Theorem tmdgsum 23154

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 7263	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝐵 ↑m 𝑤) = (𝐵 ↑m ∅))
2	1	mpteq1d 5165	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)))
3		xpeq1 5594	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = (∅ × {𝐽}))
4		0xp 5675	. . . . . . . . . 10 ⊢ (∅ × {𝐽}) = ∅
5	3, 4	eqtrdi 2795	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = ∅)
6	5	fveq2d 6760	. . . . . . . 8 ⊢ (𝑤 = ∅ → (∏t‘(𝑤 × {𝐽})) = (∏t‘∅))
7	6	oveq1d 7270	. . . . . . 7 ⊢ (𝑤 = ∅ → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘∅) Cn 𝐽))
8	2, 7	eleq12d 2833	. . . . . 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 7263	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝐵 ↑m 𝑤) = (𝐵 ↑m 𝑦))
11	10	mpteq1d 5165	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m 𝑦) ↦ (𝐺 Σg 𝑥)))
12		xpeq1 5594	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 × {𝐽}) = (𝑦 × {𝐽}))
13	12	fveq2d 6760	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝑦 × {𝐽})))
14	13	oveq1d 7270	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))
15	11, 14	eleq12d 2833	. . . . . 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 7263	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐵 ↑m 𝑤) = (𝐵 ↑m (𝑦 ∪ {𝑧})))
18	17	mpteq1d 5165	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)))
19		xpeq1 5594	. . . . . . . . 9 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 × {𝐽}) = ((𝑦 ∪ {𝑧}) × {𝐽}))
20	19	fveq2d 6760	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏t‘(𝑤 × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
21	20	oveq1d 7270	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
22	18, 21	eleq12d 2833	. . . . . 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 7263	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝐵 ↑m 𝑤) = (𝐵 ↑m 𝐴))
25	24	mpteq1d 5165	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 ∈ (𝐵 ↑m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)))
26		xpeq1 5594	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑤 × {𝐽}) = (𝐴 × {𝐽}))
27	26	fveq2d 6760	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝐴 × {𝐽})))
28	27	oveq1d 7270	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
29	25, 28	eleq12d 2833	. . . . . 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 8611	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → 𝑥 Fn ∅)
32		fn0 6548	. . . . . . . . . 10 ⊢ (𝑥 Fn ∅ ↔ 𝑥 = ∅)
33	31, 32	sylib 217	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → 𝑥 = ∅)
34	33	oveq2d 7271	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → (𝐺 Σg 𝑥) = (𝐺 Σg ∅))
35		eqid 2738	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
36	35	gsum0 18283	. . . . . . . 8 ⊢ (𝐺 Σg ∅) = (0g‘𝐺)
37	34, 36	eqtrdi 2795	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) → (𝐺 Σg 𝑥) = (0g‘𝐺))
38	37	mpteq2ia 5173	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵 ↑m ∅) ↦ (0g‘𝐺))
39		0ex 5226	. . . . . . . 8 ⊢ ∅ ∈ V
40		tmdgsum.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘𝐺)
41		tmdgsum.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
42	40, 41	tmdtopon 23140	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
43	42	adantl 481	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐽 ∈ (TopOn‘𝐵))
44	4	fveq2i 6759	. . . . . . . . . 10 ⊢ (∏t‘(∅ × {𝐽})) = (∏t‘∅)
45	44	eqcomi 2747	. . . . . . . . 9 ⊢ (∏t‘∅) = (∏t‘(∅ × {𝐽}))
46	45	pttoponconst 22656	. . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘∅) ∈ (TopOn‘(𝐵 ↑m ∅)))
47	39, 43, 46	sylancr 586	. . . . . . 7 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (∏t‘∅) ∈ (TopOn‘(𝐵 ↑m ∅)))
48		tmdmnd 23134	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
49	48	adantl 481	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐺 ∈ Mnd)
50	41, 35	mndidcl 18315	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵)
51	49, 50	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (0g‘𝐺) ∈ 𝐵)
52	47, 43, 51	cnmptc 22721	. . . . . 6 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m ∅) ↦ (0g‘𝐺)) ∈ ((∏t‘∅) Cn 𝐽))
53	38, 52	eqeltrid 2843	. . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑m ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽))
54		oveq2 7263	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑤))
55	54	cbvmptv 5183	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) = (𝑤 ∈ (𝐵 ↑m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑤))
56		eqid 2738	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
57		simpl1l 1222	. . . . . . . . . . . 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 1197	. . . . . . . . . . . . . 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 8788	. . . . . . . . . . . . . 14 ⊢ {𝑧} ∈ Fin
60		unfi 8917	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
61	58, 59, 60	sylancl 585	. . . . . . . . . . . . 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 8595	. . . . . . . . . . . . 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 6771	. . . . . . . . . . . . 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 9068	. . . . . . . . . . . 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 1225	. . . . . . . . . . . . 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 4644	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
69	67, 68	sylibr 233	. . . . . . . . . . . 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 2739	. . . . . . . . . . . 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 19444	. . . . . . . . . . 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 5170	. . . . . . . . . 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 2790	. . . . . . . . 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 1196	. . . . . . . . . 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 2738	. . . . . . . . . . . 12 ⊢ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
77	76	pttoponconst 22656	. . . . . . . . . . 11 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵 ↑m (𝑦 ∪ {𝑧}))))
78	61, 75, 77	syl2anc 583	. . . . . . . . . 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 21971	. . . . . . . . . . . . . 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 5165	. . . . . . . . . . . 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 21970	. . . . . . . . . . . . . . 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 6647	. . . . . . . . . . . . . 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 4102	. . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . 14 ⊢ ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = ∪ (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
89		xpssres 5917	. . . . . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . . 15 ⊢ (𝑦 × {𝐽}) = (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦)
92	91	fveq2i 6759	. . . . . . . . . . . . . 14 ⊢ (∏t‘(𝑦 × {𝐽})) = (∏t‘(((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦))
93	88, 76, 92	ptrescn 22698	. . . . . . . . . . . . 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 1369	. . . . . . . . . . . 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 2839	. . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (∏t‘(𝑦 × {𝐽})) = (∏t‘(𝑦 × {𝐽}))
97	96	pttoponconst 22656	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵 ↑m 𝑦)))
98	58, 75, 97	syl2anc 583	. . . . . . . . . . 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 1136	. . . . . . . . . . 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 7263	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑤 ↾ 𝑦) → (𝐺 Σg 𝑥) = (𝐺 Σg (𝑤 ↾ 𝑦)))
101	78, 95, 98, 99, 100	cnmpt11 22722	. . . . . . . . . 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 6819	. . . . . . . . . . . . . . . 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 5879	. . . . . . . . . . . . . . 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 4103	. . . . . . . . . . . . . . . 16 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
105		resmpt 5934	. . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . 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 7271	. . . . . . . . . . . . 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 19317	. . . . . . . . . . . . . . 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 3426	. . . . . . . . . . . . . . 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 4595	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ {𝑧}
114		elun2 4107	. . . . . . . . . . . . . . . 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 6944	. . . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (𝑤‘𝑘) = (𝑤‘𝑧))
118	41, 117	gsumsn 19470	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑧 ∈ V ∧ (𝑤‘𝑧) ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘))) = (𝑤‘𝑧))
119	110, 112, 116, 118	syl3anc 1369	. . . . . . . . . . . . 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 2778	. . . . . . . . . . . 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 5170	. . . . . . . . . . 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 5165	. . . . . . . . . . . . 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 22670	. . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 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 2839	. . . . . . . . . . . 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 7059	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
128	83, 123, 127	syl2anc 583	. . . . . . . . . . . . 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 7271	. . . . . . . . . . . 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 2841	. . . . . . . . . . 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 2839	. . . . . . . . . 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 23146	. . . . . . . . 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 2839	. . . . . . . 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 1119	. . . . . . 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 8910	. . . 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 1115	. 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 22714	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
142	140, 141	sylan 579	. . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
143	142	3adant1 1128	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
144	143	oveq1d 7270	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → ((𝐽 ↑ko 𝒫 𝐴) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
145	139, 144	eleqtrrd 2842	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 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836   ↦ cmpt 5153   × cxp 5578   ↾ cres 5582   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  Fincfn 8691  Basecbs 16840  +_gcplusg 16888  TopOpenctopn 17049  ∏_tcpt 17066  0_gc0g 17067   Σ_g cgsu 17068  Mndcmnd 18300  CMndccmn 19301  Topctop 21950  TopOnctopon 21967   Cn ccn 22283   ↑_ko cxko 22620  TopMndctmd 23129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-fi 9100  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-rest 17050  df-0g 17069  df-gsum 17070  df-topgen 17071  df-pt 17072  df-mre 17212  df-mrc 17213  df-acs 17215  df-plusf 18240  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-mulg 18616  df-cntz 18838  df-cmn 19303  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-cn 22286  df-cnp 22287  df-cmp 22446  df-tx 22621  df-xko 22622  df-tmd 23131

This theorem is referenced by:  tmdgsum2  23155