Theorem seq1st 16605

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 14051	. . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) Fn (ℤ≥‘𝑀))
3	2	adantr 480	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) Fn (ℤ≥‘𝑀))
4		seqfn 14051	. . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}) Fn (ℤ≥‘𝑀))
5	4	adantr 480	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}) Fn (ℤ≥‘𝑀))
6		fveq2 6907	. . . . . . . 8 ⊢ (𝑦 = 𝑀 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀))
7		fveq2 6907	. . . . . . . 8 ⊢ (𝑦 = 𝑀 → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀))
8	6, 7	eqeq12d 2751	. . . . . . 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 340	. . . . . 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 6907	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥))
11		fveq2 6907	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
12	10, 11	eqeq12d 2751	. . . . . . 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 340	. . . . . 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 6907	. . . . . . . 8 ⊢ (𝑦 = (𝑥 + 1) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)))
15		fveq2 6907	. . . . . . . 8 ⊢ (𝑦 = (𝑥 + 1) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)))
16	14, 15	eqeq12d 2751	. . . . . . 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 340	. . . . . 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 14052	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
19	18	adantr 480	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
20		seq1 14052	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀) = ({⟨𝑀, 𝐴⟩}‘𝑀))
21	20	adantr 480	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀) = ({⟨𝑀, 𝐴⟩}‘𝑀))
22		id 22	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
23		uzid 12891	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
24		algrf.1	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ≥‘𝑀)
25	23, 24	eleqtrrdi 2850	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ 𝑍)
26		fvconst2g 7222	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑀 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑀) = 𝐴)
27	22, 25, 26	syl2anr 597	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → ((𝑍 × {𝐴})‘𝑀) = 𝐴)
28		fvsng 7200	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → ({⟨𝑀, 𝐴⟩}‘𝑀) = 𝐴)
29	27, 28	eqtr4d 2778	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → ((𝑍 × {𝐴})‘𝑀) = ({⟨𝑀, 𝐴⟩}‘𝑀))
30	21, 29	eqtr4d 2778	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
31	19, 30	eqtr4d 2778	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀))
32	31	ex 412	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀)))
33		fveq2 6907	. . . . . . . . 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 14054	. . . . . . . . . . . 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 6920	. . . . . . . . . . . . 13 ⊢ (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) ∈ V
36		fvex 6920	. . . . . . . . . . . . 13 ⊢ ((𝑍 × {𝐴})‘(𝑥 + 1)) ∈ V
37	35, 36	opco1i 8149	. . . . . . . . . . . 12 ⊢ ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝑥 + 1))) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥))
38	34, 37	eqtrdi 2791	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥)))
39		seqp1 14054	. . . . . . . . . . . 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 6920	. . . . . . . . . . . . 13 ⊢ (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥) ∈ V
41		fvex 6920	. . . . . . . . . . . . 13 ⊢ ({⟨𝑀, 𝐴⟩}‘(𝑥 + 1)) ∈ V
42	40, 41	opco1i 8149	. . . . . . . . . . . 12 ⊢ ((seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)(𝐹 ∘ 1st )({⟨𝑀, 𝐴⟩}‘(𝑥 + 1))) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
43	39, 42	eqtrdi 2791	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)))
44	38, 43	eqeq12d 2751	. . . . . . . . . 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 481	. . . . . . . . 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 246	. . . . . . . 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 413	. . . . . . 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 12946	. . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)))
50	49	impcom 407	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
51	50	adantll 714	. . 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 7054	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) = seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}))
53	1, 52	eqtrid 2787	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → 𝑅 = seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {csn 4631  ⟨cop 4637   × cxp 5687   ∘ ccom 5693   Fn wfn 6558  ‘cfv 6563  (class class class)co 7431  1^st c1st 8011  1c1 11154   + caddc 11156  ℤcz 12611  ℤ_≥cuz 12876  seqcseq 14039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-seq 14040

This theorem is referenced by: (None)