tsmsxp - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tsmsxp		Structured version Visualization version GIF version

Theorem tsmsxp 24049

Description: Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 19913 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 24047 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 23973	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ TopMnd)
4	3	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝐺 ∈ TopMnd)
5		simp2 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑢 ∈ (TopOpen‘𝐺))
6		eqid 2730	. . . . . . . . . . . . 13 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
7		tsmsxp.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐺)
8	6, 7	tmdtopon 23975	. . . . . . . . . . . 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 22821	. . . . . . . . . . 11 ⊢ (((TopOpen‘𝐺) ∈ (TopOn‘𝐵) ∧ 𝑢 ∈ (TopOpen‘𝐺)) → 𝑢 ⊆ 𝐵)
11	9, 5, 10	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑢 ⊆ 𝐵)
12		simp3 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑥 ∈ 𝑢)
13	11, 12	sseldd 3950	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝑥 ∈ 𝐵)
14		tmdmnd 23969	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
15	4, 14	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → 𝐺 ∈ Mnd)
16		eqid 2730	. . . . . . . . . . 11 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
17	7, 16	mndidcl 18683	. . . . . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
20	7, 19, 16	mndrid 18689	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) = 𝑥)
21	15, 13, 20	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) = 𝑥)
22	21, 12	eqeltrd 2829	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (𝑥(+_g‘𝐺)(0_g‘𝐺)) ∈ 𝑢)
23	7, 6, 19	tmdcn2 23983	. . . . . . . . 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 1379	. . . . . . . 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 3095	. . . . . . . . 9 ⊢ ((∀𝑣 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ ∃𝑣 ∈ (TopOpen‘𝐺)∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → ∃𝑣 ∈ (TopOpen‘𝐺)((𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ ∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)))
26		simp31 1210	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → 𝑥 ∈ 𝑣)
27		elfpw 9312	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ (𝑦 ⊆ (𝐴 × 𝐶) ∧ 𝑦 ∈ Fin))
28	27	simplbi 497	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ⊆ (𝐴 × 𝐶))
29	28	ad2antrl 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → 𝑦 ⊆ (𝐴 × 𝐶))
30		dmss 5869	. . . . . . . . . . . . . . . . . . . 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 6147	. . . . . . . . . . . . . . . . . . 19 ⊢ dom (𝐴 × 𝐶) ⊆ 𝐴
33	31, 32	sstrdi 3962	. . . . . . . . . . . . . . . . . 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 4168	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ∈ Fin)
35	34	ad2antrl 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → 𝑦 ∈ Fin)
36		dmfi 9293	. . . . . . . . . . . . . . . . . . 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 9312	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↔ (dom 𝑦 ⊆ 𝐴 ∧ dom 𝑦 ∈ Fin))
39	33, 37, 38	sylanbrc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → dom 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
40		eqid 2730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (._g‘𝐺) = (._g‘𝐺)
41		simpl11 1249	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝜑)
42		tsmsxp.g	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐺 ∈ CMnd)
43	41, 42	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝐺 ∈ CMnd)
44	41, 3	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝐺 ∈ TopMnd)
45		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
46	45	elin2d 4171	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝑏 ∈ Fin)
47		simpl2r 1228	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝑡 ∈ (TopOpen‘𝐺))
48	44, 14	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → 𝐺 ∈ Mnd)
49	48, 17	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → (0_g‘𝐺) ∈ 𝐵)
50		hashcl 14328	. . . . . . . . . . . . . . . . . . . . . . . . 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 19040	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Mnd ∧ (♯‘𝑏) ∈ ℕ₀) → ((♯‘𝑏)(._g‘𝐺)(0_g‘𝐺)) = (0_g‘𝐺))
53	48, 51, 52	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → ((♯‘𝑏)(._g‘𝐺)(0_g‘𝐺)) = (0_g‘𝐺))
54		simpl32 1256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → (0_g‘𝐺) ∈ 𝑡)
55	53, 54	eqeltrd 2829	. . . . . . . . . . . . . . . . . . . . . 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 23990	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → ∃𝑠 ∈ (TopOpen‘𝐺)((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))
57		simp111 1303	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝜑)
58	57, 42	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝐺 ∈ CMnd)
59	57, 1	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝐺 ∈ TopGrp)
60		tsmsxp.a	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
61	57, 60	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝐴 ∈ 𝑉)
62		tsmsxp.c	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
63	57, 62	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝐶 ∈ 𝑊)
64		tsmsxp.f	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)
65	57, 64	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
66		tsmsxp.h	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
67	57, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝐻:𝐴⟶𝐵)
68		tsmsxp.1	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))
69	57, 68	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))
70		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (-_g‘𝐺) = (-_g‘𝐺)
71		simp3l 1202	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝑠 ∈ (TopOpen‘𝐺))
72		simp3rl 1247	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → (0_g‘𝐺) ∈ 𝑠)
73		simp2rl 1243	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
74		simp2rr 1244	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → dom 𝑦 ⊆ 𝑏)
75		simp2ll 1241	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) → 𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
76	7, 58, 59, 61, 63, 65, 67, 69, 6, 16, 19, 70, 71, 72, 73, 74, 75	tsmsxplem1 24047	. . . . . . . . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐺 ∈ CMnd)
78	59	3adant3r 1182	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐺 ∈ TopGrp)
79	61	3adant3r 1182	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐴 ∈ 𝑉)
80	63	3adant3r 1182	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐶 ∈ 𝑊)
81	65	3adant3r 1182	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
82	67	3adant3r 1182	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐻:𝐴⟶𝐵)
83	41	3adant3 1132	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝜑)
84	83, 68	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))
85		simp3ll 1245	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑠 ∈ (TopOpen‘𝐺))
86	72	3adant3r 1182	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (0_g‘𝐺) ∈ 𝑠)
87		simp2rl 1243	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
88		simp133 1311	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)
89		simp3rl 1247	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑛 ∈ (𝒫 𝐶 ∩ Fin))
90		simp2ll 1241	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
91		simp2rr 1244	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → dom 𝑦 ⊆ 𝑏)
92		simp3rr 1248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠))
93	92	simpld 494	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ran 𝑦 ⊆ 𝑛)
94		relxp 5659	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ Rel (𝐴 × 𝐶)
95		relss 5747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6244	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((dom 𝑦 ⊆ 𝑏 ∧ ran 𝑦 ⊆ 𝑛) → (dom 𝑦 × ran 𝑦) ⊆ (𝑏 × 𝑛))
100	98, 99	sylan9ss 3963	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ (dom 𝑦 ⊆ 𝑏 ∧ ran 𝑦 ⊆ 𝑛)) → 𝑦 ⊆ (𝑏 × 𝑛))
101	90, 91, 93, 100	syl12anc 836	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑦 ⊆ (𝑏 × 𝑛))
102	92	simprd 495	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)
103		sseq2 3976	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ (𝑏 × 𝑛)))
104		reseq2 5948	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝐹 ↾ 𝑧) = (𝐹 ↾ (𝑏 × 𝑛)))
105	104	oveq2d 7406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑏 × 𝑛) → (𝐺 Σ_g (𝐹 ↾ 𝑧)) = (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))))
106	105	eleq1d 2814	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑏 × 𝑛) → ((𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣 ↔ (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣))
107	103, 106	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑏 × 𝑛) → ((𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣) ↔ (𝑦 ⊆ (𝑏 × 𝑛) → (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣)))
108		simp2lr 1242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))
109		elfpw 9312	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑏 ⊆ 𝐴 ∧ 𝑏 ∈ Fin))
110		elfpw 9312	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑛 ⊆ 𝐶 ∧ 𝑛 ∈ Fin))
111		xpss12 5656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑏 ⊆ 𝐴 ∧ 𝑛 ⊆ 𝐶) → (𝑏 × 𝑛) ⊆ (𝐴 × 𝐶))
112		xpfi 9276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑏 ∈ Fin ∧ 𝑛 ∈ Fin) → (𝑏 × 𝑛) ∈ Fin)
113	111, 112	anim12i 613	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑏 ⊆ 𝐴 ∧ 𝑛 ⊆ 𝐶) ∧ (𝑏 ∈ Fin ∧ 𝑛 ∈ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
114	113	an4s 660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑏 ⊆ 𝐴 ∧ 𝑏 ∈ Fin) ∧ (𝑛 ⊆ 𝐶 ∧ 𝑛 ∈ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
115	109, 110, 114	syl2anb 598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑛 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
116		elfpw 9312	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
117	115, 116	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑛 ∈ (𝒫 𝐶 ∩ Fin)) → (𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
118	87, 89, 117	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
119	107, 108, 118	rspcdva 3592	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (𝑦 ⊆ (𝑏 × 𝑛) → (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣))
120	101, 119	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (𝐺 Σ_g (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣)
121		simp3lr 1246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))
122	121	simprd 495	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)
123		oveq2 7398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑔 = ℎ → (𝐺 Σ_g 𝑔) = (𝐺 Σ_g ℎ))
124	123	eleq1d 2814	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑔 = ℎ → ((𝐺 Σ_g 𝑔) ∈ 𝑡 ↔ (𝐺 Σ_g ℎ) ∈ 𝑡))
125	124	cbvralvw 3216	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡 ↔ ∀ℎ ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g ℎ) ∈ 𝑡)
126	122, 125	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀ℎ ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g ℎ) ∈ 𝑡)
127	7, 77, 78, 79, 80, 81, 82, 84, 6, 16, 19, 70, 85, 86, 87, 88, 89, 101, 102, 120, 126	tsmsxplem2 24048	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → (((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) → (((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠))) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
129	128	exp4a 431	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → (((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) → ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡)) → ((𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))))
130	129	3imp1 1348	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0_g‘𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠 ↑_m 𝑏)(𝐺 Σ_g 𝑔) ∈ 𝑡))) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝑏 ((𝐻‘𝑥)(-_g‘𝐺)(𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠))) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)
131	76, 130	rexlimddv 3141	. . . . . . . . . . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . . . . . . . . . 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 3141	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏))) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)
134	133	anassrs 467	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦 ⊆ 𝑏)) → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)
135	134	expr 456	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → (dom 𝑦 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))
136	135	ralrimiva 3126	. . . . . . . . . . . . . . . . 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 3975	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = dom 𝑦 → (𝑎 ⊆ 𝑏 ↔ dom 𝑦 ⊆ 𝑏))
138	137	rspceaimv 3597	. . . . . . . . . . . . . . . . 17 ⊢ ((dom 𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∧ ∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(dom 𝑦 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))
139	39, 136, 138	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))
140	139	rexlimdvaa 3136	. . . . . . . . . . . . . . 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 1121	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺))) → ((𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢) → ((𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))))
143	142	anassrs 467	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘𝐺)) ∧ 𝑡 ∈ (TopOpen‘𝐺)) → ((𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢) → ((𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))))
144	143	rexlimdva 3135	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘𝐺)) → (∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢) → ((𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))))
145	144	impcomd 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘𝐺)) → (((𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ ∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
146	145	rexlimdva 3135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (∃𝑣 ∈ (TopOpen‘𝐺)((𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ ∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
147	25, 146	syl5 34	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → ((∀𝑣 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) ∧ ∃𝑣 ∈ (TopOpen‘𝐺)∃𝑡 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 ∧ (0_g‘𝐺) ∈ 𝑡 ∧ ∀𝑐 ∈ 𝑣 ∀𝑑 ∈ 𝑡 (𝑐(+_g‘𝐺)𝑑) ∈ 𝑢)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
148	24, 147	mpan2d 694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥 ∈ 𝑢) → (∀𝑣 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))
149	148	3expia 1121	. . . . . 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 3134	. . . 4 ⊢ (𝜑 → (∀𝑣 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣)) → ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢))))
152	151	anim2d 612	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ ∀𝑣 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑣))) → (𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))))
153		eqid 2730	. . . 4 ⊢ (𝒫 (𝐴 × 𝐶) ∩ Fin) = (𝒫 (𝐴 × 𝐶) ∩ Fin)
154		tgptps 23974	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
155	1, 154	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TopSp)
156	60, 62	xpexd 7730	. . . 4 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V)
157	7, 6, 153, 42, 155, 156, 64	eltsms 24027	. . 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 2730	. . . 4 ⊢ (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin)
159	7, 6, 158, 42, 155, 60, 66	eltsms 24027	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐻) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g (𝐻 ↾ 𝑏)) ∈ 𝑢)))))
160	152, 157, 159	3imtr4d 294	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ (𝐺 tsums 𝐻)))
161	160	ssrdv 3955	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ (𝐺 tsums 𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ∩ cin 3916 ⊆ wss 3917 𝒫 cpw 4566 {csn 4592 ↦ cmpt 5191 × cxp 5639 dom cdm 5641 ran crn 5642 ↾ cres 5643 Rel wrel 5646 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ↑_m cmap 8802 Fincfn 8921 ℕ₀cn0 12449 ♯chash 14302 Basecbs 17186 +_gcplusg 17227 TopOpenctopn 17391 0_gc0g 17409 Σ_g cgsu 17410 Mndcmnd 18668 -_gcsg 18874 ._gcmg 19006 CMndccmn 19717 TopOnctopon 22804 TopSpctps 22826 TopMndctmd 23964 TopGrpctgp 23965 tsums ctsu 24020

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623 df-seq 13974 df-hash 14303 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-rest 17392 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-mre 17554 df-mrc 17555 df-acs 17557 df-plusf 18573 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-mulg 19007 df-ghm 19152 df-cntz 19256 df-cmn 19719 df-abl 19720 df-fbas 21268 df-fg 21269 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-ntr 22914 df-nei 22992 df-cn 23121 df-cnp 23122 df-cmp 23281 df-tx 23456 df-xko 23457 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-tmd 23966 df-tgp 23967 df-tsms 24021

This theorem is referenced by: (None)

W3C validator