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Theorem iseqeq3 8896
 Description: Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
Assertion
Ref Expression
iseqeq3 (𝐹 = 𝐺 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))

Proof of Theorem iseqeq3
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 903 . . . . . . . 8 ((𝐹 = 𝐺 x (ℤ𝑀) y 𝑆) → 𝐹 = 𝐺)
21fveq1d 5123 . . . . . . 7 ((𝐹 = 𝐺 x (ℤ𝑀) y 𝑆) → (𝐹‘(x + 1)) = (𝐺‘(x + 1)))
32oveq2d 5471 . . . . . 6 ((𝐹 = 𝐺 x (ℤ𝑀) y 𝑆) → (y + (𝐹‘(x + 1))) = (y + (𝐺‘(x + 1))))
43opeq2d 3547 . . . . 5 ((𝐹 = 𝐺 x (ℤ𝑀) y 𝑆) → ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩ = ⟨(x + 1), (y + (𝐺‘(x + 1)))⟩)
54mpt2eq3dva 5511 . . . 4 (𝐹 = 𝐺 → (x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩) = (x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐺‘(x + 1)))⟩))
6 fveq1 5120 . . . . 5 (𝐹 = 𝐺 → (𝐹𝑀) = (𝐺𝑀))
76opeq2d 3547 . . . 4 (𝐹 = 𝐺 → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑀, (𝐺𝑀)⟩)
8 freceq1 5919 . . . . 5 ((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩) = (x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐺‘(x + 1)))⟩) → frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐺‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
9 freceq2 5920 . . . . 5 (⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑀, (𝐺𝑀)⟩ → frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐺‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐺‘(x + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
108, 9sylan9eq 2089 . . . 4 (((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩) = (x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐺‘(x + 1)))⟩) 𝑀, (𝐹𝑀)⟩ = ⟨𝑀, (𝐺𝑀)⟩) → frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐺‘(x + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
115, 7, 10syl2anc 391 . . 3 (𝐹 = 𝐺 → frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐺‘(x + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
1211rneqd 4506 . 2 (𝐹 = 𝐺 → ran frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = ran frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐺‘(x + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
13 df-iseq 8893 . 2 seq𝑀( + , 𝐹, 𝑆) = ran frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
14 df-iseq 8893 . 2 seq𝑀( + , 𝐺, 𝑆) = ran frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐺‘(x + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩)
1512, 13, 143eqtr4g 2094 1 (𝐹 = 𝐺 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ⟨cop 3370  ran crn 4289  ‘cfv 4845  (class class class)co 5455   ↦ cmpt2 5457  freccfrec 5917  1c1 6712   + caddc 6714  ℤ≥cuz 8249  seqcseq 8892 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-iota 4810  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-recs 5861  df-frec 5918  df-iseq 8893 This theorem is referenced by:  expival  8911
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