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Theorem ialgrp1 10572
Description: The value of the algorithm iterator 𝑅 at (𝐾 + 1). (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
algrf.1 𝑍 = (ℤ𝑀)
algrf.2 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)
algrf.3 (𝜑𝑀 ∈ ℤ)
algrf.4 (𝜑𝐴𝑆)
algrf.5 (𝜑𝐹:𝑆𝑆)
algrf.s (𝜑𝑆𝑉)
Assertion
Ref Expression
ialgrp1 ((𝜑𝐾𝑍) → (𝑅‘(𝐾 + 1)) = (𝐹‘(𝑅𝐾)))

Proof of Theorem ialgrp1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 algrf.2 . . . 4 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)
21fveq1i 5210 . . 3 (𝑅‘(𝐾 + 1)) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘(𝐾 + 1))
3 simpr 108 . . . . 5 ((𝜑𝐾𝑍) → 𝐾𝑍)
4 algrf.1 . . . . 5 𝑍 = (ℤ𝑀)
53, 4syl6eleq 2172 . . . 4 ((𝜑𝐾𝑍) → 𝐾 ∈ (ℤ𝑀))
6 algrf.4 . . . . . 6 (𝜑𝐴𝑆)
76adantr 270 . . . . 5 ((𝜑𝐾𝑍) → 𝐴𝑆)
84, 7ialgrlemconst 10569 . . . 4 (((𝜑𝐾𝑍) ∧ 𝑥 ∈ (ℤ𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)
9 algrf.5 . . . . . 6 (𝜑𝐹:𝑆𝑆)
109adantr 270 . . . . 5 ((𝜑𝐾𝑍) → 𝐹:𝑆𝑆)
1110ialgrlem1st 10568 . . . 4 (((𝜑𝐾𝑍) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)
125, 8, 11iseqp1 9538 . . 3 ((𝜑𝐾𝑍) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘(𝐾 + 1)) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))))
132, 12syl5eq 2126 . 2 ((𝜑𝐾𝑍) → (𝑅‘(𝐾 + 1)) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))))
145, 8, 11iseqcl 9537 . . . 4 ((𝜑𝐾𝑍) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾) ∈ 𝑆)
154peano2uzs 8753 . . . . . 6 (𝐾𝑍 → (𝐾 + 1) ∈ 𝑍)
16 fvconst2g 5407 . . . . . 6 ((𝐴𝑆 ∧ (𝐾 + 1) ∈ 𝑍) → ((𝑍 × {𝐴})‘(𝐾 + 1)) = 𝐴)
176, 15, 16syl2an 283 . . . . 5 ((𝜑𝐾𝑍) → ((𝑍 × {𝐴})‘(𝐾 + 1)) = 𝐴)
1817, 7eqeltrd 2156 . . . 4 ((𝜑𝐾𝑍) → ((𝑍 × {𝐴})‘(𝐾 + 1)) ∈ 𝑆)
19 algrflemg 5882 . . . 4 (((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾) ∈ 𝑆 ∧ ((𝑍 × {𝐴})‘(𝐾 + 1)) ∈ 𝑆) → ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)))
2014, 18, 19syl2anc 403 . . 3 ((𝜑𝐾𝑍) → ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)))
211fveq1i 5210 . . . 4 (𝑅𝐾) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)
2221fveq2i 5212 . . 3 (𝐹‘(𝑅𝐾)) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾))
2320, 22syl6reqr 2133 . 2 ((𝜑𝐾𝑍) → (𝐹‘(𝑅𝐾)) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))))
2413, 23eqtr4d 2117 1 ((𝜑𝐾𝑍) → (𝑅‘(𝐾 + 1)) = (𝐹‘(𝑅𝐾)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  {csn 3406   × cxp 4369  ccom 4375  wf 4928  cfv 4932  (class class class)co 5543  1st c1st 5796  1c1 7044   + caddc 7046  cz 8432  cuz 8700  seqcseq 9521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-iseq 9522
This theorem is referenced by:  ialginv  10573  ialgcvg  10574  ialgcvga  10577  ialgfx  10578
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