
52		hashcl 13009	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 ∈ Fin → (#‘𝑏) ∈ ℕ₀)
53	48, 52	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 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))) → (#‘𝑏) ∈ ℕ₀)
54	7, 40, 16	mulgnn0z 17390	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ Mnd ∧ (#‘𝑏) ∈ ℕ₀) → ((#‘𝑏)(._g‘𝐺)(0_g‘𝐺)) = (0_g‘𝐺))
55	50, 53, 54	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . 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)(._g‘𝐺)(0_g‘𝐺)) = (0_g‘𝐺))
56		simpl32 1136	. . . . . . . . . . . . . . . . . . . . . . . 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))) → (0_g‘𝐺) ∈ 𝑡)
57	55, 56	eqeltrd 2688	. . . . . . . . . . . . . . . . . . . . . . 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)) ∈ 𝑡)
58	6, 7, 40, 43, 44, 48, 49, 51, 57	tmdgsum2 21710	. . . . . . . . . . . . . . . . . . . . . 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))
59		simp111 1183	. . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝜑)
60	59, 42	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝐺 ∈ CMnd)
61	59, 1	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝐺 ∈ TopGrp)
62		tsmsxp.a	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
63	59, 62	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝐴 ∈ 𝑉)
64		tsmsxp.c	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
65	59, 64	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝐶 ∈ 𝑊)
66		tsmsxp.f	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)
67	59, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
68		tsmsxp.h	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
69	59, 68	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝐻:𝐴⟶𝐵)
70		tsmsxp.1	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))
71	59, 70	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))
72		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (-_g‘𝐺) = (-_g‘𝐺)
73		simp3l 1082	. . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑠 ∈ (TopOpen‘𝐺))
74		simp3rl 1127	. . . . . . . . . . . . . . . . . . . . . . . . 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))) → (0_g‘𝐺) ∈ 𝑠)
75		simp2rl 1123	. . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
76		simp2rr 1124	. . . . . . . . . . . . . . . . . . . . . . . . 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))) → dom 𝑦 ⊆ 𝑏)
77		simp2ll 1121	. . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
78	7, 60, 61, 63, 65, 67, 69, 71, 6, 16, 19, 72, 73, 74, 75, 76, 77	tsmsxplem1 21766	. . . . . . . . . . . . . . . . . . . . . . . 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))
79	43	3adant3 1074	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ CMnd)
80	61	3adant3r 1315	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ TopGrp)
81	63	3adant3r 1315	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝐴 ∈ 𝑉)
82	65	3adant3r 1315	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝐶 ∈ 𝑊)
83	67	3adant3r 1315	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
84	69	3adant3r 1315	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝐻:𝐴⟶𝐵)
85	41	3adant3 1074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝜑)
86	85, 70	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))
87		simp3ll 1125	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑠 ∈ (TopOpen‘𝐺))
88	74	3adant3r 1315	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (0_g‘𝐺) ∈ 𝑠)
89		simp2rl 1123	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
90		simp133 1191	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)
91		simp3rl 1127	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑛 ∈ (𝒫 𝐶 ∩ Fin))
92		simp2ll 1121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
93		simp2rr 1124	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → dom 𝑦 ⊆ 𝑏)
94		simp3rr 1128	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((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))
95	94	simpld 474	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
96		relxp 5150	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ Rel (𝐴 × 𝐶)
97		relss 5129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 ⊆ (𝐴 × 𝐶) → (Rel (𝐴 × 𝐶) → Rel 𝑦))
98	28, 96, 97	mpisyl 21	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → Rel 𝑦)
99		relssdmrn 5573	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (Rel 𝑦 → 𝑦 ⊆ (dom 𝑦 × ran 𝑦))
100	98, 99	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ⊆ (dom 𝑦 × ran 𝑦))
101		xpss12 5148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((dom 𝑦 ⊆ 𝑏 ∧ ran 𝑦 ⊆ 𝑛) → (dom 𝑦 × ran 𝑦) ⊆ (𝑏 × 𝑛))
102	100, 101	sylan9ss 3581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ (dom 𝑦 ⊆ 𝑏 ∧ ran 𝑦 ⊆ 𝑛)) → 𝑦 ⊆ (𝑏 × 𝑛))
103	92, 93, 95, 102	syl12anc 1316	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
104	94	simprd 478	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)
105		elfpw 8151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑛 ⊆ 𝐶 ∧ 𝑛 ∈ Fin))
106		xpss12 5148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑏 ⊆ 𝐴 ∧ 𝑛 ⊆ 𝐶) → (𝑏 × 𝑛) ⊆ (𝐴 × 𝐶))
107		xpfi 8116	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑏 ∈ Fin ∧ 𝑛 ∈ Fin) → (𝑏 × 𝑛) ∈ Fin)
108	106, 107	anim12i 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑏 ⊆ 𝐴 ∧ 𝑛 ⊆ 𝐶) ∧ (𝑏 ∈ Fin ∧ 𝑛 ∈ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
109	108	an4s 865	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑏 ⊆ 𝐴 ∧ 𝑏 ∈ Fin) ∧ (𝑛 ⊆ 𝐶 ∧ 𝑛 ∈ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
110	46, 105, 109	syl2anb 495	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑛 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
111		elfpw 8151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
112	110, 111	sylibr 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑛 ∈ (𝒫 𝐶 ∩ Fin)) → (𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
113	89, 91, 112	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
114		simp2lr 1122	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
115		sseq2 3590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ (𝑏 × 𝑛)))
116		reseq2 5312	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝐹 ↾ 𝑧) = (𝐹 ↾ (𝑏 × 𝑛)))
117	116	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝐺 Σ_g (𝐹 ↾ 𝑧)) = (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))))
118	117	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑏 × 𝑛) → ((𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣 ↔ (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣))
119	115, 118	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑏 × 𝑛) → ((𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣) ↔ (𝑦 ⊆ (𝑏 × 𝑛) → (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣)))
120	119	rspcv 3278	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → (∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣) → (𝑦 ⊆ (𝑏 × 𝑛) → (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣)))
121	113, 114, 103, 120	syl3c 64	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣)
122		simp3lr 1126	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((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))
123	122	simprd 478	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ∀𝑔 ∈ (𝑠 ↑_𝑚 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)
124		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑔 = ℎ → (𝐺 Σ_g 𝑔) = (𝐺 Σ_g ℎ))
125	124	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑔 = ℎ → ((𝐺 Σ_g 𝑔) ∈ 𝑡 ↔ (𝐺 Σ_g ℎ) ∈ 𝑡))
126	125	cbvralv 3147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∀𝑔 ∈ (𝑠 ↑_𝑚 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡 ↔ ∀ℎ ∈ (𝑠 ↑_𝑚 𝑏)(𝐺 Σ_g ℎ) ∈ 𝑡)
127	123, 126	sylib 207	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ∀ℎ ∈ (𝑠 ↑_𝑚 𝑏)(𝐺 Σ_g ℎ) ∈ 𝑡)
128	7, 79, 80, 81, 82, 83, 84, 86, 6, 16, 19, 72, 87, 88, 89, 90, 91, 103, 104, 121, 127	tsmsxplem2 21767	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 (𝐻 ↾ 𝑏)) ∈ 𝑢)
129	128	3exp 1256	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))
130	129	exp4a 631	. . . . . . . . . . . . . . . . . . . . . . . . 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))))
131	130	3imp1 1272	. . . . . . . . . . . . . . . . . . . . . . . 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)
132	78, 131	rexlimddv 3017	. . . . . . . . . . . . . . . . . . . . . . 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))) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)
133	132	3expa 1257	. . . . . . . . . . . . . . . . . . . . . 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)
134	58, 133	rexlimddv 3017	. . . . . . . . . . . . . . . . . . . . 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))) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)
135	134	anassrs 678	. . . . . . . . . . . . . . . . . . . 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)
136	135	expr 641	. . . . . . . . . . . . . . . . . . 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))
137	136	ralrimiva 2949	. . . . . . . . . . . . . . . . . 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))
138		sseq1 3589	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = dom 𝑦 → (𝑎 ⊆ 𝑏 ↔ dom 𝑦 ⊆ 𝑏))
139	138	imbi1d 330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = dom 𝑦 → ((𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢) ↔ (dom 𝑦 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
140	139	ralbidv 2969	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = dom 𝑦 → (∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢) ↔ ∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(dom 𝑦 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
141	140	rspcev 3282	. . . . . . . . . . . . . . . . . 18 ⊢ ((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))
142	39, 137, 141	syl2anc 691	. . . . . . . . . . . . . . . . 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))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))
143	142	rexlimdvaa 3014	. . . . . . . . . . . . . . . 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)))
144	26, 143	embantd 57	. . . . . . . . . . . . . . 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 → ∃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	3expia 1259	. . . . . . . . . . . . . 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))))
146	145	anassrs 678	. . . . . . . . . . . . 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))))
147	146	rexlimdva 3013	. . . . . . . . . . . 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))))
148	147	com23 84	. . . . . . . . . . 11 ⊢ (((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))))
149	148	impd 446	. . . . . . . . . 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)))
150	149	rexlimdva 3013	. . . . . . . . 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)))
151	25, 150	syl5 33	. . . . . . . 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)))
152	24, 151	mpan2d 706	. . . . . . 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)))
153	152	3expia 1259	. . . . . 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))))
154	153	com23 84	. . . . 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))))
155	154	ralrimdva 2952	. . . 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))))
156	155	anim2d 587	. . 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)))))
157		eqid 2610	. . . 4 ⊢ (𝒫 (𝐴 × 𝐶) ∩ Fin) = (𝒫 (𝐴 × 𝐶) ∩ Fin)
158		tgptps 21694	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
159	1, 158	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TopSp)
160		xpexg 6858	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 × 𝐶) ∈ V)
161	62, 64, 160	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V)
162	7, 6, 157, 42, 159, 161, 66	eltsms 21746	. . 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)))))
163		eqid 2610	. . . 4 ⊢ (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin)
164	7, 6, 163, 42, 159, 62, 68	eltsms 21746	. . 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)))))
165	156, 162, 164	3imtr4d 282	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ (𝐺 tsums 𝐻)))
166	165	ssrdv 3574	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ (𝐺 tsums 𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 {csn 4125 ↦ cmpt 4643 × cxp 5036 dom cdm 5038 ran crn 5039 ↾ cres 5040 Rel wrel 5043 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑_𝑚 cmap 7744 Fincfn 7841 ℕ₀cn0 11169 #chash 12979 Basecbs 15695 +_gcplusg 15768 TopOpenctopn 15905 0_gc0g 15923 Σ_g cgsu 15924 Mndcmnd 17117 -_gcsg 17247 ._gcmg 17363 CMndccmn 18016 TopOnctopon 20518 TopSpctps 20519 TopMndctmd 21684 TopGrpctgp 21685 tsums ctsu 21739

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-rest 15906 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-mre 16069 df-mrc 16070 df-acs 16072 df-plusf 17064 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-fbas 19564 df-fg 19565 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-ntr 20634 df-nei 20712 df-cn 20841 df-cnp 20842 df-cmp 21000 df-tx 21175 df-xko 21176 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tmd 21686 df-tgp 21687 df-tsms 21740

This theorem is referenced by: (None)

W3C validator