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Theorem iseqeq1 8854
Description: Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
Assertion
Ref Expression
iseqeq1 (𝑀 = 𝑁 → seq𝑀( + , 𝐹, 𝑆) = seq𝑁( + , 𝐹, 𝑆))

Proof of Theorem iseqeq1
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . . 6 (𝑀 = 𝑁𝑀 = 𝑁)
2 fveq2 5121 . . . . . 6 (𝑀 = 𝑁 → (𝐹𝑀) = (𝐹𝑁))
31, 2opeq12d 3548 . . . . 5 (𝑀 = 𝑁 → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐹𝑁)⟩)
4 freceq2 5920 . . . . 5 (⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐹𝑁)⟩ → frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
53, 4syl 14 . . . 4 (𝑀 = 𝑁 → frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
6 fveq2 5121 . . . . . 6 (𝑀 = 𝑁 → (ℤ𝑀) = (ℤ𝑁))
7 eqid 2037 . . . . . 6 𝑆 = 𝑆
8 mpt2eq12 5507 . . . . . 6 (((ℤ𝑀) = (ℤ𝑁) 𝑆 = 𝑆) → (x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩) = (x (ℤ𝑁), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩))
96, 7, 8sylancl 392 . . . . 5 (𝑀 = 𝑁 → (x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩) = (x (ℤ𝑁), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩))
10 freceq1 5919 . . . . 5 ((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩) = (x (ℤ𝑁), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩) → frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩) = frec((x (ℤ𝑁), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
119, 10syl 14 . . . 4 (𝑀 = 𝑁 → frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩) = frec((x (ℤ𝑁), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
125, 11eqtrd 2069 . . 3 (𝑀 = 𝑁 → frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((x (ℤ𝑁), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
1312rneqd 4506 . 2 (𝑀 = 𝑁 → ran frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = ran frec((x (ℤ𝑁), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
14 df-iseq 8853 . 2 seq𝑀( + , 𝐹, 𝑆) = ran frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
15 df-iseq 8853 . 2 seq𝑁( + , 𝐹, 𝑆) = ran frec((x (ℤ𝑁), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩)
1613, 14, 153eqtr4g 2094 1 (𝑀 = 𝑁 → seq𝑀( + , 𝐹, 𝑆) = seq𝑁( + , 𝐹, 𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cop 3370  ran crn 4289  cfv 4845  (class class class)co 5455  cmpt2 5457  freccfrec 5917  1c1 6672   + caddc 6674  cuz 8209  seqcseq 8852
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-bnd 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-oprab 5459  df-mpt2 5460  df-recs 5861  df-frec 5918  df-iseq 8853
This theorem is referenced by: (None)
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