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Theorem cpnord 25099

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 6774	. . . . . 6 ⊢ (𝑛 = 𝑀 → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘𝑀))
2	1	sseq1d 3952	. . . . 5 ⊢ (𝑛 = 𝑀 → (((𝓑C^𝑛‘𝑆)‘𝑛) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀) ↔ ((𝓑C^𝑛‘𝑆)‘𝑀) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)))
3	2	imbi2d 341	. . . 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 6774	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘𝑚))
5	4	sseq1d 3952	. . . . 5 ⊢ (𝑛 = 𝑚 → (((𝓑C^𝑛‘𝑆)‘𝑛) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀) ↔ ((𝓑C^𝑛‘𝑆)‘𝑚) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)))
6	5	imbi2d 341	. . . 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 6774	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)))
8	7	sseq1d 3952	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (((𝓑C^𝑛‘𝑆)‘𝑛) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀) ↔ ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)))
9	8	imbi2d 341	. . . 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 6774	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘𝑁))
11	10	sseq1d 3952	. . . . 5 ⊢ (𝑛 = 𝑁 → (((𝓑C^𝑛‘𝑆)‘𝑛) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀) ↔ ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑀)))
12	11	imbi2d 341	. . . 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 3943	. . . . 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 768	. . . . . . . . . . 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 25068	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
17	16	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → 𝑆 ⊆ ℂ)
18	17	adantr 481	. . . . . . . . . . . 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 772	. . . . . . . . . . . . . 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 12657	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → 𝑚 ∈ ℕ₀)
21	20	adantll 711	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → 𝑚 ∈ ℕ₀)
22	21	adantr 481	. . . . . . . . . . . . . 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 25091	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 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 25092	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑚 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝑓)‘𝑚) ⊆ dom 𝑓)
26	19, 15, 22, 25	syl3anc 1370	. . . . . . . . . . . . . . 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 25089	. . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . 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 24056	. . . . . . . . . . . . . . . . . 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 6611	. . . . . . . . . . . . . . . 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 10952	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ ∈ V
35		elpm2g 8632	. . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . . 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 25065	. . . . . . . . . . . . . . . 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 3960	. . . . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . 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 6586	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 25085	. . . . . . . . . . . 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 1372	. . . . . . . . . . 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 512	. . . . . . . . . 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 413	. . . . . . . . 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 12273	. . . . . . . . . . 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 25098	. . . . . . . . . 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 25098	. . . . . . . . . 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 3927	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑚))
57		sstr2 3928	. . . . . . 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 414	. . . . 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 12646	. . 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 1116	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 {cpr 4563 dom cdm 5589 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_pm cpm 8616 ℂcc 10869 ℝcr 10870 1c1 10872 + caddc 10874 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 –cn→ccncf 24039 D cdv 25027 D^𝑛 cdvn 25028 𝓑C^𝑛ccpn 25029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-dvn 25032 df-cpn 25033

This theorem is referenced by: cpncn 25100 c1lip2 25162

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