rmxfval - Mathbox for Stefan O'Rear

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Theorem rmxfval 41632

Description: Value of the X sequence. Not used after rmxyval 41644 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)

**Assertion**
Ref	Expression
rmxfval	⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 X_rm 𝑁) = (1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))

Distinct variable groups: 𝐴,𝑏 𝑁,𝑏

Proof of Theorem rmxfval Dummy variables 𝑛 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7415	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎↑2) = (𝐴↑2))
2	1	fvoveq1d 7430	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (√‘((𝑎↑2) − 1)) = (√‘((𝐴↑2) − 1)))
3	2	oveq1d 7423	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))
4	3	oveq2d 7424	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏))) = ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))
5	4	mpteq2dv 5250	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏)))) = (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))))
6	5	cnveqd 5875	. . . . 5 ⊢ (𝑎 = 𝐴 → ^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏)))) = ^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))))
7	6	adantr 481	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 = 𝑁) → ^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏)))) = ^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))))
8		id 22	. . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
9	8, 2	oveq12d 7426	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 + (√‘((𝑎↑2) − 1))) = (𝐴 + (√‘((𝐴↑2) − 1))))
10		id 22	. . . . 5 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
11	9, 10	oveqan12d 7427	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 = 𝑁) → ((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))
12	7, 11	fveq12d 6898	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 = 𝑁) → (^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛)) = (^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)))
13	12	fveq2d 6895	. 2 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 = 𝑁) → (1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))) = (1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))
14		df-rmx 41630	. 2 ⊢ X_rm = (𝑎 ∈ (ℤ_≥‘2), 𝑛 ∈ ℤ ↦ (1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
15		fvex 6904	. 2 ⊢ (1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) ∈ V
16	13, 14, 15	ovmpoa 7562	1 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 X_rm 𝑁) = (1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ↦ cmpt 5231 × cxp 5674 ^◡ccnv 5675 ‘cfv 6543 (class class class)co 7408 1^st c1st 7972 2^nd c2nd 7973 1c1 11110 + caddc 11112 · cmul 11114 − cmin 11443 2c2 12266 ℕ₀cn0 12471 ℤcz 12557 ℤ_≥cuz 12821 ↑cexp 14026 √csqrt 15179 X_rm crmx 41628

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-rmx 41630

This theorem is referenced by: rmxyval 41644

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