Theorem tmdgsum 22808

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 7164	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝐵 ↑m 𝑤) = (𝐵 ↑m ∅))
2	1	mpteq1d 5125	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)))
3		xpeq1 5542	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = (∅ × {𝐽}))
4		0xp 5623	. . . . . . . . . 10 ⊢ (∅ × {𝐽}) = ∅
5	3, 4	eqtrdi 2809	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = ∅)
6	5	fveq2d 6667	. . . . . . . 8 ⊢ (𝑤 = ∅ → (∏t‘(𝑤 × {𝐽})) = (∏t‘∅))
7	6	oveq1d 7171	. . . . . . 7 ⊢ (𝑤 = ∅ → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘∅) Cn 𝐽))
8	2, 7	eleq12d 2846	. . . . . 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 344	. . . . 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 7164	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝐵 ↑m 𝑤) = (𝐵 ↑m 𝑦))
11	10	mpteq1d 5125	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)))
12		xpeq1 5542	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 × {𝐽}) = (𝑦 × {𝐽}))
13	12	fveq2d 6667	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝑦 × {𝐽})))
14	13	oveq1d 7171	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))
15	11, 14	eleq12d 2846	. . . . . 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 344	. . . . 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 7164	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐵 ↑m 𝑤) = (𝐵 ↑m (𝑦 ∪ {𝑧})))
18	17	mpteq1d 5125	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)))
19		xpeq1 5542	. . . . . . . . 9 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 × {𝐽}) = ((𝑦 ∪ {𝑧}) × {𝐽}))
20	19	fveq2d 6667	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏t‘(𝑤 × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
21	20	oveq1d 7171	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
22	18, 21	eleq12d 2846	. . . . . 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 344	. . . . 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 7164	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝐵 ↑m 𝑤) = (𝐵 ↑m 𝐴))
25	24	mpteq1d 5125	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)))
26		xpeq1 5542	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑤 × {𝐽}) = (𝐴 × {𝐽}))
27	26	fveq2d 6667	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝐴 × {𝐽})))
28	27	oveq1d 7171	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
29	25, 28	eleq12d 2846	. . . . . 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 344	. . . . 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 8460	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → 𝑥 Fn ∅)
32		fn0 6467	. . . . . . . . . 10 ⊢ (𝑥 Fn ∅ ↔ 𝑥 = ∅)
33	31, 32	sylib 221	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → 𝑥 = ∅)
34	33	oveq2d 7172	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → (𝐺 Σg 𝑥) = (𝐺 Σg ∅))
35		eqid 2758	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
36	35	gsum0 17973	. . . . . . . 8 ⊢ (𝐺 Σg ∅) = (0g‘𝐺)
37	34, 36	eqtrdi 2809	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → (𝐺 Σg 𝑥) = (0g‘𝐺))
38	37	mpteq2ia 5127	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m ∅) ↦ (0g‘𝐺))
39		0ex 5181	. . . . . . . 8 ⊢ ∅ ∈ V
40		tmdgsum.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘𝐺)
41		tmdgsum.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
42	40, 41	tmdtopon 22794	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
43	42	adantl 485	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐽 ∈ (TopOn‘𝐵))
44	4	fveq2i 6666	. . . . . . . . . 10 ⊢ (∏t‘(∅ × {𝐽})) = (∏t‘∅)
45	44	eqcomi 2767	. . . . . . . . 9 ⊢ (∏t‘∅) = (∏t‘(∅ × {𝐽}))
46	45	pttoponconst 22310	. . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘∅) ∈ (TopOn‘(𝐵 ↑m ∅)))
47	39, 43, 46	sylancr 590	. . . . . . 7 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (∏t‘∅) ∈ (TopOn‘(𝐵 ↑m ∅)))
48		tmdmnd 22788	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
49	48	adantl 485	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐺 ∈ Mnd)
50	41, 35	mndidcl 18005	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵)
51	49, 50	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (0g‘𝐺) ∈ 𝐵)
52	47, 43, 51	cnmptc 22375	. . . . . 6 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m ∅) ↦ (0g‘𝐺)) ∈ ((∏t‘∅) Cn 𝐽))
53	38, 52	eqeltrid 2856	. . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽))
54		oveq2 7164	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑤))
55	54	cbvmptv 5139	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) = (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑤))
56		eqid 2758	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
57		simpl1l 1221	. . . . . . . . . . . 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 1196	. . . . . . . . . . . . . 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 8627	. . . . . . . . . . . . . 14 ⊢ {𝑧} ∈ Fin
60		unfi 8754	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
61	58, 59, 60	sylancl 589	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 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 8444	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) → 𝑤:(𝑦 ∪ {𝑧})⟶𝐵)
64	63	adantl 485	. . . . . . . . . . . 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 6678	. . . . . . . . . . . . 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 8889	. . . . . . . . . . . 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 1224	. . . . . . . . . . . . 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 4607	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
69	67, 68	sylibr 237	. . . . . . . . . . . 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 2759	. . . . . . . . . . . 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 19129	. . . . . . . . . . 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 5131	. . . . . . . . . 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	syl5eq 2805	. . . . . . . . 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 1195	. . . . . . . . . 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 2758	. . . . . . . . . . . 12 ⊢ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
77	76	pttoponconst 22310	. . . . . . . . . . 11 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵 ↑m (𝑦 ∪ {𝑧}))))
78	61, 75, 77	syl2anc 587	. . . . . . . . . 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 21627	. . . . . . . . . . . . . 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 5125	. . . . . . . . . . . 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 21626	. . . . . . . . . . . . . . 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 6558	. . . . . . . . . . . . . 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 4079	. . . . . . . . . . . . . 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 2758	. . . . . . . . . . . . . 14 ⊢ ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
89		xpssres 5865	. . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . 15 ⊢ (𝑦 × {𝐽}) = (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦)
92	91	fveq2i 6666	. . . . . . . . . . . . . 14 ⊢ (∏t‘(𝑦 × {𝐽})) = (∏t‘(((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦))
93	88, 76, 92	ptrescn 22352	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 2852	. . . . . . . . . . 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 2758	. . . . . . . . . . . . 13 ⊢ (∏t‘(𝑦 × {𝐽})) = (∏t‘(𝑦 × {𝐽}))
97	96	pttoponconst 22310	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵 ↑m 𝑦)))
98	58, 75, 97	syl2anc 587	. . . . . . . . . . 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 1135	. . . . . . . . . . 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 7164	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑤 ↾ 𝑦) → (𝐺 Σg 𝑥) = (𝐺 Σg (𝑤 ↾ 𝑦)))
101	78, 95, 98, 99, 100	cnmpt11 22376	. . . . . . . . . 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 6726	. . . . . . . . . . . . . . . 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 5827	. . . . . . . . . . . . . . 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 4080	. . . . . . . . . . . . . . . 16 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
105		resmpt 5882	. . . . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . . 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 7172	. . . . . . . . . . . . 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 19002	. . . . . . . . . . . . . . 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 3413	. . . . . . . . . . . . . . 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 4562	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ {𝑧}
114		elun2 4084	. . . . . . . . . . . . . . . 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	ffvelrnd 6849	. . . . . . . . . . . . . 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 6663	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (𝑤‘𝑘) = (𝑤‘𝑧))
118	41, 117	gsumsn 19155	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑧 ∈ V ∧ (𝑤‘𝑧) ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘))) = (𝑤‘𝑧))
119	110, 112, 116, 118	syl3anc 1368	. . . . . . . . . . . . 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 2793	. . . . . . . . . . . 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 5131	. . . . . . . . . . 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 5125	. . . . . . . . . . . . 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 22324	. . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . 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 2852	. . . . . . . . . . . 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 6961	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
128	83, 123, 127	syl2anc 587	. . . . . . . . . . . . 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 7172	. . . . . . . . . . . 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 2854	. . . . . . . . . . 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 2852	. . . . . . . . . 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 22800	. . . . . . . . 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 2852	. . . . . . . 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 1118	. . . . . . 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 417	. . . . . 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 8749	. . . 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 1114	. 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 22368	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
142	140, 141	sylan 583	. . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
143	142	3adant1 1127	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
144	143	oveq1d 7171	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → ((𝐽 ↑ko 𝒫 𝐴) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
145	139, 144	eleqtrrd 2855	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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∪ cun 3858   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801   ↦ cmpt 5116   × cxp 5526   ↾ cres 5530   Fn wfn 6335  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156   ↑_m cmap 8422  Fincfn 8540  Basecbs 16554  +_gcplusg 16636  TopOpenctopn 16766  ∏_tcpt 16783  0_gc0g 16784   Σ_g cgsu 16785  Mndcmnd 17990  CMndccmn 18986  Topctop 21606  TopOnctopon 21623   Cn ccn 21937   ↑_ko cxko 22274  TopMndctmd 22783

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 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-isom 6349  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7411  df-om 7586  df-1st 7699  df-2nd 7700  df-supp 7842  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-er 8305  df-map 8424  df-ixp 8493  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-fsupp 8880  df-fi 8921  df-oi 9020  df-card 9414  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-2 11750  df-n0 11948  df-z 12034  df-uz 12296  df-fz 12953  df-fzo 13096  df-seq 13432  df-hash 13754  df-ndx 16557  df-slot 16558  df-base 16560  df-sets 16561  df-ress 16562  df-plusg 16649  df-rest 16767  df-0g 16786  df-gsum 16787  df-topgen 16788  df-pt 16789  df-mre 16928  df-mrc 16929  df-acs 16931  df-plusf 17930  df-mgm 17931  df-sgrp 17980  df-mnd 17991  df-submnd 18036  df-mulg 18305  df-cntz 18527  df-cmn 18988  df-top 21607  df-topon 21624  df-topsp 21646  df-bases 21659  df-cn 21940  df-cnp 21941  df-cmp 22100  df-tx 22275  df-xko 22276  df-tmd 22785

This theorem is referenced by:  tmdgsum2  22809