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Theorem cpnord 25902

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 6841	. . . . . 6 ⊢ (𝑛 = 𝑀 → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘𝑀))
2	1	sseq1d 3954	. . . . 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 6841	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘𝑚))
5	4	sseq1d 3954	. . . . 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 6841	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)))
8	7	sseq1d 3954	. . . . 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 6841	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘𝑁))
11	10	sseq1d 3954	. . . . 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 3945	. . . . 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 771	. . . . . . . . . . 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 25871	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
17	16	ad2antrr 727	. . . . . . . . . . . . 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 775	. . . . . . . . . . . . . 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 12867	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → 𝑚 ∈ ℕ₀)
21	20	adantll 715	. . . . . . . . . . . . . . 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 25894	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 25895	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝑓 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑚 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝑓)‘𝑚) ⊆ dom 𝑓)
26	19, 15, 22, 25	syl3anc 1374	. . . . . . . . . . . . . . 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 25892	. . . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . . . . 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 24860	. . . . . . . . . . . . . . . . . 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 6679	. . . . . . . . . . . . . . . 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 11119	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ ∈ V
35		elpm2g 8791	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) → (𝑓 ∈ (ℂ ↑_pm 𝑆) ↔ (𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ 𝑆)))
36	34, 19, 35	sylancr 588	. . . . . . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . . . 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 25868	. . . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . . 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 3940	. . . . . . . . . . . . . 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 6653	. . . . . . . . . . . . 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 25888	. . . . . . . . . . . 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 1376	. . . . . . . . . . 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 12477	. . . . . . . . . . 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 25901	. . . . . . . . . 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 585	. . . . . . . . 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 25901	. . . . . . . . . 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 585	. . . . . . . . 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 3928	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑚))
57		sstr2 3929	. . . . . . 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 12856	. . 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 1118	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 1087 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ⊆ wss 3890 {cpr 4570 dom cdm 5631 ⟶wf 6495 ‘cfv 6499 (class class class)co 7367 ↑_pm cpm 8774 ℂcc 11036 ℝcr 11037 1c1 11039 + caddc 11041 ℕ₀cn0 12437 ℤcz 12524 ℤ_≥cuz 12788 –cn→ccncf 24843 D cdv 25830 D^𝑛 cdvn 25831 𝓑C^𝑛ccpn 25832

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

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 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-icc 13305 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-dvn 25835 df-cpn 25836

This theorem is referenced by: cpncn 25903 c1lip2 25965

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