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Theorem tmdgsum 23591

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 Σ_g 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 7414	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝐵 ↑_m 𝑤) = (𝐵 ↑_m ∅))
2	1	mpteq1d 5243	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) = (𝑥 ∈ (𝐵 ↑_m ∅) ↦ (𝐺 Σ_g 𝑥)))
3		xpeq1 5690	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = (∅ × {𝐽}))
4		0xp 5773	. . . . . . . . . 10 ⊢ (∅ × {𝐽}) = ∅
5	3, 4	eqtrdi 2789	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = ∅)
6	5	fveq2d 6893	. . . . . . . 8 ⊢ (𝑤 = ∅ → (∏_t‘(𝑤 × {𝐽})) = (∏_t‘∅))
7	6	oveq1d 7421	. . . . . . 7 ⊢ (𝑤 = ∅ → ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏_t‘∅) Cn 𝐽))
8	2, 7	eleq12d 2828	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵 ↑_m ∅) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘∅) Cn 𝐽)))
9	8	imbi2d 341	. . . . 5 ⊢ (𝑤 = ∅ → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m ∅) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘∅) Cn 𝐽))))
10		oveq2 7414	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝐵 ↑_m 𝑤) = (𝐵 ↑_m 𝑦))
11	10	mpteq1d 5243	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) = (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)))
12		xpeq1 5690	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 × {𝐽}) = (𝑦 × {𝐽}))
13	12	fveq2d 6893	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (∏_t‘(𝑤 × {𝐽})) = (∏_t‘(𝑦 × {𝐽})))
14	13	oveq1d 7421	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽))
15	11, 14	eleq12d 2828	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)))
16	15	imbi2d 341	. . . . 5 ⊢ (𝑤 = 𝑦 → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽))))
17		oveq2 7414	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐵 ↑_m 𝑤) = (𝐵 ↑_m (𝑦 ∪ {𝑧})))
18	17	mpteq1d 5243	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) = (𝑥 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 𝑥)))
19		xpeq1 5690	. . . . . . . . 9 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 × {𝐽}) = ((𝑦 ∪ {𝑧}) × {𝐽}))
20	19	fveq2d 6893	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏_t‘(𝑤 × {𝐽})) = (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})))
21	20	oveq1d 7421	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
22	18, 21	eleq12d 2828	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽)))
23	22	imbi2d 341	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))))
24		oveq2 7414	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝐵 ↑_m 𝑤) = (𝐵 ↑_m 𝐴))
25	24	mpteq1d 5243	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) = (𝑥 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑥)))
26		xpeq1 5690	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑤 × {𝐽}) = (𝐴 × {𝐽}))
27	26	fveq2d 6893	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (∏_t‘(𝑤 × {𝐽})) = (∏_t‘(𝐴 × {𝐽})))
28	27	oveq1d 7421	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏_t‘(𝐴 × {𝐽})) Cn 𝐽))
29	25, 28	eleq12d 2828	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝐴 × {𝐽})) Cn 𝐽)))
30	29	imbi2d 341	. . . . 5 ⊢ (𝑤 = 𝐴 → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝐴 × {𝐽})) Cn 𝐽))))
31		elmapfn 8856	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑_m ∅) → 𝑥 Fn ∅)
32		fn0 6679	. . . . . . . . . 10 ⊢ (𝑥 Fn ∅ ↔ 𝑥 = ∅)
33	31, 32	sylib 217	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ↑_m ∅) → 𝑥 = ∅)
34	33	oveq2d 7422	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑_m ∅) → (𝐺 Σ_g 𝑥) = (𝐺 Σ_g ∅))
35		eqid 2733	. . . . . . . . 9 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
36	35	gsum0 18600	. . . . . . . 8 ⊢ (𝐺 Σ_g ∅) = (0_g‘𝐺)
37	34, 36	eqtrdi 2789	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ↑_m ∅) → (𝐺 Σ_g 𝑥) = (0_g‘𝐺))
38	37	mpteq2ia 5251	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ↑_m ∅) ↦ (𝐺 Σ_g 𝑥)) = (𝑥 ∈ (𝐵 ↑_m ∅) ↦ (0_g‘𝐺))
39		0ex 5307	. . . . . . . 8 ⊢ ∅ ∈ V
40		tmdgsum.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘𝐺)
41		tmdgsum.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
42	40, 41	tmdtopon 23577	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
43	42	adantl 483	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐽 ∈ (TopOn‘𝐵))
44	4	fveq2i 6892	. . . . . . . . . 10 ⊢ (∏_t‘(∅ × {𝐽})) = (∏_t‘∅)
45	44	eqcomi 2742	. . . . . . . . 9 ⊢ (∏_t‘∅) = (∏_t‘(∅ × {𝐽}))
46	45	pttoponconst 23093	. . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏_t‘∅) ∈ (TopOn‘(𝐵 ↑_m ∅)))
47	39, 43, 46	sylancr 588	. . . . . . 7 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (∏_t‘∅) ∈ (TopOn‘(𝐵 ↑_m ∅)))
48		tmdmnd 23571	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
49	48	adantl 483	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐺 ∈ Mnd)
50	41, 35	mndidcl 18637	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → (0_g‘𝐺) ∈ 𝐵)
51	49, 50	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (0_g‘𝐺) ∈ 𝐵)
52	47, 43, 51	cnmptc 23158	. . . . . 6 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m ∅) ↦ (0_g‘𝐺)) ∈ ((∏_t‘∅) Cn 𝐽))
53	38, 52	eqeltrid 2838	. . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m ∅) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘∅) Cn 𝐽))
54		oveq2 7414	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝐺 Σ_g 𝑥) = (𝐺 Σ_g 𝑤))
55	54	cbvmptv 5261	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 𝑥)) = (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 𝑤))
56		eqid 2733	. . . . . . . . . . . 12 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
57		simpl1l 1225	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 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 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑦 ∈ Fin)
59		snfi 9041	. . . . . . . . . . . . . 14 ⊢ {𝑧} ∈ Fin
60		unfi 9169	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
61	58, 59, 60	sylancl 587	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑦 ∪ {𝑧}) ∈ Fin)
62	61	adantr 482	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → (𝑦 ∪ {𝑧}) ∈ Fin)
63		elmapi 8840	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) → 𝑤:(𝑦 ∪ {𝑧})⟶𝐵)
64	63	adantl 483	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → 𝑤:(𝑦 ∪ {𝑧})⟶𝐵)
65		fvexd 6904	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → (0_g‘𝐺) ∈ V)
66	64, 62, 65	fdmfifsupp 9370	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → 𝑤 finSupp (0_g‘𝐺))
67		simpl2r 1228	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → ¬ 𝑧 ∈ 𝑦)
68		disjsn 4715	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
69	67, 68	sylibr 233	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → (𝑦 ∩ {𝑧}) = ∅)
70		eqidd 2734	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 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 19791	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → (𝐺 Σ_g 𝑤) = ((𝐺 Σ_g (𝑤 ↾ 𝑦))(+_g‘𝐺)(𝐺 Σ_g (𝑤 ↾ {𝑧}))))
72	71	mpteq2dva 5248	. . . . . . . . . 10 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 𝑤)) = (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ ((𝐺 Σ_g (𝑤 ↾ 𝑦))(+_g‘𝐺)(𝐺 Σ_g (𝑤 ↾ {𝑧})))))
73	55, 72	eqtrid 2785	. . . . . . . . 9 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑥 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 𝑥)) = (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ ((𝐺 Σ_g (𝑤 ↾ 𝑦))(+_g‘𝐺)(𝐺 Σ_g (𝑤 ↾ {𝑧})))))
74		simp1r 1199	. . . . . . . . . 10 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 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 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐽 ∈ (TopOn‘𝐵))
76		eqid 2733	. . . . . . . . . . . 12 ⊢ (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽}))
77	76	pttoponconst 23093	. . . . . . . . . . 11 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵 ↑_m (𝑦 ∪ {𝑧}))))
78	61, 75, 77	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵 ↑_m (𝑦 ∪ {𝑧}))))
79		toponuni 22408	. . . . . . . . . . . . . 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 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝐵 ↑_m (𝑦 ∪ {𝑧})) = ∪ (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})))
81	80	mpteq1d 5243	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝑤 ↾ 𝑦)) = (𝑤 ∈ ∪ (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤 ↾ 𝑦)))
82		topontop 22407	. . . . . . . . . . . . . . 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 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐽 ∈ Top)
84		fconst6g 6778	. . . . . . . . . . . . . 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 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top)
86		ssun1 4172	. . . . . . . . . . . . . 14 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
87	86	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑦 ⊆ (𝑦 ∪ {𝑧}))
88		eqid 2733	. . . . . . . . . . . . . 14 ⊢ ∪ (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) = ∪ (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽}))
89		xpssres 6017	. . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑦 × {𝐽}) = (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦)
92	91	fveq2i 6892	. . . . . . . . . . . . . 14 ⊢ (∏_t‘(𝑦 × {𝐽})) = (∏_t‘(((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦))
93	88, 76, 92	ptrescn 23135	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top ∧ 𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑤 ∈ ∪ (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤 ↾ 𝑦)) ∈ ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (∏_t‘(𝑦 × {𝐽}))))
94	61, 85, 87, 93	syl3anc 1372	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 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 2834	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝑤 ↾ 𝑦)) ∈ ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (∏_t‘(𝑦 × {𝐽}))))
96		eqid 2733	. . . . . . . . . . . . 13 ⊢ (∏_t‘(𝑦 × {𝐽})) = (∏_t‘(𝑦 × {𝐽}))
97	96	pttoponconst 23093	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏_t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵 ↑_m 𝑦)))
98	58, 75, 97	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (∏_t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵 ↑_m 𝑦)))
99		simp3 1139	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽))
100		oveq2 7414	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑤 ↾ 𝑦) → (𝐺 Σ_g 𝑥) = (𝐺 Σ_g (𝑤 ↾ 𝑦)))
101	78, 95, 98, 99, 100	cnmpt11 23159	. . . . . . . . . 10 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g (𝑤 ↾ 𝑦))) ∈ ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
102	64	feqmptd 6958	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → 𝑤 = (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤‘𝑘)))
103	102	reseq1d 5979	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → (𝑤 ↾ {𝑧}) = ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤‘𝑘)) ↾ {𝑧}))
104		ssun2 4173	. . . . . . . . . . . . . . . 16 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
105		resmpt 6036	. . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → (𝑤 ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘)))
108	107	oveq2d 7422	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → (𝐺 Σ_g (𝑤 ↾ {𝑧})) = (𝐺 Σ_g (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘))))
109		cmnmnd 19660	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
110	57, 109	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → 𝐺 ∈ Mnd)
111		vex 3479	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
112	111	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → 𝑧 ∈ V)
113		vsnid 4665	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ {𝑧}
114		elun2 4177	. . . . . . . . . . . . . . . 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 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
116	64, 115	ffvelcdmd 7085	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → (𝑤‘𝑧) ∈ 𝐵)
117		fveq2 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (𝑤‘𝑘) = (𝑤‘𝑧))
118	41, 117	gsumsn 19817	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑧 ∈ V ∧ (𝑤‘𝑧) ∈ 𝐵) → (𝐺 Σ_g (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘))) = (𝑤‘𝑧))
119	110, 112, 116, 118	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → (𝐺 Σ_g (𝑘 ∈ {𝑧} ↦ (𝑤‘𝑘))) = (𝑤‘𝑧))
120	108, 119	eqtrd 2773	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧}))) → (𝐺 Σ_g (𝑤 ↾ {𝑧})) = (𝑤‘𝑧))
121	120	mpteq2dva 5248	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g (𝑤 ↾ {𝑧}))) = (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝑤‘𝑧)))
122	80	mpteq1d 5243	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 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 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
124	88, 76	ptpjcn 23107	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (𝑤 ∈ ∪ (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤‘𝑧)) ∈ ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)))
125	61, 85, 123, 124	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 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 2834	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝑤‘𝑧)) ∈ ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)))
127		fvconst2g 7200	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
128	83, 123, 127	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
129	128	oveq2d 7422	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 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 2836	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝑤‘𝑧)) ∈ ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
131	121, 130	eqeltrd 2834	. . . . . . . . . 10 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g (𝑤 ↾ {𝑧}))) ∈ ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
132	40, 56, 74, 78, 101, 131	cnmpt1plusg 23583	. . . . . . . . 9 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ ((𝐺 Σ_g (𝑤 ↾ 𝑦))(+_g‘𝐺)(𝐺 Σ_g (𝑤 ↾ {𝑧})))) ∈ ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
133	73, 132	eqeltrd 2834	. . . . . . . 8 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑥 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
134	133	3expia 1122	. . . . . . 7 ⊢ (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽) → (𝑥 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽)))
135	134	expcom 415	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → ((𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽) → (𝑥 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 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 Σ_g 𝑥)) ∈ ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽)) → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))))
137	9, 16, 23, 30, 53, 136	findcard2s 9162	. . . 4 ⊢ (𝐴 ∈ Fin → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝐴 × {𝐽})) Cn 𝐽)))
138	137	com12 32	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝐴 ∈ Fin → (𝑥 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝐴 × {𝐽})) Cn 𝐽)))
139	138	3impia 1118	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘(𝐴 × {𝐽})) Cn 𝐽))
140	42, 82	syl 17	. . . . 5 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ Top)
141		xkopt 23151	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽 ↑_ko 𝒫 𝐴) = (∏_t‘(𝐴 × {𝐽})))
142	140, 141	sylan 581	. . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ↑_ko 𝒫 𝐴) = (∏_t‘(𝐴 × {𝐽})))
143	142	3adant1 1131	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ↑_ko 𝒫 𝐴) = (∏_t‘(𝐴 × {𝐽})))
144	143	oveq1d 7421	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → ((𝐽 ↑_ko 𝒫 𝐴) Cn 𝐽) = ((∏_t‘(𝐴 × {𝐽})) Cn 𝐽))
145	139, 144	eleqtrrd 2837	1 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑥)) ∈ ((𝐽 ↑_ko 𝒫 𝐴) Cn 𝐽))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {csn 4628 ∪ cuni 4908 ↦ cmpt 5231 × cxp 5674 ↾ cres 5678 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ↑_m cmap 8817 Fincfn 8936 Basecbs 17141 +_gcplusg 17194 TopOpenctopn 17364 ∏_tcpt 17381 0_gc0g 17382 Σ_g cgsu 17383 Mndcmnd 18622 CMndccmn 19643 Topctop 22387 TopOnctopon 22404 Cn ccn 22720 ↑_ko cxko 23057 TopMndctmd 23566

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-rest 17365 df-0g 17384 df-gsum 17385 df-topgen 17386 df-pt 17387 df-mre 17527 df-mrc 17528 df-acs 17530 df-plusf 18557 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-submnd 18669 df-mulg 18946 df-cntz 19176 df-cmn 19645 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-cn 22723 df-cnp 22724 df-cmp 22883 df-tx 23058 df-xko 23059 df-tmd 23568

This theorem is referenced by: tmdgsum2 23592

W3C validator