Theorem psrplusgg 14945

Description: The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
psrplusg.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psrplusg.b	⊢ 𝐵 = (Base‘𝑆)
psrplusg.a	⊢ + = (+g‘𝑅)
psrplusg.p	⊢ ✚ = (+g‘𝑆)

Assertion

Ref	Expression
psrplusgg	⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ✚ = ( ∘𝑓 + ↾ (𝐵 × 𝐵)))

Proof of Theorem psrplusgg

Dummy variables 𝑓 𝑔 𝑘 𝑥 ℎ 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrplusg.s	. . . 4 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		eqid 2234	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		psrplusg.a	. . . 4 ⊢ + = (+g‘𝑅)
4		eqid 2234	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
5		eqid 2234	. . . 4 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
6		eqid 2234	. . . 4 ⊢ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
7		psrplusg.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
8		simpl 109	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐼 ∈ 𝑉)
9		simpr 110	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑅 ∈ 𝑊)
10	1, 2, 6, 7, 8, 9	psrbasg 14941	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐵 = ((Base‘𝑅) ↑𝑚 {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}))
11		eqid 2234	. . . 4 ⊢ ( ∘𝑓 + ↾ (𝐵 × 𝐵)) = ( ∘𝑓 + ↾ (𝐵 × 𝐵))
12		eqid 2234	. . . 4 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥))))))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))
13		eqid 2234	. . . 4 ⊢ (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓)) = (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓))
14		eqidd 2235	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})) = (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})))
15	1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 8, 9	psrval 14926	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑆 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩}))
16	15	fveq2d 5679	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (+g‘𝑆) = (+g‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})))
17		psrplusg.p	. . 3 ⊢ ✚ = (+g‘𝑆)
18	17	a1i 9	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ✚ = (+g‘𝑆))
19		plusgslid 13409	. . 3 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
20		basfn 13355	. . . . . 6 ⊢ Base Fn V
21		fnpsr 14927	. . . . . . . 8 ⊢ mPwSer Fn (V × V)
22	8	elexd 2829	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐼 ∈ V)
23	9	elexd 2829	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑅 ∈ V)
24		fnovex 6091	. . . . . . . 8 ⊢ (( mPwSer Fn (V × V) ∧ 𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) ∈ V)
25	21, 22, 23, 24	mp3an2i 1379	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐼 mPwSer 𝑅) ∈ V)
26	1, 25	eqeltrid 2321	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑆 ∈ V)
27		funfvex 5692	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
28	27	funfni 5463	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
29	20, 26, 28	sylancr 414	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (Base‘𝑆) ∈ V)
30	7, 29	eqeltrid 2321	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐵 ∈ V)
31	30, 30	ofmresex 6343	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ( ∘𝑓 + ↾ (𝐵 × 𝐵)) ∈ V)
32		mpoexga 6421	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥))))))) ∈ V)
33	30, 30, 32	syl2anc 411	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥))))))) ∈ V)
34		funfvex 5692	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
35	34	funfni 5463	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
36	20, 23, 35	sylancr 414	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (Base‘𝑅) ∈ V)
37		mpoexga 6421	. . . . 5 ⊢ (((Base‘𝑅) ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓)) ∈ V)
38	36, 30, 37	syl2anc 411	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓)) ∈ V)
39		fnmap 6902	. . . . . . . 8 ⊢ ↑𝑚 Fn (V × V)
40		nn0ex 9519	. . . . . . . . 9 ⊢ ℕ0 ∈ V
41	40	a1i 9	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ℕ0 ∈ V)
42		fnovex 6091	. . . . . . . 8 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℕ0 ∈ V ∧ 𝐼 ∈ V) → (ℕ0 ↑𝑚 𝐼) ∈ V)
43	39, 41, 22, 42	mp3an2i 1379	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (ℕ0 ↑𝑚 𝐼) ∈ V)
44		rabexg 4260	. . . . . . 7 ⊢ ((ℕ0 ↑𝑚 𝐼) ∈ V → {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V)
45	43, 44	syl 14	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V)
46		topnfn 13541	. . . . . . . 8 ⊢ TopOpen Fn V
47		funfvex 5692	. . . . . . . . 9 ⊢ ((Fun TopOpen ∧ 𝑅 ∈ dom TopOpen) → (TopOpen‘𝑅) ∈ V)
48	47	funfni 5463	. . . . . . . 8 ⊢ ((TopOpen Fn V ∧ 𝑅 ∈ V) → (TopOpen‘𝑅) ∈ V)
49	46, 23, 48	sylancr 414	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (TopOpen‘𝑅) ∈ V)
50		snexg 4302	. . . . . . 7 ⊢ ((TopOpen‘𝑅) ∈ V → {(TopOpen‘𝑅)} ∈ V)
51	49, 50	syl 14	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → {(TopOpen‘𝑅)} ∈ V)
52		xpexg 4869	. . . . . 6 ⊢ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V ∧ {(TopOpen‘𝑅)} ∈ V) → ({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}) ∈ V)
53	45, 51, 52	syl2anc 411	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}) ∈ V)
54		ptex 13561	. . . . 5 ⊢ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}) ∈ V → (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})) ∈ V)
55	53, 54	syl 14	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})) ∈ V)
56	30, 31, 33, 9, 38, 55	psrvalstrd 14928	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩}) Struct ⟨1, 9⟩)
57		plusgndxnn 13408	. . . . 5 ⊢ (+g‘ndx) ∈ ℕ
58		opexg 4349	. . . . 5 ⊢ (((+g‘ndx) ∈ ℕ ∧ ( ∘𝑓 + ↾ (𝐵 × 𝐵)) ∈ V) → ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩ ∈ V)
59	57, 31, 58	sylancr 414	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩ ∈ V)
60		snsstp2 3850	. . . . . 6 ⊢ {⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩}
61		ssun1 3386	. . . . . 6 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})
62	60, 61	sstri 3251	. . . . 5 ⊢ {⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})
63		snssg 3833	. . . . 5 ⊢ (⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩ ∈ V → (⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩ ∈ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩}) ↔ {⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})))
64	62, 63	mpbiri 168	. . . 4 ⊢ (⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩ ∈ V → ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩ ∈ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩}))
65	59, 64	syl 14	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩ ∈ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩}))
66	19, 56, 31, 65	opelstrsl 13411	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ( ∘𝑓 + ↾ (𝐵 × 𝐵)) = (+g‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})))
67	16, 18, 66	3eqtr4d 2277	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ✚ = ( ∘𝑓 + ↾ (𝐵 × 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  {crab 2526  Vcvv 2815   ∪ cun 3212   ⊆ wss 3214  {csn 3694  {ctp 3696  ⟨cop 3697   class class class wbr 4114   ↦ cmpt 4176   × cxp 4752  ^◡ccnv 4753   ↾ cres 4756   “ cima 4757   Fn wfn 5352  ‘cfv 5357  (class class class)co 6058   ∈ cmpo 6060   ∘_𝑓 cof 6273   ∘_𝑟 cofr 6274   ↑_𝑚 cmap 6895  Fincfn 6988  1c1 8144   ≤ cle 8325   − cmin 8460  ℕcn 9254  9c9 9312  ℕ₀cn0 9513  ndxcnx 13293  Basecbs 13296  +_gcplusg 13374  ._rcmulr 13375  Scalarcsca 13377   ·_𝑠 cvsca 13378  TopSetcts 13380  TopOpenctopn 13537  ∏_tcpt 13552   Σ_g cgsu 13554   mPwSer cmps 14921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275  df-1st 6347  df-2nd 6348  df-map 6897  df-ixp 6947  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-struct 13298  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-mulr 13388  df-sca 13390  df-vsca 13391  df-tset 13393  df-rest 13538  df-topn 13539  df-topgen 13557  df-pt 13558  df-psr 14923

This theorem is referenced by:  psradd  14946