Theorem psrplusg 21896

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
psrplusg	⊢ ✚ = ( ∘f + ↾ (𝐵 × 𝐵))

Proof of Theorem psrplusg

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

Step	Hyp	Ref	Expression
1		psrplusg.s	. . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		eqid 2737	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		psrplusg.a	. . . . 5 ⊢ + = (+g‘𝑅)
4		eqid 2737	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
5		eqid 2737	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
6		eqid 2737	. . . . 5 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
7		psrplusg.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
8		simpl 482	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐼 ∈ V)
9	1, 2, 6, 7, 8	psrbas 21893	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}))
10		eqid 2737	. . . . 5 ⊢ ( ∘f + ↾ (𝐵 × 𝐵)) = ( ∘f + ↾ (𝐵 × 𝐵))
11		eqid 2737	. . . . 5 ⊢ (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)))))))
12		eqid 2737	. . . . 5 ⊢ (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))
13		eqidd 2738	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (∏t‘({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})) = (∏t‘({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})))
14		simpr 484	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
15	1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 8, 14	psrval 21875	. . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 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 6839	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (+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	7	fvexi 6849	. . . . 5 ⊢ 𝐵 ∈ V
19	18, 18	xpex 7700	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
20		ofexg 7629	. . . 4 ⊢ ((𝐵 × 𝐵) ∈ V → ( ∘f + ↾ (𝐵 × 𝐵)) ∈ V)
21		psrvalstr 21876	. . . . 5 ⊢ ({⟨(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⟩
22		plusgid 17208	. . . . 5 ⊢ +g = Slot (+g‘ndx)
23		snsstp2 4774	. . . . . 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)))))))⟩}
24		ssun1 4131	. . . . . 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)}))⟩})
25	23, 24	sstri 3944	. . . . 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)}))⟩})
26	21, 22, 25	strfv 17134	. . . 4 ⊢ (( ∘f + ↾ (𝐵 × 𝐵)) ∈ V → ( ∘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)}))⟩})))
27	19, 20, 26	mp2b 10	. . 3 ⊢ ( ∘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)}))⟩}))
28	16, 17, 27	3eqtr4g 2797	. 2 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → ✚ = ( ∘f + ↾ (𝐵 × 𝐵)))
29		reldmpsr 21874	. . . . . . 7 ⊢ Rel dom mPwSer
30	29	ovprc 7398	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
31	1, 30	eqtrid 2784	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
32	31	fveq2d 6839	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (+g‘𝑆) = (+g‘∅))
33	22	str0 17120	. . . 4 ⊢ ∅ = (+g‘∅)
34	32, 17, 33	3eqtr4g 2797	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ✚ = ∅)
35	31	fveq2d 6839	. . . . . . . 8 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝑆) = (Base‘∅))
36		base0 17145	. . . . . . . 8 ⊢ ∅ = (Base‘∅)
37	35, 7, 36	3eqtr4g 2797	. . . . . . 7 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
38	37	xpeq2d 5655	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐵 × 𝐵) = (𝐵 × ∅))
39		xp0 5725	. . . . . 6 ⊢ (𝐵 × ∅) = ∅
40	38, 39	eqtrdi 2788	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐵 × 𝐵) = ∅)
41	40	reseq2d 5939	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ∘f + ↾ (𝐵 × 𝐵)) = ( ∘f + ↾ ∅))
42		res0 5943	. . . 4 ⊢ ( ∘f + ↾ ∅) = ∅
43	41, 42	eqtrdi 2788	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ∘f + ↾ (𝐵 × 𝐵)) = ∅)
44	34, 43	eqtr4d 2775	. 2 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ✚ = ( ∘f + ↾ (𝐵 × 𝐵)))
45	28, 44	pm2.61i 182	1 ⊢ ✚ = ( ∘f + ↾ (𝐵 × 𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3400  Vcvv 3441   ∪ cun 3900  ∅c0 4286  {csn 4581  {ctp 4585  ⟨cop 4587   class class class wbr 5099   ↦ cmpt 5180   × cxp 5623  ^◡ccnv 5624   ↾ cres 5627   “ cima 5628  ‘cfv 6493  (class class class)co 7360   ∈ cmpo 7362   ∘_f cof 7622   ∘_r cofr 7623   ↑_m cmap 8767  Fincfn 8887  1c1 11031   ≤ cle 11171   − cmin 11368  ℕcn 12149  9c9 12211  ℕ₀cn0 12405  ndxcnx 17124  Basecbs 17140  +_gcplusg 17181  ._rcmulr 17182  Scalarcsca 17184   ·_𝑠 cvsca 17185  TopSetcts 17187  TopOpenctopn 17345  ∏_tcpt 17362   Σ_g cgsu 17364   mPwSer cmps 21864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12406  df-z 12493  df-uz 12756  df-fz 13428  df-struct 17078  df-slot 17113  df-ndx 17125  df-base 17141  df-plusg 17194  df-mulr 17195  df-sca 17197  df-vsca 17198  df-tset 17200  df-psr 21869

This theorem is referenced by:  psradd  21897  psrmulr  21902  psrsca  21907  psrvscafval  21908  psrplusgpropd  22180  ply1plusgfvi  22186