Theorem tmdgsum 24155

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 7404	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝐵 ↑m 𝑤) = (𝐵 ↑m ∅))
2	1	mpteq1d 5190	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)))
3		xpeq1 5661	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = (∅ × {𝐽}))
4		0xp 5746	. . . . . . . . . 10 ⊢ (∅ × {𝐽}) = ∅
5	3, 4	eqtrdi 2813	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = ∅)
6	5	fveq2d 6871	. . . . . . . 8 ⊢ (𝑤 = ∅ → (∏t‘(𝑤 × {𝐽})) = (∏t‘∅))
7	6	oveq1d 7411	. . . . . . 7 ⊢ (𝑤 = ∅ → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘∅) Cn 𝐽))
8	2, 7	eleq12d 2856	. . . . . 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 342	. . . . 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 7404	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝐵 ↑m 𝑤) = (𝐵 ↑m 𝑦))
11	10	mpteq1d 5190	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)))
12		xpeq1 5661	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 × {𝐽}) = (𝑦 × {𝐽}))
13	12	fveq2d 6871	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝑦 × {𝐽})))
14	13	oveq1d 7411	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))
15	11, 14	eleq12d 2856	. . . . . 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 342	. . . . 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 7404	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐵 ↑m 𝑤) = (𝐵 ↑m (𝑦 ∪ {𝑧})))
18	17	mpteq1d 5190	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)))
19		xpeq1 5661	. . . . . . . . 9 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 × {𝐽}) = ((𝑦 ∪ {𝑧}) × {𝐽}))
20	19	fveq2d 6871	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏t‘(𝑤 × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
21	20	oveq1d 7411	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
22	18, 21	eleq12d 2856	. . . . . 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 342	. . . . 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 7404	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝐵 ↑m 𝑤) = (𝐵 ↑m 𝐴))
25	24	mpteq1d 5190	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)))
26		xpeq1 5661	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑤 × {𝐽}) = (𝐴 × {𝐽}))
27	26	fveq2d 6871	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝐴 × {𝐽})))
28	27	oveq1d 7411	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
29	25, 28	eleq12d 2856	. . . . . 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 342	. . . . 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 8846	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → 𝑥 Fn ∅)
32		fn0 6652	. . . . . . . . . 10 ⊢ (𝑥 Fn ∅ ↔ 𝑥 = ∅)
33	31, 32	sylib 220	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → 𝑥 = ∅)
34	33	oveq2d 7412	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → (𝐺 Σg 𝑥) = (𝐺 Σg ∅))
35		eqid 2762	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
36	35	gsum0 18718	. . . . . . . 8 ⊢ (𝐺 Σg ∅) = (0g‘𝐺)
37	34, 36	eqtrdi 2813	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → (𝐺 Σg 𝑥) = (0g‘𝐺))
38	37	mpteq2ia 5195	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m ∅) ↦ (0g‘𝐺))
39		0ex 5257	. . . . . . . 8 ⊢ ∅ ∈ V
40		tmdgsum.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘𝐺)
41		tmdgsum.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
42	40, 41	tmdtopon 24141	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
43	42	adantl 485	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐽 ∈ (TopOn‘𝐵))
44	4	fveq2i 6870	. . . . . . . . . 10 ⊢ (∏t‘(∅ × {𝐽})) = (∏t‘∅)
45	44	eqcomi 2771	. . . . . . . . 9 ⊢ (∏t‘∅) = (∏t‘(∅ × {𝐽}))
46	45	pttoponconst 23657	. . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘∅) ∈ (TopOn‘(𝐵 ↑m ∅)))
47	39, 43, 46	sylancr 596	. . . . . . 7 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (∏t‘∅) ∈ (TopOn‘(𝐵 ↑m ∅)))
48		tmdmnd 24135	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
49	48	adantl 485	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐺 ∈ Mnd)
50	41, 35	mndidcl 18783	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵)
51	49, 50	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (0g‘𝐺) ∈ 𝐵)
52	47, 43, 51	cnmptc 23722	. . . . . 6 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m ∅) ↦ (0g‘𝐺)) ∈ ((∏t‘∅) Cn 𝐽))
53	38, 52	eqeltrid 2866	. . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽))
54		oveq2 7404	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑤))
55	54	cbvmptv 5204	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) = (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑤))
56		eqid 2762	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
57		simpl1l 1238	. . . . . . . . . . . 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 1213	. . . . . . . . . . . . . 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 9024	. . . . . . . . . . . . . 14 ⊢ {𝑧} ∈ Fin
60		unfi 9139	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
61	58, 59, 60	sylancl 595	. . . . . . . . . . . . 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 8830	. . . . . . . . . . . . 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 6882	. . . . . . . . . . . . 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 9321	. . . . . . . . . . . 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 1241	. . . . . . . . . . . . 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 4670	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
69	67, 68	sylibr 236	. . . . . . . . . . . 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 2763	. . . . . . . . . . . 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 19968	. . . . . . . . . . 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 5193	. . . . . . . . . 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 2809	. . . . . . . . 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 1212	. . . . . . . . . 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 2762	. . . . . . . . . . . 12 ⊢ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
77	76	pttoponconst 23657	. . . . . . . . . . 11 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵 ↑m (𝑦 ∪ {𝑧}))))
78	61, 75, 77	syl2anc 593	. . . . . . . . . 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 22974	. . . . . . . . . . . . . 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 5190	. . . . . . . . . . . 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 22973	. . . . . . . . . . . . . . 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 6753	. . . . . . . . . . . . . 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 4130	. . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . 14 ⊢ ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
89		xpssres 6004	. . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 15 ⊢ (𝑦 × {𝐽}) = (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦)
92	91	fveq2i 6870	. . . . . . . . . . . . . 14 ⊢ (∏t‘(𝑦 × {𝐽})) = (∏t‘(((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦))
93	88, 76, 92	ptrescn 23699	. . . . . . . . . . . . 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 1390	. . . . . . . . . . . 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 2862	. . . . . . . . . . 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 2762	. . . . . . . . . . . . 13 ⊢ (∏t‘(𝑦 × {𝐽})) = (∏t‘(𝑦 × {𝐽}))
97	96	pttoponconst 23657	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵 ↑m 𝑦)))
98	58, 75, 97	syl2anc 593	. . . . . . . . . . 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 1151	. . . . . . . . . . 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 7404	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑤 ↾ 𝑦) → (𝐺 Σg 𝑥) = (𝐺 Σg (𝑤 ↾ 𝑦)))
101	78, 95, 98, 99, 100	cnmpt11 23723	. . . . . . . . . 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 6935	. . . . . . . . . . . . . . . 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 5964	. . . . . . . . . . . . . . 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 4131	. . . . . . . . . . . . . . . 16 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
105		resmpt 6026	. . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . 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 7412	. . . . . . . . . . . . 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 19837	. . . . . . . . . . . . . . 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 3458	. . . . . . . . . . . . . . 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 4622	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ {𝑧}
114		elun2 4135	. . . . . . . . . . . . . . . 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 7066	. . . . . . . . . . . . . 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 6867	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (𝑤‘𝑘) = (𝑤‘𝑧))
118	41, 117	gsumsn 19994	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑧 ∈ V ∧ (𝑤‘𝑧) ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘))) = (𝑤‘𝑧))
119	110, 112, 116, 118	syl3anc 1390	. . . . . . . . . . . . 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 2797	. . . . . . . . . . . 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 5193	. . . . . . . . . . 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 5190	. . . . . . . . . . . . 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 23671	. . . . . . . . . . . . . 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 1390	. . . . . . . . . . . . 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 2862	. . . . . . . . . . . 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 7186	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
128	83, 123, 127	syl2anc 593	. . . . . . . . . . . . 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 7412	. . . . . . . . . . . 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 2864	. . . . . . . . . . 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 2862	. . . . . . . . . 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 24147	. . . . . . . . 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 2862	. . . . . . . 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 1134	. . . . . . 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 9134	. . . 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 1130	. 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 23715	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
142	140, 141	sylan 589	. . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
143	142	3adant1 1143	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
144	143	oveq1d 7411	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → ((𝐽 ↑ko 𝒫 𝐴) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
145	139, 144	eleqtrrd 2865	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 1098   = wceq 1560   ∈ wcel 2142  Vcvv 3454   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4555  {csn 4582  ∪ cuni 4865   ↦ cmpt 5181   × cxp 5645   ↾ cres 5649   Fn wfn 6516  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   ↑_m cmap 8808  Fincfn 8927  Basecbs 17245  +_gcplusg 17286  TopOpenctopn 17450  ∏_tcpt 17467  0_gc0g 17468   Σ_g cgsu 17469  Mndcmnd 18768  CMndccmn 19820  Topctop 22953  TopOnctopon 22970   Cn ccn 23284   ↑_ko cxko 23621  TopMndctmd 24130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9308  df-fi 9357  df-oi 9458  df-card 9897  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-n0 12482  df-z 12569  df-uz 12840  df-fz 13513  df-fzo 13660  df-seq 14015  df-hash 14344  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-ress 17267  df-plusg 17299  df-rest 17451  df-0g 17470  df-gsum 17471  df-topgen 17472  df-pt 17473  df-mre 17614  df-mrc 17615  df-acs 17617  df-plusf 18673  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-submnd 18818  df-mulg 19110  df-cntz 19357  df-cmn 19822  df-top 22954  df-topon 22971  df-topsp 22993  df-bases 23006  df-cn 23287  df-cnp 23288  df-cmp 23447  df-tx 23622  df-xko 23623  df-tmd 24132

This theorem is referenced by:  tmdgsum2  24156