Theorem tmdgsum 23980

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 7357	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝐵 ↑m 𝑤) = (𝐵 ↑m ∅))
2	1	mpteq1d 5182	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)))
3		xpeq1 5633	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = (∅ × {𝐽}))
4		0xp 5718	. . . . . . . . . 10 ⊢ (∅ × {𝐽}) = ∅
5	3, 4	eqtrdi 2780	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = ∅)
6	5	fveq2d 6826	. . . . . . . 8 ⊢ (𝑤 = ∅ → (∏t‘(𝑤 × {𝐽})) = (∏t‘∅))
7	6	oveq1d 7364	. . . . . . 7 ⊢ (𝑤 = ∅ → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘∅) Cn 𝐽))
8	2, 7	eleq12d 2822	. . . . . 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 7357	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝐵 ↑m 𝑤) = (𝐵 ↑m 𝑦))
11	10	mpteq1d 5182	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)))
12		xpeq1 5633	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 × {𝐽}) = (𝑦 × {𝐽}))
13	12	fveq2d 6826	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝑦 × {𝐽})))
14	13	oveq1d 7364	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))
15	11, 14	eleq12d 2822	. . . . . 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 7357	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐵 ↑m 𝑤) = (𝐵 ↑m (𝑦 ∪ {𝑧})))
18	17	mpteq1d 5182	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)))
19		xpeq1 5633	. . . . . . . . 9 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 × {𝐽}) = ((𝑦 ∪ {𝑧}) × {𝐽}))
20	19	fveq2d 6826	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏t‘(𝑤 × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
21	20	oveq1d 7364	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
22	18, 21	eleq12d 2822	. . . . . 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 7357	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝐵 ↑m 𝑤) = (𝐵 ↑m 𝐴))
25	24	mpteq1d 5182	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)))
26		xpeq1 5633	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑤 × {𝐽}) = (𝐴 × {𝐽}))
27	26	fveq2d 6826	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝐴 × {𝐽})))
28	27	oveq1d 7364	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
29	25, 28	eleq12d 2822	. . . . . 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 8792	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → 𝑥 Fn ∅)
32		fn0 6613	. . . . . . . . . 10 ⊢ (𝑥 Fn ∅ ↔ 𝑥 = ∅)
33	31, 32	sylib 218	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → 𝑥 = ∅)
34	33	oveq2d 7365	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → (𝐺 Σg 𝑥) = (𝐺 Σg ∅))
35		eqid 2729	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
36	35	gsum0 18558	. . . . . . . 8 ⊢ (𝐺 Σg ∅) = (0g‘𝐺)
37	34, 36	eqtrdi 2780	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → (𝐺 Σg 𝑥) = (0g‘𝐺))
38	37	mpteq2ia 5187	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m ∅) ↦ (0g‘𝐺))
39		0ex 5246	. . . . . . . 8 ⊢ ∅ ∈ V
40		tmdgsum.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘𝐺)
41		tmdgsum.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
42	40, 41	tmdtopon 23966	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
43	42	adantl 481	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐽 ∈ (TopOn‘𝐵))
44	4	fveq2i 6825	. . . . . . . . . 10 ⊢ (∏t‘(∅ × {𝐽})) = (∏t‘∅)
45	44	eqcomi 2738	. . . . . . . . 9 ⊢ (∏t‘∅) = (∏t‘(∅ × {𝐽}))
46	45	pttoponconst 23482	. . . . . . . 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 23960	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
49	48	adantl 481	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐺 ∈ Mnd)
50	41, 35	mndidcl 18623	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵)
51	49, 50	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (0g‘𝐺) ∈ 𝐵)
52	47, 43, 51	cnmptc 23547	. . . . . 6 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m ∅) ↦ (0g‘𝐺)) ∈ ((∏t‘∅) Cn 𝐽))
53	38, 52	eqeltrid 2832	. . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽))
54		oveq2 7357	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑤))
55	54	cbvmptv 5196	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) = (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑤))
56		eqid 2729	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
57		simpl1l 1225	. . . . . . . . . . . 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 1200	. . . . . . . . . . . . . 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 8968	. . . . . . . . . . . . . 14 ⊢ {𝑧} ∈ Fin
60		unfi 9085	. . . . . . . . . . . . . 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 8776	. . . . . . . . . . . . 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 6837	. . . . . . . . . . . . 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 9265	. . . . . . . . . . . 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 1228	. . . . . . . . . . . . 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 4663	. . . . . . . . . . . . 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 2730	. . . . . . . . . . . 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 19807	. . . . . . . . . . 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 5185	. . . . . . . . . 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 2776	. . . . . . . . 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 1199	. . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
77	76	pttoponconst 23482	. . . . . . . . . . 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 22799	. . . . . . . . . . . . . 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 5182	. . . . . . . . . . . 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 22798	. . . . . . . . . . . . . . 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 6713	. . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . 14 ⊢ ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
89		xpssres 5969	. . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑦 × {𝐽}) = (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦)
92	91	fveq2i 6825	. . . . . . . . . . . . . 14 ⊢ (∏t‘(𝑦 × {𝐽})) = (∏t‘(((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦))
93	88, 76, 92	ptrescn 23524	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 2828	. . . . . . . . . . 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 2729	. . . . . . . . . . . . 13 ⊢ (∏t‘(𝑦 × {𝐽})) = (∏t‘(𝑦 × {𝐽}))
97	96	pttoponconst 23482	. . . . . . . . . . . 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 7357	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑤 ↾ 𝑦) → (𝐺 Σg 𝑥) = (𝐺 Σg (𝑤 ↾ 𝑦)))
101	78, 95, 98, 99, 100	cnmpt11 23548	. . . . . . . . . 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 6891	. . . . . . . . . . . . . . . 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 5929	. . . . . . . . . . . . . . 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 4130	. . . . . . . . . . . . . . . 16 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
105		resmpt 5988	. . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . 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 7365	. . . . . . . . . . . . 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 19676	. . . . . . . . . . . . . . 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 3440	. . . . . . . . . . . . . . 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 4615	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ {𝑧}
114		elun2 4134	. . . . . . . . . . . . . . . 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 7019	. . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (𝑤‘𝑘) = (𝑤‘𝑧))
118	41, 117	gsumsn 19833	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑧 ∈ V ∧ (𝑤‘𝑧) ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘))) = (𝑤‘𝑧))
119	110, 112, 116, 118	syl3anc 1373	. . . . . . . . . . . . 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 2764	. . . . . . . . . . . 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 5185	. . . . . . . . . . 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 5182	. . . . . . . . . . . . 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 23496	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 2828	. . . . . . . . . . . 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 7138	. . . . . . . . . . . . . 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 7365	. . . . . . . . . . . 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 2830	. . . . . . . . . . 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 2828	. . . . . . . . . 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 23972	. . . . . . . . 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 2828	. . . . . . . 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 9079	. . . 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 23540	. . . . 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 7364	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → ((𝐽 ↑ko 𝒫 𝐴) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
145	139, 144	eleqtrrd 2831	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 1540   ∈ wcel 2109  Vcvv 3436   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  {csn 4577  ∪ cuni 4858   ↦ cmpt 5173   × cxp 5617   ↾ cres 5621   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753  Fincfn 8872  Basecbs 17120  +_gcplusg 17161  TopOpenctopn 17325  ∏_tcpt 17342  0_gc0g 17343   Σ_g cgsu 17344  Mndcmnd 18608  CMndccmn 19659  Topctop 22778  TopOnctopon 22795   Cn ccn 23109   ↑_ko cxko 23446  TopMndctmd 23955

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-fi 9301  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-fzo 13558  df-seq 13909  df-hash 14238  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-rest 17326  df-0g 17345  df-gsum 17346  df-topgen 17347  df-pt 17348  df-mre 17488  df-mrc 17489  df-acs 17491  df-plusf 18513  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-submnd 18658  df-mulg 18947  df-cntz 19196  df-cmn 19661  df-top 22779  df-topon 22796  df-topsp 22818  df-bases 22831  df-cn 23112  df-cnp 23113  df-cmp 23272  df-tx 23447  df-xko 23448  df-tmd 23957

This theorem is referenced by:  tmdgsum2  23981