Theorem tmdgsum 24104

Description: In a topological monoid, the group sum operation is a continuous function from the function space to the base topology. This theorem is not true when 𝐴 is infinite, because in this case for any basic open set of the domain one of the factors will be the whole space, so by varying the value of the functions to sum at this index, one can achieve any desired sum. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)

Hypotheses

Ref	Expression
tmdgsum.j	⊢ 𝐽 = (TopOpen‘𝐺)
tmdgsum.b	⊢ 𝐵 = (Base‘𝐺)

Assertion

Ref	Expression
tmdgsum	⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((𝐽 ↑ko 𝒫 𝐴) Cn 𝐽))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝐵   𝑥,𝐺

Proof of Theorem tmdgsum

Dummy variables 𝑘 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7440	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝐵 ↑m 𝑤) = (𝐵 ↑m ∅))
2	1	mpteq1d 5236	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)))
3		xpeq1 5698	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = (∅ × {𝐽}))
4		0xp 5783	. . . . . . . . . 10 ⊢ (∅ × {𝐽}) = ∅
5	3, 4	eqtrdi 2792	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = ∅)
6	5	fveq2d 6909	. . . . . . . 8 ⊢ (𝑤 = ∅ → (∏t‘(𝑤 × {𝐽})) = (∏t‘∅))
7	6	oveq1d 7447	. . . . . . 7 ⊢ (𝑤 = ∅ → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘∅) Cn 𝐽))
8	2, 7	eleq12d 2834	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽)))
9	8	imbi2d 340	. . . . 5 ⊢ (𝑤 = ∅ → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽))))
10		oveq2 7440	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝐵 ↑m 𝑤) = (𝐵 ↑m 𝑦))
11	10	mpteq1d 5236	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)))
12		xpeq1 5698	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 × {𝐽}) = (𝑦 × {𝐽}))
13	12	fveq2d 6909	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝑦 × {𝐽})))
14	13	oveq1d 7447	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))
15	11, 14	eleq12d 2834	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)))
16	15	imbi2d 340	. . . . 5 ⊢ (𝑤 = 𝑦 → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))))
17		oveq2 7440	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐵 ↑m 𝑤) = (𝐵 ↑m (𝑦 ∪ {𝑧})))
18	17	mpteq1d 5236	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)))
19		xpeq1 5698	. . . . . . . . 9 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 × {𝐽}) = ((𝑦 ∪ {𝑧}) × {𝐽}))
20	19	fveq2d 6909	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏t‘(𝑤 × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
21	20	oveq1d 7447	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
22	18, 21	eleq12d 2834	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽)))
23	22	imbi2d 340	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))))
24		oveq2 7440	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝐵 ↑m 𝑤) = (𝐵 ↑m 𝐴))
25	24	mpteq1d 5236	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)))
26		xpeq1 5698	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑤 × {𝐽}) = (𝐴 × {𝐽}))
27	26	fveq2d 6909	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝐴 × {𝐽})))
28	27	oveq1d 7447	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
29	25, 28	eleq12d 2834	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽)))
30	29	imbi2d 340	. . . . 5 ⊢ (𝑤 = 𝐴 → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))))
31		elmapfn 8906	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → 𝑥 Fn ∅)
32		fn0 6698	. . . . . . . . . 10 ⊢ (𝑥 Fn ∅ ↔ 𝑥 = ∅)
33	31, 32	sylib 218	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → 𝑥 = ∅)
34	33	oveq2d 7448	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → (𝐺 Σg 𝑥) = (𝐺 Σg ∅))
35		eqid 2736	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
36	35	gsum0 18698	. . . . . . . 8 ⊢ (𝐺 Σg ∅) = (0g‘𝐺)
37	34, 36	eqtrdi 2792	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → (𝐺 Σg 𝑥) = (0g‘𝐺))
38	37	mpteq2ia 5244	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m ∅) ↦ (0g‘𝐺))
39		0ex 5306	. . . . . . . 8 ⊢ ∅ ∈ V
40		tmdgsum.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘𝐺)
41		tmdgsum.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
42	40, 41	tmdtopon 24090	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
43	42	adantl 481	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐽 ∈ (TopOn‘𝐵))
44	4	fveq2i 6908	. . . . . . . . . 10 ⊢ (∏t‘(∅ × {𝐽})) = (∏t‘∅)
45	44	eqcomi 2745	. . . . . . . . 9 ⊢ (∏t‘∅) = (∏t‘(∅ × {𝐽}))
46	45	pttoponconst 23606	. . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘∅) ∈ (TopOn‘(𝐵 ↑m ∅)))
47	39, 43, 46	sylancr 587	. . . . . . 7 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (∏t‘∅) ∈ (TopOn‘(𝐵 ↑m ∅)))
48		tmdmnd 24084	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
49	48	adantl 481	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐺 ∈ Mnd)
50	41, 35	mndidcl 18763	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵)
51	49, 50	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (0g‘𝐺) ∈ 𝐵)
52	47, 43, 51	cnmptc 23671	. . . . . 6 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m ∅) ↦ (0g‘𝐺)) ∈ ((∏t‘∅) Cn 𝐽))
53	38, 52	eqeltrid 2844	. . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽))
54		oveq2 7440	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑤))
55	54	cbvmptv 5254	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) = (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑤))
56		eqid 2736	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
57		simpl1l 1224	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → 𝐺 ∈ CMnd)
58		simp2l 1199	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑦 ∈ Fin)
59		snfi 9084	. . . . . . . . . . . . . 14 ⊢ {𝑧} ∈ Fin
60		unfi 9212	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
61	58, 59, 60	sylancl 586	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑦 ∪ {𝑧}) ∈ Fin)
62	61	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → (𝑦 ∪ {𝑧}) ∈ Fin)
63		elmapi 8890	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) → 𝑤:(𝑦 ∪ {𝑧})⟶𝐵)
64	63	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → 𝑤:(𝑦 ∪ {𝑧})⟶𝐵)
65		fvexd 6920	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → (0g‘𝐺) ∈ V)
66	64, 62, 65	fdmfifsupp 9416	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → 𝑤 finSupp (0g‘𝐺))
67		simpl2r 1227	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → ¬ 𝑧 ∈ 𝑦)
68		disjsn 4710	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
69	67, 68	sylibr 234	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → (𝑦 ∩ {𝑧}) = ∅)
70		eqidd 2737	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
71	41, 35, 56, 57, 62, 64, 66, 69, 70	gsumsplit 19947	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → (𝐺 Σg 𝑤) = ((𝐺 Σg (𝑤 ↾ 𝑦))(+g‘𝐺)(𝐺 Σg (𝑤 ↾ {𝑧}))))
72	71	mpteq2dva 5241	. . . . . . . . . 10 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑤)) = (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ ((𝐺 Σg (𝑤 ↾ 𝑦))(+g‘𝐺)(𝐺 Σg (𝑤 ↾ {𝑧})))))
73	55, 72	eqtrid 2788	. . . . . . . . 9 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) = (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ ((𝐺 Σg (𝑤 ↾ 𝑦))(+g‘𝐺)(𝐺 Σg (𝑤 ↾ {𝑧})))))
74		simp1r 1198	. . . . . . . . . 10 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐺 ∈ TopMnd)
75	74, 42	syl 17	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐽 ∈ (TopOn‘𝐵))
76		eqid 2736	. . . . . . . . . . . 12 ⊢ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
77	76	pttoponconst 23606	. . . . . . . . . . 11 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵 ↑m (𝑦 ∪ {𝑧}))))
78	61, 75, 77	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵 ↑m (𝑦 ∪ {𝑧}))))
79		toponuni 22921	. . . . . . . . . . . . . 14 ⊢ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵 ↑m (𝑦 ∪ {𝑧}))) → (𝐵 ↑m (𝑦 ∪ {𝑧})) = ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
80	78, 79	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝐵 ↑m (𝑦 ∪ {𝑧})) = ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
81	80	mpteq1d 5236	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝑤 ↾ 𝑦)) = (𝑤 ∈ ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤 ↾ 𝑦)))
82		topontop 22920	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
83	74, 42, 82	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐽 ∈ Top)
84		fconst6g 6796	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top)
85	83, 84	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top)
86		ssun1 4177	. . . . . . . . . . . . . 14 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
87	86	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑦 ⊆ (𝑦 ∪ {𝑧}))
88		eqid 2736	. . . . . . . . . . . . . 14 ⊢ ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
89		xpssres 6035	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦) = (𝑦 × {𝐽}))
90	86, 89	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦) = (𝑦 × {𝐽})
91	90	eqcomi 2745	. . . . . . . . . . . . . . 15 ⊢ (𝑦 × {𝐽}) = (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦)
92	91	fveq2i 6908	. . . . . . . . . . . . . 14 ⊢ (∏t‘(𝑦 × {𝐽})) = (∏t‘(((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦))
93	88, 76, 92	ptrescn 23648	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top ∧ 𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑤 ∈ ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤 ↾ 𝑦)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (∏t‘(𝑦 × {𝐽}))))
94	61, 85, 87, 93	syl3anc 1372	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤 ↾ 𝑦)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (∏t‘(𝑦 × {𝐽}))))
95	81, 94	eqeltrd 2840	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝑤 ↾ 𝑦)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (∏t‘(𝑦 × {𝐽}))))
96		eqid 2736	. . . . . . . . . . . . 13 ⊢ (∏t‘(𝑦 × {𝐽})) = (∏t‘(𝑦 × {𝐽}))
97	96	pttoponconst 23606	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵 ↑m 𝑦)))
98	58, 75, 97	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (∏t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵 ↑m 𝑦)))
99		simp3 1138	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))
100		oveq2 7440	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑤 ↾ 𝑦) → (𝐺 Σg 𝑥) = (𝐺 Σg (𝑤 ↾ 𝑦)))
101	78, 95, 98, 99, 100	cnmpt11 23672	. . . . . . . . . 10 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg (𝑤 ↾ 𝑦))) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
102	64	feqmptd 6976	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → 𝑤 = (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤‘𝑘)))
103	102	reseq1d 5995	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → (𝑤 ↾ {𝑧}) = ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤‘𝑘)) ↾ {𝑧}))
104		ssun2 4178	. . . . . . . . . . . . . . . 16 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
105		resmpt 6054	. . . . . . . . . . . . . . . 16 ⊢ ({𝑧} ⊆ (𝑦 ∪ {𝑧}) → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤‘𝑘)) ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘)))
106	104, 105	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤‘𝑘)) ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘))
107	103, 106	eqtrdi 2792	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → (𝑤 ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘)))
108	107	oveq2d 7448	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → (𝐺 Σg (𝑤 ↾ {𝑧})) = (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘))))
109		cmnmnd 19816	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
110	57, 109	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → 𝐺 ∈ Mnd)
111		vex 3483	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
112	111	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → 𝑧 ∈ V)
113		vsnid 4662	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ {𝑧}
114		elun2 4182	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
115	113, 114	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
116	64, 115	ffvelcdmd 7104	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → (𝑤‘𝑧) ∈ 𝐵)
117		fveq2 6905	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (𝑤‘𝑘) = (𝑤‘𝑧))
118	41, 117	gsumsn 19973	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑧 ∈ V ∧ (𝑤‘𝑧) ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘))) = (𝑤‘𝑧))
119	110, 112, 116, 118	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘))) = (𝑤‘𝑧))
120	108, 119	eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧}))) → (𝐺 Σg (𝑤 ↾ {𝑧})) = (𝑤‘𝑧))
121	120	mpteq2dva 5241	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg (𝑤 ↾ {𝑧}))) = (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝑤‘𝑧)))
122	80	mpteq1d 5236	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝑤‘𝑧)) = (𝑤 ∈ ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤‘𝑧)))
123	113, 114	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
124	88, 76	ptpjcn 23620	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (𝑤 ∈ ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤‘𝑧)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)))
125	61, 85, 123, 124	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤‘𝑧)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)))
126	122, 125	eqeltrd 2840	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝑤‘𝑧)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)))
127		fvconst2g 7223	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
128	83, 123, 127	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
129	128	oveq2d 7448	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)) = ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
130	126, 129	eleqtrd 2842	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝑤‘𝑧)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
131	121, 130	eqeltrd 2840	. . . . . . . . . 10 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg (𝑤 ↾ {𝑧}))) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
132	40, 56, 74, 78, 101, 131	cnmpt1plusg 24096	. . . . . . . . 9 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ ((𝐺 Σg (𝑤 ↾ 𝑦))(+g‘𝐺)(𝐺 Σg (𝑤 ↾ {𝑧})))) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
133	73, 132	eqeltrd 2840	. . . . . . . 8 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
134	133	3expia 1121	. . . . . . 7 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽) → (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽)))
135	134	expcom 413	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → ((𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽) → (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))))
136	135	a2d 29	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))))
137	9, 16, 23, 30, 53, 136	findcard2s 9206	. . . 4 ⊢ (𝐴 ∈ Fin → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽)))
138	137	com12 32	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝐴 ∈ Fin → (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽)))
139	138	3impia 1117	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
140	42, 82	syl 17	. . . . 5 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ Top)
141		xkopt 23664	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
142	140, 141	sylan 580	. . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
143	142	3adant1 1130	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
144	143	oveq1d 7447	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → ((𝐽 ↑ko 𝒫 𝐴) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
145	139, 144	eleqtrrd 2843	1 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((𝐽 ↑ko 𝒫 𝐴) Cn 𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  Vcvv 3479   ∪ cun 3948   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  𝒫 cpw 4599  {csn 4625  ∪ cuni 4906   ↦ cmpt 5224   × cxp 5682   ↾ cres 5686   Fn wfn 6555  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ↑_m cmap 8867  Fincfn 8986  Basecbs 17248  +_gcplusg 17298  TopOpenctopn 17467  ∏_tcpt 17484  0_gc0g 17485   Σ_g cgsu 17486  Mndcmnd 18748  CMndccmn 19799  Topctop 22900  TopOnctopon 22917   Cn ccn 23233   ↑_ko cxko 23570  TopMndctmd 24079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-fi 9452  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-rest 17468  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-mre 17630  df-mrc 17631  df-acs 17633  df-plusf 18653  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-submnd 18798  df-mulg 19087  df-cntz 19336  df-cmn 19801  df-top 22901  df-topon 22918  df-topsp 22940  df-bases 22954  df-cn 23236  df-cnp 23237  df-cmp 23396  df-tx 23571  df-xko 23572  df-tmd 24081

This theorem is referenced by:  tmdgsum2  24105