Theorem seq1st 15917

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 13384	. . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) Fn (ℤ≥‘𝑀))
3	2	adantr 483	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) Fn (ℤ≥‘𝑀))
4		seqfn 13384	. . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}) Fn (ℤ≥‘𝑀))
5	4	adantr 483	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}) Fn (ℤ≥‘𝑀))
6		fveq2 6672	. . . . . . . 8 ⊢ (𝑦 = 𝑀 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀))
7		fveq2 6672	. . . . . . . 8 ⊢ (𝑦 = 𝑀 → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀))
8	6, 7	eqeq12d 2839	. . . . . . 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 343	. . . . . 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 6672	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥))
11		fveq2 6672	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
12	10, 11	eqeq12d 2839	. . . . . . 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 343	. . . . . 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 6672	. . . . . . . 8 ⊢ (𝑦 = (𝑥 + 1) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)))
15		fveq2 6672	. . . . . . . 8 ⊢ (𝑦 = (𝑥 + 1) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)))
16	14, 15	eqeq12d 2839	. . . . . . 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 343	. . . . . 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 13385	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
19	18	adantr 483	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
20		seq1 13385	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀) = ({⟨𝑀, 𝐴⟩}‘𝑀))
21	20	adantr 483	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀) = ({⟨𝑀, 𝐴⟩}‘𝑀))
22		id 22	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
23		uzid 12261	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
24		algrf.1	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ≥‘𝑀)
25	23, 24	eleqtrrdi 2926	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ 𝑍)
26		fvconst2g 6966	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑀 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑀) = 𝐴)
27	22, 25, 26	syl2anr 598	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → ((𝑍 × {𝐴})‘𝑀) = 𝐴)
28		fvsng 6944	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → ({⟨𝑀, 𝐴⟩}‘𝑀) = 𝐴)
29	27, 28	eqtr4d 2861	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → ((𝑍 × {𝐴})‘𝑀) = ({⟨𝑀, 𝐴⟩}‘𝑀))
30	21, 29	eqtr4d 2861	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
31	19, 30	eqtr4d 2861	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀))
32	31	ex 415	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀)))
33		fveq2 6672	. . . . . . . . 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 13387	. . . . . . . . . . . 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 6685	. . . . . . . . . . . . 13 ⊢ (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) ∈ V
36		fvex 6685	. . . . . . . . . . . . 13 ⊢ ((𝑍 × {𝐴})‘(𝑥 + 1)) ∈ V
37	35, 36	algrflem 7821	. . . . . . . . . . . 12 ⊢ ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝑥 + 1))) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥))
38	34, 37	syl6eq 2874	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥)))
39		seqp1 13387	. . . . . . . . . . . 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 6685	. . . . . . . . . . . . 13 ⊢ (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥) ∈ V
41		fvex 6685	. . . . . . . . . . . . 13 ⊢ ({⟨𝑀, 𝐴⟩}‘(𝑥 + 1)) ∈ V
42	40, 41	algrflem 7821	. . . . . . . . . . . 12 ⊢ ((seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)(𝐹 ∘ 1st )({⟨𝑀, 𝐴⟩}‘(𝑥 + 1))) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
43	39, 42	syl6eq 2874	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)))
44	38, 43	eqeq12d 2839	. . . . . . . . . 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 484	. . . . . . . . 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	syl5ibr 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 416	. . . . . . 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 12309	. . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (𝐴 ∈ 𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)))
50	49	impcom 410	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
51	50	adantll 712	. . 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 6807	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) = seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}))
53	1, 52	syl5eq 2870	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉) → 𝑅 = seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {csn 4569  ⟨cop 4575   × cxp 5555   ∘ ccom 5561   Fn wfn 6352  ‘cfv 6357  (class class class)co 7158  1^st c1st 7689  1c1 10540   + caddc 10542  ℤcz 11984  ℤ_≥cuz 12246  seqcseq 13372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-n0 11901  df-z 11985  df-uz 12247  df-seq 13373

This theorem is referenced by: (None)