tmdgsum - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tmdgsum		Structured version Visualization version GIF version

Theorem tmdgsum 23819

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 7419	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝐵 ↑_m 𝑤) = (𝐵 ↑_m ∅))
2	1	mpteq1d 5242	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) = (𝑥 ∈ (𝐵 ↑_m ∅) ↦ (𝐺 Σ_g 𝑥)))
3		xpeq1 5689	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = (∅ × {𝐽}))
4		0xp 5773	. . . . . . . . . 10 ⊢ (∅ × {𝐽}) = ∅
5	3, 4	eqtrdi 2786	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 × {𝐽}) = ∅)
6	5	fveq2d 6894	. . . . . . . 8 ⊢ (𝑤 = ∅ → (∏_t‘(𝑤 × {𝐽})) = (∏_t‘∅))
7	6	oveq1d 7426	. . . . . . 7 ⊢ (𝑤 = ∅ → ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏_t‘∅) Cn 𝐽))
8	2, 7	eleq12d 2825	. . . . . 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 339	. . . . 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 7419	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝐵 ↑_m 𝑤) = (𝐵 ↑_m 𝑦))
11	10	mpteq1d 5242	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) = (𝑥 ∈ (𝐵 ↑_m 𝑦) ↦ (𝐺 Σ_g 𝑥)))
12		xpeq1 5689	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 × {𝐽}) = (𝑦 × {𝐽}))
13	12	fveq2d 6894	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (∏_t‘(𝑤 × {𝐽})) = (∏_t‘(𝑦 × {𝐽})))
14	13	oveq1d 7426	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏_t‘(𝑦 × {𝐽})) Cn 𝐽))
15	11, 14	eleq12d 2825	. . . . . 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 339	. . . . 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 7419	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐵 ↑_m 𝑤) = (𝐵 ↑_m (𝑦 ∪ {𝑧})))
18	17	mpteq1d 5242	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) = (𝑥 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 𝑥)))
19		xpeq1 5689	. . . . . . . . 9 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 × {𝐽}) = ((𝑦 ∪ {𝑧}) × {𝐽}))
20	19	fveq2d 6894	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏_t‘(𝑤 × {𝐽})) = (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})))
21	20	oveq1d 7426	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
22	18, 21	eleq12d 2825	. . . . . 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 339	. . . . 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 7419	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝐵 ↑_m 𝑤) = (𝐵 ↑_m 𝐴))
25	24	mpteq1d 5242	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 ∈ (𝐵 ↑_m 𝑤) ↦ (𝐺 Σ_g 𝑥)) = (𝑥 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑥)))
26		xpeq1 5689	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑤 × {𝐽}) = (𝐴 × {𝐽}))
27	26	fveq2d 6894	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (∏_t‘(𝑤 × {𝐽})) = (∏_t‘(𝐴 × {𝐽})))
28	27	oveq1d 7426	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((∏_t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏_t‘(𝐴 × {𝐽})) Cn 𝐽))
29	25, 28	eleq12d 2825	. . . . . 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 339	. . . . 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 8861	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑_m ∅) → 𝑥 Fn ∅)
32		fn0 6680	. . . . . . . . . 10 ⊢ (𝑥 Fn ∅ ↔ 𝑥 = ∅)
33	31, 32	sylib 217	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ↑_m ∅) → 𝑥 = ∅)
34	33	oveq2d 7427	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑_m ∅) → (𝐺 Σ_g 𝑥) = (𝐺 Σ_g ∅))
35		eqid 2730	. . . . . . . . 9 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
36	35	gsum0 18609	. . . . . . . 8 ⊢ (𝐺 Σ_g ∅) = (0_g‘𝐺)
37	34, 36	eqtrdi 2786	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ↑_m ∅) → (𝐺 Σ_g 𝑥) = (0_g‘𝐺))
38	37	mpteq2ia 5250	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ↑_m ∅) ↦ (𝐺 Σ_g 𝑥)) = (𝑥 ∈ (𝐵 ↑_m ∅) ↦ (0_g‘𝐺))
39		0ex 5306	. . . . . . . 8 ⊢ ∅ ∈ V
40		tmdgsum.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘𝐺)
41		tmdgsum.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
42	40, 41	tmdtopon 23805	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
43	42	adantl 480	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐽 ∈ (TopOn‘𝐵))
44	4	fveq2i 6893	. . . . . . . . . 10 ⊢ (∏_t‘(∅ × {𝐽})) = (∏_t‘∅)
45	44	eqcomi 2739	. . . . . . . . 9 ⊢ (∏_t‘∅) = (∏_t‘(∅ × {𝐽}))
46	45	pttoponconst 23321	. . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏_t‘∅) ∈ (TopOn‘(𝐵 ↑_m ∅)))
47	39, 43, 46	sylancr 585	. . . . . . 7 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (∏_t‘∅) ∈ (TopOn‘(𝐵 ↑_m ∅)))
48		tmdmnd 23799	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
49	48	adantl 480	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐺 ∈ Mnd)
50	41, 35	mndidcl 18674	. . . . . . . 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 23386	. . . . . 6 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m ∅) ↦ (0_g‘𝐺)) ∈ ((∏_t‘∅) Cn 𝐽))
53	38, 52	eqeltrid 2835	. . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵 ↑_m ∅) ↦ (𝐺 Σ_g 𝑥)) ∈ ((∏_t‘∅) Cn 𝐽))
54		oveq2 7419	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝐺 Σ_g 𝑥) = (𝐺 Σ_g 𝑤))
55	54	cbvmptv 5260	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 𝑥)) = (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σ_g 𝑤))
56		eqid 2730	. . . . . . . . . . . 12 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
57		simpl1l 1222	. . . . . . . . . . . 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 1197	. . . . . . . . . . . . . 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 9046	. . . . . . . . . . . . . 14 ⊢ {𝑧} ∈ Fin
60		unfi 9174	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
61	58, 59, 60	sylancl 584	. . . . . . . . . . . . 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 479	. . . . . . . . . . . 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 8845	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐵 ↑_m (𝑦 ∪ {𝑧})) → 𝑤:(𝑦 ∪ {𝑧})⟶𝐵)
64	63	adantl 480	. . . . . . . . . . . 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 6905	. . . . . . . . . . . . 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 9375	. . . . . . . . . . . 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 1225	. . . . . . . . . . . . 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 4714	. . . . . . . . . . . . 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 2731	. . . . . . . . . . . 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 19837	. . . . . . . . . . 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 5247	. . . . . . . . . 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 2782	. . . . . . . . 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 1196	. . . . . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽}))
77	76	pttoponconst 23321	. . . . . . . . . . 11 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏_t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵 ↑_m (𝑦 ∪ {𝑧}))))
78	61, 75, 77	syl2anc 582	. . . . . . . . . 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 22636	. . . . . . . . . . . . . 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 5242	. . . . . . . . . . . 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 22635	. . . . . . . . . . . . . . 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 6779	. . . . . . . . . . . . . 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 4171	. . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑦 × {𝐽}) = (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦)
92	91	fveq2i 6893	. . . . . . . . . . . . . 14 ⊢ (∏_t‘(𝑦 × {𝐽})) = (∏_t‘(((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦))
93	88, 76, 92	ptrescn 23363	. . . . . . . . . . . . 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 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 2831	. . . . . . . . . . 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 2730	. . . . . . . . . . . . 13 ⊢ (∏_t‘(𝑦 × {𝐽})) = (∏_t‘(𝑦 × {𝐽}))
97	96	pttoponconst 23321	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏_t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵 ↑_m 𝑦)))
98	58, 75, 97	syl2anc 582	. . . . . . . . . . 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 1136	. . . . . . . . . . 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 7419	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑤 ↾ 𝑦) → (𝐺 Σ_g 𝑥) = (𝐺 Σ_g (𝑤 ↾ 𝑦)))
101	78, 95, 98, 99, 100	cnmpt11 23387	. . . . . . . . . 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 6959	. . . . . . . . . . . . . . . 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 4172	. . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . 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 19706	. . . . . . . . . . . . . . 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 3476	. . . . . . . . . . . . . . 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 4664	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ {𝑧}
114		elun2 4176	. . . . . . . . . . . . . . . 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 7086	. . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (𝑤‘𝑘) = (𝑤‘𝑧))
118	41, 117	gsumsn 19863	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑧 ∈ V ∧ (𝑤‘𝑧) ∈ 𝐵) → (𝐺 Σ_g (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 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 2770	. . . . . . . . . . . 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 5247	. . . . . . . . . . 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 5242	. . . . . . . . . . . . 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 23335	. . . . . . . . . . . . . 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 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 2831	. . . . . . . . . . . 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 7204	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
128	83, 123, 127	syl2anc 582	. . . . . . . . . . . . 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 7427	. . . . . . . . . . . 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 2833	. . . . . . . . . . 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 2831	. . . . . . . . . 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 23811	. . . . . . . . 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 2831	. . . . . . . 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 1119	. . . . . . 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 412	. . . . . 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 9167	. . . 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 1115	. 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 23379	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽 ↑_ko 𝒫 𝐴) = (∏_t‘(𝐴 × {𝐽})))
142	140, 141	sylan 578	. . . 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 7426	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → ((𝐽 ↑_ko 𝒫 𝐴) Cn 𝐽) = ((∏_t‘(𝐴 × {𝐽})) Cn 𝐽))
145	139, 144	eleqtrrd 2834	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 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 {csn 4627 ∪ cuni 4907 ↦ cmpt 5230 × cxp 5673 ↾ cres 5677 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ↑_m cmap 8822 Fincfn 8941 Basecbs 17148 +_gcplusg 17201 TopOpenctopn 17371 ∏_tcpt 17388 0_gc0g 17389 Σ_g cgsu 17390 Mndcmnd 18659 CMndccmn 19689 Topctop 22615 TopOnctopon 22632 Cn ccn 22948 ↑_ko cxko 23285 TopMndctmd 23794

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-rest 17372 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-mre 17534 df-mrc 17535 df-acs 17537 df-plusf 18564 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cn 22951 df-cnp 22952 df-cmp 23111 df-tx 23286 df-xko 23287 df-tmd 23796

This theorem is referenced by: tmdgsum2 23820

W3C validator