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Theorem tsmsxp 22760

Description: Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 19089 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 22758 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 22684	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ TopMnd)
4	3	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝐺 ∈ TopMnd)
5		simp2 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑢 ∈ (TopOpen‘𝐺))
6		eqid 2798	. . . . . . . . . . . . 13 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
7		tsmsxp.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐺)
8	6, 7	tmdtopon 22686	. . . . . . . . . . . 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 21532	. . . . . . . . . . 11 ⊢ (((TopOpen‘𝐺) ∈ (TopOn‘𝐵) ∧ 𝑢 ∈ (TopOpen‘𝐺)) → 𝑢 ⊆ 𝐵)
11	9, 5, 10	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑢 ⊆ 𝐵)
12		simp3 1135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑥 ∈ 𝑢)
13	11, 12	sseldd 3916	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑥 ∈ 𝐵)
14		tmdmnd 22680	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
15	4, 14	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝐺 ∈ Mnd)
16		eqid 2798	. . . . . . . . . . 11 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
17	7, 16	mndidcl 17918	. . . . . . . . . 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 2798	. . . . . . . . . . . 12 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
20	7, 19, 16	mndrid 17924	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) = 𝑥)
21	15, 13, 20	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) = 𝑥)
22	21, 12	eqeltrd 2890	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) ∈ 𝑢)
23	7, 6, 19	tmdcn2 22694	. . . . . . . . 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 1374	. . . . . . . 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 3216	. . . . . . . . 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 1206	. . . . . . . . . . . . . . 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 8810	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ (𝑦 ⊆ (𝐴 × 𝐶) ∧ 𝑦 ∈ Fin))
28	27	simplbi 501	. . . . . . . . . . . . . . . . . . . . 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 5735	. . . . . . . . . . . . . . . . . . . 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 5995	. . . . . . . . . . . . . . . . . . 19 ⊢ dom (𝐴 × 𝐶) ⊆ 𝐴
33	31, 32	sstrdi 3927	. . . . . . . . . . . . . . . . . 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 4123	. . . . . . . . . . . . . . . . . . . 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 8786	. . . . . . . . . . . . . . . . . . 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 8810	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↔ (dom 𝑦 ⊆ 𝐴 ∧ dom 𝑦 ∈ Fin))
39	33, 37, 38	sylanbrc 586	. . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (._g‘𝐺) = (._g‘𝐺)
41		simpl11 1245	. . . . . . . . . . . . . . . . . . . . . . 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 4126	. . . . . . . . . . . . . . . . . . . . . 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 1224	. . . . . . . . . . . . . . . . . . . . . 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 13713	. . . . . . . . . . . . . . . . . . . . . . . . 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 18246	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Mnd ∧ (♯‘𝑏) ∈ ℕ₀) → ((♯‘𝑏)(._g‘𝐺)(0_g‘𝐺)) = (0_g‘𝐺))
53	48, 51, 52	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . 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 1252	. . . . . . . . . . . . . . . . . . . . . . 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 2890	. . . . . . . . . . . . . . . . . . . . . 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 22701	. . . . . . . . . . . . . . . . . . . . 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 1299	. . . . . . . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (-_g‘𝐺) = (-_g‘𝐺)
71		simp3l 1198	. . . . . . . . . . . . . . . . . . . . . . . 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 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))) → (0_g‘𝐺) ∈ 𝑠)
73		simp2rl 1239	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
74		simp2rr 1240	. . . . . . . . . . . . . . . . . . . . . . . 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 1237	. . . . . . . . . . . . . . . . . . . . . . . 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 22758	. . . . . . . . . . . . . . . . . . . . . . 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 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1178	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1178	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1178	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1178	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1178	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1241	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑠 ∈ (TopOpen‘𝐺))
86	72	3adant3r 1178	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1239	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1307	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 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)))) → 𝑛 ∈ (𝒫 𝐶 ∩ Fin))
90		simp2ll 1237	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1240	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1244	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠))
93	92	simpld 498	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5537	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ Rel (𝐴 × 𝐶)
95		relss 5620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6088	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5534	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((dom 𝑦 ⊆ 𝑏 ∧ ran 𝑦 ⊆ 𝑛) → (dom 𝑦 × ran 𝑦) ⊆ (𝑏 × 𝑛))
100	98, 99	sylan9ss 3928	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ (dom 𝑦 ⊆ 𝑏 ∧ ran 𝑦 ⊆ 𝑛)) → 𝑦 ⊆ (𝑏 × 𝑛))
101	90, 91, 93, 100	syl12anc 835	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ (𝑏 × 𝑛)))
104		reseq2 5813	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝐹 ↾ 𝑧) = (𝐹 ↾ (𝑏 × 𝑛)))
105	104	oveq2d 7151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝐺 Σ_g (𝐹 ↾ 𝑧)) = (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))))
106	105	eleq1d 2874	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑏 × 𝑛) → ((𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣 ↔ (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣))
107	103, 106	imbi12d 348	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑏 × 𝑛) → ((𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣) ↔ (𝑦 ⊆ (𝑏 × 𝑛) → (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣)))
108		simp2lr 1238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8810	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑏 ⊆ 𝐴 ∧ 𝑏 ∈ Fin))
110		elfpw 8810	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑛 ⊆ 𝐶 ∧ 𝑛 ∈ Fin))
111		xpss12 5534	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑏 ⊆ 𝐴 ∧ 𝑛 ⊆ 𝐶) → (𝑏 × 𝑛) ⊆ (𝐴 × 𝐶))
112		xpfi 8773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑏 ∈ Fin ∧ 𝑛 ∈ Fin) → (𝑏 × 𝑛) ∈ Fin)
113	111, 112	anim12i 615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 600	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑛 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
116		elfpw 8810	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
117	115, 116	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑛 ∈ (𝒫 𝐶 ∩ Fin)) → (𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
118	87, 89, 117	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3573	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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)))) → ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))
122	121	simprd 499	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7143	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑔 = ℎ → (𝐺 Σ_g 𝑔) = (𝐺 Σ_g ℎ))
124	123	eleq1d 2874	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑔 = ℎ → ((𝐺 Σ_g 𝑔) ∈ 𝑡 ↔ (𝐺 Σ_g ℎ) ∈ 𝑡))
125	124	cbvralvw 3396	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡 ↔ ∀ℎ ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g ℎ) ∈ 𝑡)
126	122, 125	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 22759	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1116	. . . . . . . . . . . . . . . . . . . . . . . . 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 435	. . . . . . . . . . . . . . . . . . . . . . . 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 1344	. . . . . . . . . . . . . . . . . . . . . . 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 3250	. . . . . . . . . . . . . . . . . . . . . 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 1115	. . . . . . . . . . . . . . . . . . . . 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 3250	. . . . . . . . . . . . . . . . . . . 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 471	. . . . . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . . . . 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 3149	. . . . . . . . . . . . . . . . 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 3940	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = dom 𝑦 → (𝑎 ⊆ 𝑏 ↔ dom 𝑦 ⊆ 𝑏))
138	137	rspceaimv 3576	. . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . 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 3244	. . . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . 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 471	. . . . . . . . . . . 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 3243	. . . . . . . . . . 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 415	. . . . . . . . . 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 3243	. . . . . . . . 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 1118	. . . . . 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 3154	. . . 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 614	. . 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 2798	. . . 4 ⊢ (𝒫 (𝐴 × 𝐶) ∩ Fin) = (𝒫 (𝐴 × 𝐶) ∩ Fin)
154		tgptps 22685	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
155	1, 154	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TopSp)
156	60, 62	xpexd 7454	. . . 4 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V)
157	7, 6, 153, 42, 155, 156, 64	eltsms 22738	. . 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 2798	. . . 4 ⊢ (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin)
159	7, 6, 158, 42, 155, 60, 66	eltsms 22738	. . 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 297	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ (𝐺 tsums 𝐻)))
161	160	ssrdv 3921	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ (𝐺 tsums 𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 {csn 4525 ↦ cmpt 5110 × cxp 5517 dom cdm 5519 ran crn 5520 ↾ cres 5521 Rel wrel 5524 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑_m cmap 8389 Fincfn 8492 ℕ₀cn0 11885 ♯chash 13686 Basecbs 16475 +_gcplusg 16557 TopOpenctopn 16687 0_gc0g 16705 Σ_g cgsu 16706 Mndcmnd 17903 -_gcsg 18097 ._gcmg 18216 CMndccmn 18898 TopOnctopon 21515 TopSpctps 21537 TopMndctmd 22675 TopGrpctgp 22676 tsums ctsu 22731

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-rest 16688 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-mre 16849 df-mrc 16850 df-acs 16852 df-plusf 17843 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-fbas 20088 df-fg 20089 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-ntr 21625 df-nei 21703 df-cn 21832 df-cnp 21833 df-cmp 21992 df-tx 22167 df-xko 22168 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-tmd 22677 df-tgp 22678 df-tsms 22732

This theorem is referenced by: (None)

W3C validator