Theorem psrplusgg 14306

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 2196	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		psrplusg.a	. . . 4 ⊢ + = (+g‘𝑅)
4		eqid 2196	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
5		eqid 2196	. . . 4 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
6		eqid 2196	. . . 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 14303	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐵 = ((Base‘𝑅) ↑𝑚 {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}))
11		eqid 2196	. . . 4 ⊢ ( ∘𝑓 + ↾ (𝐵 × 𝐵)) = ( ∘𝑓 + ↾ (𝐵 × 𝐵))
12		eqid 2196	. . . 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 2196	. . . 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 2197	. . . 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 14296	. . 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 5565	. 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 12815	. . 3 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
20		basfn 12761	. . . . . 6 ⊢ Base Fn V
21		fnpsr 14297	. . . . . . . 8 ⊢ mPwSer Fn (V × V)
22	8	elexd 2776	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐼 ∈ V)
23	9	elexd 2776	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑅 ∈ V)
24		fnovex 5958	. . . . . . . 8 ⊢ (( mPwSer Fn (V × V) ∧ 𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) ∈ V)
25	21, 22, 23, 24	mp3an2i 1353	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐼 mPwSer 𝑅) ∈ V)
26	1, 25	eqeltrid 2283	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑆 ∈ V)
27		funfvex 5578	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
28	27	funfni 5361	. . . . . 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 2283	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐵 ∈ V)
31	30, 30	ofmresex 6203	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ( ∘𝑓 + ↾ (𝐵 × 𝐵)) ∈ V)
32		mpoexga 6279	. . . . 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 5578	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
35	34	funfni 5361	. . . . . 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 6279	. . . . 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 6723	. . . . . . . 8 ⊢ ↑𝑚 Fn (V × V)
40		nn0ex 9272	. . . . . . . . 9 ⊢ ℕ0 ∈ V
41	40	a1i 9	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ℕ0 ∈ V)
42		fnovex 5958	. . . . . . . 8 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℕ0 ∈ V ∧ 𝐼 ∈ V) → (ℕ0 ↑𝑚 𝐼) ∈ V)
43	39, 41, 22, 42	mp3an2i 1353	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (ℕ0 ↑𝑚 𝐼) ∈ V)
44		rabexg 4177	. . . . . . 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 12946	. . . . . . . 8 ⊢ TopOpen Fn V
47		funfvex 5578	. . . . . . . . 9 ⊢ ((Fun TopOpen ∧ 𝑅 ∈ dom TopOpen) → (TopOpen‘𝑅) ∈ V)
48	47	funfni 5361	. . . . . . . 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 4218	. . . . . . 7 ⊢ ((TopOpen‘𝑅) ∈ V → {(TopOpen‘𝑅)} ∈ V)
51	49, 50	syl 14	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → {(TopOpen‘𝑅)} ∈ V)
52		xpexg 4778	. . . . . 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 12966	. . . . 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 14298	. . 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 12814	. . . . 5 ⊢ (+g‘ndx) ∈ ℕ
58		opexg 4262	. . . . 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 3774	. . . . . 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 3327	. . . . . 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 3193	. . . . 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 3757	. . . . 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 12817	. 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 2239	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ✚ = ( ∘𝑓 + ↾ (𝐵 × 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  {crab 2479  Vcvv 2763   ∪ cun 3155   ⊆ wss 3157  {csn 3623  {ctp 3625  ⟨cop 3626   class class class wbr 4034   ↦ cmpt 4095   × cxp 4662  ^◡ccnv 4663   ↾ cres 4666   “ cima 4667   Fn wfn 5254  ‘cfv 5259  (class class class)co 5925   ∈ cmpo 5927   ∘_𝑓 cof 6137   ∘_𝑟 cofr 6138   ↑_𝑚 cmap 6716  Fincfn 6808  1c1 7897   ≤ cle 8079   − cmin 8214  ℕcn 9007  9c9 9065  ℕ₀cn0 9266  ndxcnx 12700  Basecbs 12703  +_gcplusg 12780  ._rcmulr 12781  Scalarcsca 12783   ·_𝑠 cvsca 12784  TopSetcts 12786  TopOpenctopn 12942  ∏_tcpt 12957   Σ_g cgsu 12959   mPwSer cmps 14293

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-tp 3631  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-of 6139  df-1st 6207  df-2nd 6208  df-map 6718  df-ixp 6767  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-9 9073  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101  df-struct 12705  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-mulr 12794  df-sca 12796  df-vsca 12797  df-tset 12799  df-rest 12943  df-topn 12944  df-topgen 12962  df-pt 12963  df-psr 14294

This theorem is referenced by:  psradd  14307