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Theorem seq1st 15904
 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 = (ℤ𝑀)
algrf.2 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))
Assertion
Ref Expression
seq1st ((𝑀 ∈ ℤ ∧ 𝐴𝑉) → 𝑅 = seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}))

Proof of Theorem seq1st
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 algrf.2 . 2 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))
2 seqfn 13376 . . . 4 (𝑀 ∈ ℤ → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) Fn (ℤ𝑀))
32adantr 484 . . 3 ((𝑀 ∈ ℤ ∧ 𝐴𝑉) → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) Fn (ℤ𝑀))
4 seqfn 13376 . . . 4 (𝑀 ∈ ℤ → seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}) Fn (ℤ𝑀))
54adantr 484 . . 3 ((𝑀 ∈ ℤ ∧ 𝐴𝑉) → seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}) Fn (ℤ𝑀))
6 fveq2 6652 . . . . . . . 8 (𝑦 = 𝑀 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀))
7 fveq2 6652 . . . . . . . 8 (𝑦 = 𝑀 → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀))
86, 7eqeq12d 2838 . . . . . . 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)))
98imbi2d 344 . . . . . 6 (𝑦 = 𝑀 → ((𝐴𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦)) ↔ (𝐴𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀))))
10 fveq2 6652 . . . . . . . 8 (𝑦 = 𝑥 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥))
11 fveq2 6652 . . . . . . . 8 (𝑦 = 𝑥 → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
1210, 11eqeq12d 2838 . . . . . . 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)))
1312imbi2d 344 . . . . . 6 (𝑦 = 𝑥 → ((𝐴𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦)) ↔ (𝐴𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))))
14 fveq2 6652 . . . . . . . 8 (𝑦 = (𝑥 + 1) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)))
15 fveq2 6652 . . . . . . . 8 (𝑦 = (𝑥 + 1) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)))
1614, 15eqeq12d 2838 . . . . . . 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))))
1716imbi2d 344 . . . . . 6 (𝑦 = (𝑥 + 1) → ((𝐴𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑦) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑦)) ↔ (𝐴𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)))))
18 seq1 13377 . . . . . . . . 9 (𝑀 ∈ ℤ → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
1918adantr 484 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝐴𝑉) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
20 seq1 13377 . . . . . . . . . 10 (𝑀 ∈ ℤ → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀) = ({⟨𝑀, 𝐴⟩}‘𝑀))
2120adantr 484 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝐴𝑉) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀) = ({⟨𝑀, 𝐴⟩}‘𝑀))
22 id 22 . . . . . . . . . . 11 (𝐴𝑉𝐴𝑉)
23 uzid 12246 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
24 algrf.1 . . . . . . . . . . . 12 𝑍 = (ℤ𝑀)
2523, 24eleqtrrdi 2925 . . . . . . . . . . 11 (𝑀 ∈ ℤ → 𝑀𝑍)
26 fvconst2g 6946 . . . . . . . . . . 11 ((𝐴𝑉𝑀𝑍) → ((𝑍 × {𝐴})‘𝑀) = 𝐴)
2722, 25, 26syl2anr 599 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝐴𝑉) → ((𝑍 × {𝐴})‘𝑀) = 𝐴)
28 fvsng 6924 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝐴𝑉) → ({⟨𝑀, 𝐴⟩}‘𝑀) = 𝐴)
2927, 28eqtr4d 2860 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝐴𝑉) → ((𝑍 × {𝐴})‘𝑀) = ({⟨𝑀, 𝐴⟩}‘𝑀))
3021, 29eqtr4d 2860 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝐴𝑉) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
3119, 30eqtr4d 2860 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝐴𝑉) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀))
3231ex 416 . . . . . 6 (𝑀 ∈ ℤ → (𝐴𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑀)))
33 fveq2 6652 . . . . . . . . 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 13379 . . . . . . . . . . . 12 (𝑥 ∈ (ℤ𝑀) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝑥 + 1))))
35 fvex 6665 . . . . . . . . . . . . 13 (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) ∈ V
36 fvex 6665 . . . . . . . . . . . . 13 ((𝑍 × {𝐴})‘(𝑥 + 1)) ∈ V
3735, 36algrflem 7806 . . . . . . . . . . . 12 ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝑥 + 1))) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥))
3834, 37syl6eq 2873 . . . . . . . . . . 11 (𝑥 ∈ (ℤ𝑀) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥)))
39 seqp1 13379 . . . . . . . . . . . 12 (𝑥 ∈ (ℤ𝑀) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)) = ((seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)(𝐹 ∘ 1st )({⟨𝑀, 𝐴⟩}‘(𝑥 + 1))))
40 fvex 6665 . . . . . . . . . . . . 13 (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥) ∈ V
41 fvex 6665 . . . . . . . . . . . . 13 ({⟨𝑀, 𝐴⟩}‘(𝑥 + 1)) ∈ V
4240, 41algrflem 7806 . . . . . . . . . . . 12 ((seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)(𝐹 ∘ 1st )({⟨𝑀, 𝐴⟩}‘(𝑥 + 1))) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
4339, 42syl6eq 2873 . . . . . . . . . . 11 (𝑥 ∈ (ℤ𝑀) → (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)))
4438, 43eqeq12d 2838 . . . . . . . . . 10 (𝑥 ∈ (ℤ𝑀) → ((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))))
4544adantl 485 . . . . . . . . 9 ((𝐴𝑉𝑥 ∈ (ℤ𝑀)) → ((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))))
4633, 45syl5ibr 249 . . . . . . . 8 ((𝐴𝑉𝑥 ∈ (ℤ𝑀)) → ((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))))
4746expcom 417 . . . . . . 7 (𝑥 ∈ (ℤ𝑀) → (𝐴𝑉 → ((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)))))
4847a2d 29 . . . . . 6 (𝑥 ∈ (ℤ𝑀) → ((𝐴𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)) → (𝐴𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝑥 + 1)) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘(𝑥 + 1)))))
499, 13, 17, 13, 32, 48uzind4 12294 . . . . 5 (𝑥 ∈ (ℤ𝑀) → (𝐴𝑉 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥)))
5049impcom 411 . . . 4 ((𝐴𝑉𝑥 ∈ (ℤ𝑀)) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
5150adantll 713 . . 3 (((𝑀 ∈ ℤ ∧ 𝐴𝑉) ∧ 𝑥 ∈ (ℤ𝑀)) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑥) = (seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩})‘𝑥))
523, 5, 51eqfnfvd 6787 . 2 ((𝑀 ∈ ℤ ∧ 𝐴𝑉) → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) = seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}))
531, 52syl5eq 2869 1 ((𝑀 ∈ ℤ ∧ 𝐴𝑉) → 𝑅 = seq𝑀((𝐹 ∘ 1st ), {⟨𝑀, 𝐴⟩}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2114  {csn 4539  ⟨cop 4545   × cxp 5530   ∘ ccom 5536   Fn wfn 6329  ‘cfv 6334  (class class class)co 7140  1st c1st 7673  1c1 10527   + caddc 10529  ℤcz 11969  ℤ≥cuz 12231  seqcseq 13364 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-seq 13365 This theorem is referenced by: (None)
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