Theorem psrplusg 19202

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 2610	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		psrplusg.a	. . . . 5 ⊢ + = (+g‘𝑅)
4		eqid 2610	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
5		eqid 2610	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
6		eqid 2610	. . . . 5 ⊢ {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
7		psrplusg.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
8		simpl 472	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐼 ∈ V)
9	1, 2, 6, 7, 8	psrbas 19199	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ((Base‘𝑅) ↑𝑚 {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}))
10		eqid 2610	. . . . 5 ⊢ ( ∘𝑓 + ↾ (𝐵 × 𝐵)) = ( ∘𝑓 + ↾ (𝐵 × 𝐵))
11		eqid 2610	. . . . 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 2610	. . . . 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 2611	. . . . 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 476	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
15	1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 8, 14	psrval 19183	. . . 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 6107	. . 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		fvex 6113	. . . . . 6 ⊢ (Base‘𝑆) ∈ V
19	7, 18	eqeltri 2684	. . . . 5 ⊢ 𝐵 ∈ V
20	19, 19	xpex 6860	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
21		ofexg 6799	. . . 4 ⊢ ((𝐵 × 𝐵) ∈ V → ( ∘𝑓 + ↾ (𝐵 × 𝐵)) ∈ V)
22		psrvalstr 19184	. . . . 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⟩
23		plusgid 15804	. . . . 5 ⊢ +g = Slot (+g‘ndx)
24		snsstp2 4288	. . . . . 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)))))))⟩}
25		ssun1 3738	. . . . . 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)}))⟩})
26	24, 25	sstri 3577	. . . . 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)}))⟩})
27	22, 23, 26	strfv 15735	. . . 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)}))⟩})))
28	20, 21, 27	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)}))⟩}))
29	16, 17, 28	3eqtr4g 2669	. 2 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → ✚ = ( ∘𝑓 + ↾ (𝐵 × 𝐵)))
30		reldmpsr 19182	. . . . . . 7 ⊢ Rel dom mPwSer
31	30	ovprc 6581	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
32	1, 31	syl5eq 2656	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
33	32	fveq2d 6107	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (+g‘𝑆) = (+g‘∅))
34	23	str0 15739	. . . 4 ⊢ ∅ = (+g‘∅)
35	33, 17, 34	3eqtr4g 2669	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ✚ = ∅)
36	32	fveq2d 6107	. . . . . . . 8 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝑆) = (Base‘∅))
37		base0 15740	. . . . . . . 8 ⊢ ∅ = (Base‘∅)
38	36, 7, 37	3eqtr4g 2669	. . . . . . 7 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
39	38	xpeq2d 5063	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐵 × 𝐵) = (𝐵 × ∅))
40		xp0 5471	. . . . . 6 ⊢ (𝐵 × ∅) = ∅
41	39, 40	syl6eq 2660	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐵 × 𝐵) = ∅)
42	41	reseq2d 5317	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ∘𝑓 + ↾ (𝐵 × 𝐵)) = ( ∘𝑓 + ↾ ∅))
43		res0 5321	. . . 4 ⊢ ( ∘𝑓 + ↾ ∅) = ∅
44	42, 43	syl6eq 2660	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ∘𝑓 + ↾ (𝐵 × 𝐵)) = ∅)
45	35, 44	eqtr4d 2647	. 2 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ✚ = ( ∘𝑓 + ↾ (𝐵 × 𝐵)))
46	29, 45	pm2.61i 175	1 ⊢ ✚ = ( ∘𝑓 + ↾ (𝐵 × 𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125  {ctp 4129  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ^◡ccnv 5037   ↾ cres 5040   “ cima 5041  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ∘_𝑓 cof 6793   ∘_𝑟 cofr 6794   ↑_𝑚 cmap 7744  Fincfn 7841  1c1 9816   ≤ cle 9954   − cmin 10145  ℕcn 10897  9c9 10954  ℕ₀cn0 11169  ndxcnx 15692  Basecbs 15695  +_gcplusg 15768  ._rcmulr 15769  Scalarcsca 15771   ·_𝑠 cvsca 15772  TopSetcts 15774  TopOpenctopn 15905  ∏_tcpt 15922   Σ_g cgsu 15924   mPwSer cmps 19172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-psr 19177

This theorem is referenced by:  psradd  19203  psrmulr  19205  psrsca  19210  psrvscafval  19211  psrplusgpropd  19427  ply1plusgfvi  19433