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Theorem cpnord 25857

Description: 𝓑C^𝑛 conditions are ordered by strength. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

**Assertion**
Ref	Expression
cpnord	⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀))

Proof of Theorem cpnord Dummy variables 𝑓 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6817	. . . . . 6 ⊢ (𝑛 = 𝑀 → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘𝑀))
2	1	sseq1d 3964	. . . . 5 ⊢ (𝑛 = 𝑀 → (((𝓑C^𝑛‘𝑆)‘𝑛) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀) ↔ ((𝓑C^𝑛‘𝑆)‘𝑀) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)))
3	2	imbi2d 340	. . . 4 ⊢ (𝑛 = 𝑀 → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → ((𝓑C^𝑛‘𝑆)‘𝑛) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)) ↔ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → ((𝓑C^𝑛‘𝑆)‘𝑀) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀))))
4		fveq2 6817	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘𝑚))
5	4	sseq1d 3964	. . . . 5 ⊢ (𝑛 = 𝑚 → (((𝓑C^𝑛‘𝑆)‘𝑛) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀) ↔ ((𝓑C^𝑛‘𝑆)‘𝑚) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)))
6	5	imbi2d 340	. . . 4 ⊢ (𝑛 = 𝑚 → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → ((𝓑C^𝑛‘𝑆)‘𝑛) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)) ↔ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → ((𝓑C^𝑛‘𝑆)‘𝑚) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀))))
7		fveq2 6817	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)))
8	7	sseq1d 3964	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (((𝓑C^𝑛‘𝑆)‘𝑛) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀) ↔ ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)))
9	8	imbi2d 340	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → ((𝓑C^𝑛‘𝑆)‘𝑛) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)) ↔ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀))))
10		fveq2 6817	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘𝑁))
11	10	sseq1d 3964	. . . . 5 ⊢ (𝑛 = 𝑁 → (((𝓑C^𝑛‘𝑆)‘𝑛) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀) ↔ ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)))
12	11	imbi2d 340	. . . 4 ⊢ (𝑛 = 𝑁 → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → ((𝓑C^𝑛‘𝑆)‘𝑛) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)) ↔ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀))))
13		ssid 3955	. . . . 5 ⊢ ((𝓑C^𝑛‘𝑆)‘𝑀) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)
14	13	2a1i 12	. . . 4 ⊢ (𝑀 ∈ ℤ → ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → ((𝓑C^𝑛‘𝑆)‘𝑀) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)))
15		simprl 770	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → 𝑓 ∈ (ℂ ↑_pm 𝑆))
16		recnprss 25825	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
17	16	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → 𝑆 ⊆ ℂ)
18	17	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → 𝑆 ⊆ ℂ)
19		simplll 774	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → 𝑆 ∈ {ℝ, ℂ})
20		eluznn0 12807	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → 𝑚 ∈ ℕ₀)
21	20	adantll 714	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → 𝑚 ∈ ℕ₀)
22	21	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → 𝑚 ∈ ℕ₀)
23		dvnf 25849	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑚 ∈ ℕ₀) → ((𝑆 D^𝑛 𝑓)‘𝑚):dom ((𝑆 D^𝑛 𝑓)‘𝑚)⟶ℂ)
24	19, 15, 22, 23	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → ((𝑆 D^𝑛 𝑓)‘𝑚):dom ((𝑆 D^𝑛 𝑓)‘𝑚)⟶ℂ)
25		dvnbss 25850	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑚 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝑓)‘𝑚) ⊆ dom 𝑓)
26	19, 15, 22, 25	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → dom ((𝑆 D^𝑛 𝑓)‘𝑚) ⊆ dom 𝑓)
27		dvnp1 25847	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑚 ∈ ℕ₀) → ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) = (𝑆 D ((𝑆 D^𝑛 𝑓)‘𝑚)))
28	18, 15, 22, 27	syl3anc 1373	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) = (𝑆 D ((𝑆 D^𝑛 𝑓)‘𝑚)))
29		simprr 772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))
30	28, 29	eqeltrrd 2830	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → (𝑆 D ((𝑆 D^𝑛 𝑓)‘𝑚)) ∈ (dom 𝑓–cn→ℂ))
31		cncff 24806	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 D ((𝑆 D^𝑛 𝑓)‘𝑚)) ∈ (dom 𝑓–cn→ℂ) → (𝑆 D ((𝑆 D^𝑛 𝑓)‘𝑚)):dom 𝑓⟶ℂ)
32	30, 31	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → (𝑆 D ((𝑆 D^𝑛 𝑓)‘𝑚)):dom 𝑓⟶ℂ)
33	32	fdmd 6657	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → dom (𝑆 D ((𝑆 D^𝑛 𝑓)‘𝑚)) = dom 𝑓)
34		cnex 11079	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ ∈ V
35		elpm2g 8763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) → (𝑓 ∈ (ℂ ↑_pm 𝑆) ↔ (𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ 𝑆)))
36	34, 19, 35	sylancr 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → (𝑓 ∈ (ℂ ↑_pm 𝑆) ↔ (𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ 𝑆)))
37	15, 36	mpbid 232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → (𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ 𝑆))
38	37	simprd 495	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → dom 𝑓 ⊆ 𝑆)
39	26, 38	sstrd 3943	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → dom ((𝑆 D^𝑛 𝑓)‘𝑚) ⊆ 𝑆)
40	18, 24, 39	dvbss 25822	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → dom (𝑆 D ((𝑆 D^𝑛 𝑓)‘𝑚)) ⊆ dom ((𝑆 D^𝑛 𝑓)‘𝑚))
41	33, 40	eqsstrrd 3968	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → dom 𝑓 ⊆ dom ((𝑆 D^𝑛 𝑓)‘𝑚))
42	26, 41	eqssd 3950	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → dom ((𝑆 D^𝑛 𝑓)‘𝑚) = dom 𝑓)
43	42	feq2d 6631	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → (((𝑆 D^𝑛 𝑓)‘𝑚):dom ((𝑆 D^𝑛 𝑓)‘𝑚)⟶ℂ ↔ ((𝑆 D^𝑛 𝑓)‘𝑚):dom 𝑓⟶ℂ))
44	24, 43	mpbid 232	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → ((𝑆 D^𝑛 𝑓)‘𝑚):dom 𝑓⟶ℂ)
45		dvcn 25843	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ ℂ ∧ ((𝑆 D^𝑛 𝑓)‘𝑚):dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ 𝑆) ∧ dom (𝑆 D ((𝑆 D^𝑛 𝑓)‘𝑚)) = dom 𝑓) → ((𝑆 D^𝑛 𝑓)‘𝑚) ∈ (dom 𝑓–cn→ℂ))
46	18, 44, 38, 33, 45	syl31anc 1375	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → ((𝑆 D^𝑛 𝑓)‘𝑚) ∈ (dom 𝑓–cn→ℂ))
47	15, 46	jca 511	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) ∧ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))) → (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘𝑚) ∈ (dom 𝑓–cn→ℂ)))
48	47	ex 412	. . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → ((𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ)) → (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘𝑚) ∈ (dom 𝑓–cn→ℂ))))
49		peano2nn0 12413	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ₀)
50	21, 49	syl 17	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → (𝑚 + 1) ∈ ℕ₀)
51		elcpn 25856	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑚 + 1) ∈ ℕ₀) → (𝑓 ∈ ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ↔ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))))
52	17, 50, 51	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → (𝑓 ∈ ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ↔ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘(𝑚 + 1)) ∈ (dom 𝑓–cn→ℂ))))
53		elcpn 25856	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑚 ∈ ℕ₀) → (𝑓 ∈ ((𝓑C^𝑛‘𝑆)‘𝑚) ↔ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘𝑚) ∈ (dom 𝑓–cn→ℂ))))
54	17, 21, 53	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → (𝑓 ∈ ((𝓑C^𝑛‘𝑆)‘𝑚) ↔ (𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝑓)‘𝑚) ∈ (dom 𝑓–cn→ℂ))))
55	48, 52, 54	3imtr4d 294	. . . . . . . 8 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → (𝑓 ∈ ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) → 𝑓 ∈ ((𝓑C^𝑛‘𝑆)‘𝑚)))
56	55	ssrdv 3938	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑚))
57		sstr2 3939	. . . . . . 7 ⊢ (((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑚) → (((𝓑C^𝑛‘𝑆)‘𝑚) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀) → ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)))
58	56, 57	syl 17	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → (((𝓑C^𝑛‘𝑆)‘𝑚) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀) → ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)))
59	58	expcom 413	. . . . 5 ⊢ (𝑚 ∈ (ℤ_≥‘𝑀) → ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → (((𝓑C^𝑛‘𝑆)‘𝑚) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀) → ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀))))
60	59	a2d 29	. . . 4 ⊢ (𝑚 ∈ (ℤ_≥‘𝑀) → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → ((𝓑C^𝑛‘𝑆)‘𝑚) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)) → ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀))))
61	3, 6, 9, 12, 14, 60	uzind4 12796	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)))
62	61	com12 32	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) → (𝑁 ∈ (ℤ_≥‘𝑀) → ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)))
63	62	3impia 1117	1 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2110 Vcvv 3434 ⊆ wss 3900 {cpr 4576 dom cdm 5614 ⟶wf 6473 ‘cfv 6477 (class class class)co 7341 ↑_pm cpm 8746 ℂcc 10996 ℝcr 10997 1c1 10999 + caddc 11001 ℕ₀cn0 12373 ℤcz 12460 ℤ_≥cuz 12724 –cn→ccncf 24789 D cdv 25784 D^𝑛 cdvn 25785 𝓑C^𝑛ccpn 25786

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076 ax-addf 11077

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9824 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-icc 13244 df-fz 13400 df-fzo 13547 df-seq 13901 df-exp 13961 df-hash 14230 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-starv 17168 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-unif 17176 df-hom 17177 df-cco 17178 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-xrs 17398 df-qtop 17403 df-imas 17404 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-submnd 18684 df-mulg 18973 df-cntz 19222 df-cmn 19687 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22802 df-topon 22819 df-topsp 22841 df-bases 22854 df-cld 22927 df-ntr 22928 df-cls 22929 df-nei 23006 df-lp 23044 df-perf 23045 df-cn 23135 df-cnp 23136 df-haus 23223 df-tx 23470 df-hmeo 23663 df-fil 23754 df-fm 23846 df-flim 23847 df-flf 23848 df-xms 24228 df-ms 24229 df-tms 24230 df-cncf 24791 df-limc 25787 df-dv 25788 df-dvn 25789 df-cpn 25790

This theorem is referenced by: cpncn 25858 c1lip2 25923

W3C validator