Theorem seq1st 16588

Description: A sequence whose iteration function ignores the second argument is only affected by the first point of the initial value function. (Contributed by Mario Carneiro, 11-Feb-2015.)

Hypotheses

Ref	Expression
algrf.1	⊢ 𝑍 = (ℤ≥‘𝑀)
algrf.2	⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))

Assertion

Ref	Expression
seq1st	⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → 𝑅 = seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}))

Proof of Theorem seq1st

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		algrf.2	. 2 ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))
2		seqfn 14023	. . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) Fn (ℤ≥‘𝑀))
3	2	adantr 484	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) Fn (ℤ≥‘𝑀))
4		seqfn 14023	. . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}) Fn (ℤ≥‘𝑀))
5	4	adantr 484	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}) Fn (ℤ≥‘𝑀))
6		fveq2 6863	. . . . . . . 8 ⊢ (𝑦 = 𝑀 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀))
7		fveq2 6863	. . . . . . . 8 ⊢ (𝑦 = 𝑀 → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀))
8	6, 7	eqeq12d 2777	. . . . . . 7 ⊢ (𝑦 = 𝑀 → ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) ↔ (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀)))
9	8	imbi2d 342	. . . . . 6 ⊢ (𝑦 = 𝑀 → ((𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦)) ↔ (𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀))))
10		fveq2 6863	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥))
11		fveq2 6863	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
12	10, 11	eqeq12d 2777	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) ↔ (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)))
13	12	imbi2d 342	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦)) ↔ (𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))))
14		fveq2 6863	. . . . . . . 8 ⊢ (𝑦 = (𝑥 + 1) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)))
15		fveq2 6863	. . . . . . . 8 ⊢ (𝑦 = (𝑥 + 1) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)))
16	14, 15	eqeq12d 2777	. . . . . . 7 ⊢ (𝑦 = (𝑥 + 1) → ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) ↔ (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1))))
17	16	imbi2d 342	. . . . . 6 ⊢ (𝑦 = (𝑥 + 1) → ((𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦)) ↔ (𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)))))
18		seq1 14024	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
19	18	adantr 484	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
20		seq1 14024	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀) = ({⟨𝑀, 𝐴⟩}‘𝑀))
21	20	adantr 484	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀) = ({⟨𝑀, 𝐴⟩}‘𝑀))
22		id 22	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
23		uzid 12851	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
24		algrf.1	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ≥‘𝑀)
25	23, 24	eleqtrrdi 2872	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ 𝑍)
26		fvconst2g 7182	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑀 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑀) = 𝐴)
27	22, 25, 26	syl2anr 606	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → ((𝑍 × {𝐴})‘𝑀) = 𝐴)
28		fvsng 7160	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → ({⟨𝑀, 𝐴⟩}‘𝑀) = 𝐴)
29	27, 28	eqtr4d 2799	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → ((𝑍 × {𝐴})‘𝑀) = ({⟨𝑀, 𝐴⟩}‘𝑀))
30	21, 29	eqtr4d 2799	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
31	19, 30	eqtr4d 2799	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀))
32	31	ex 416	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀)))
33		fveq2 6863	. . . . . . . . 9 ⊢ ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥) → (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥)) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)))
34		seqp1 14026	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝑥 + 1))))
35		fvex 6876	. . . . . . . . . . . . 13 ⊢ (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) ∈ V
36		fvex 6876	. . . . . . . . . . . . 13 ⊢ ((𝑍 × {𝐴})‘(𝑥 + 1)) ∈ V
37	35, 36	opco1i 8099	. . . . . . . . . . . 12 ⊢ ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝑥 + 1))) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥))
38	34, 37	eqtrdi 2812	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥)))
39		seqp1 14026	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)) = ((seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)(𝐹 ∘ 1st )({⟨𝑀, 𝐴⟩}‘(𝑥 + 1))))
40		fvex 6876	. . . . . . . . . . . . 13 ⊢ (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥) ∈ V
41		fvex 6876	. . . . . . . . . . . . 13 ⊢ ({⟨𝑀, 𝐴⟩}‘(𝑥 + 1)) ∈ V
42	40, 41	opco1i 8099	. . . . . . . . . . . 12 ⊢ ((seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)(𝐹 ∘ 1st )({⟨𝑀, 𝐴⟩}‘(𝑥 + 1))) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
43	39, 42	eqtrdi 2812	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)))
44	38, 43	eqeq12d 2777	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)) ↔ (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥)) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))))
45	44	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)) ↔ (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥)) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))))
46	33, 45	imbitrrid 248	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1))))
47	46	expcom 417	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (𝐴 ∈ 𝑉 → ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)))))
48	47	a2d 29	. . . . . 6 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → ((𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)) → (𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)))))
49	9, 13, 17, 13, 32, 48	uzind4 12904	. . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)))
50	49	impcom 411	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
51	50	adantll 724	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
52	3, 5, 51	eqfnfvd 7010	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) = seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}))
53	1, 52	eqtrid 2808	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → 𝑅 = seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {csn 4581  ⟨cop 4587   × cxp 5643   ∘ ccom 5649   Fn wfn 6512  ‘cfv 6517  (class class class)co 7392  1^st c1st 7964  1c1 11071   + caddc 11073  ℤcz 12565  ℤ_≥cuz 12836  seqcseq 14011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-n0 12479  df-z 12566  df-uz 12837  df-seq 14012

This theorem is referenced by: (None)