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Theorem cpnord 25897

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 6835	. . . . . 6 ⊢ (𝑛 = 𝑀 → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘𝑀))
2	1	sseq1d 3966	. . . . 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 6835	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘𝑚))
5	4	sseq1d 3966	. . . . 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 6835	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)))
8	7	sseq1d 3966	. . . . 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 6835	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝓑C^𝑛‘𝑆)‘𝑛) = ((𝓑C^𝑛‘𝑆)‘𝑁))
11	10	sseq1d 3966	. . . . 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 3957	. . . . 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 25865	. . . . . . . . . . . . . 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 12834	. . . . . . . . . . . . . . . 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 25889	. . . . . . . . . . . . . 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 25890	. . . . . . . . . . . . . . . 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 25887	. . . . . . . . . . . . . . . . . . . 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 24846	. . . . . . . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . . . 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 11111	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ ∈ V
35		elpm2g 8785	. . . . . . . . . . . . . . . . . . . . 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 3945	. . . . . . . . . . . . . . . . 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 25862	. . . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . . . 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 3952	. . . . . . . . . . . . . 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 6647	. . . . . . . . . . . . 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 25883	. . . . . . . . . . . 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 12445	. . . . . . . . . . 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 25896	. . . . . . . . . 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 25896	. . . . . . . . . 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 3940	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → ((𝓑C^𝑛‘𝑆)‘(𝑚 + 1)) ⊆ ((𝓑C^𝑛‘𝑆)‘𝑚))
57		sstr2 3941	. . . . . . 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 12823	. . 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 3441 ⊆ wss 3902 {cpr 4583 dom cdm 5625 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 ↑_pm cpm 8768 ℂcc 11028 ℝcr 11029 1c1 11031 + caddc 11033 ℕ₀cn0 12405 ℤcz 12492 ℤ_≥cuz 12755 –cn→ccncf 24829 D cdv 25824 D^𝑛 cdvn 25825 𝓑C^𝑛ccpn 25826

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-inf2 9554 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 ax-pre-sup 11108 ax-addf 11109

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-rmo 3351 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-int 4904 df-iun 4949 df-iin 4950 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-se 5579 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-isom 6502 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-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 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-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-icc 13272 df-fz 13428 df-fzo 13575 df-seq 13929 df-exp 13989 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24268 df-ms 24269 df-tms 24270 df-cncf 24831 df-limc 25827 df-dv 25828 df-dvn 25829 df-cpn 25830

This theorem is referenced by: cpncn 25898 c1lip2 25963

W3C validator