Theorem iseqeq3 9216

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.

Step	Hyp	Ref	Expression
1		simp1 904	. . . . . . . 8 ⊢ ((𝐹 = 𝐺 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆) → 𝐹 = 𝐺)
2	1	fveq1d 5180	. . . . . . 7 ⊢ ((𝐹 = 𝐺 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆) → (𝐹‘(𝑥 + 1)) = (𝐺‘(𝑥 + 1)))
3	2	oveq2d 5528	. . . . . 6 ⊢ ((𝐹 = 𝐺 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐺‘(𝑥 + 1))))
4	3	opeq2d 3556	. . . . 5 ⊢ ((𝐹 = 𝐺 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩)
5	4	mpt2eq3dva 5569	. . . 4 ⊢ (𝐹 = 𝐺 → (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩))
6		fveq1 5177	. . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑀) = (𝐺‘𝑀))
7	6	opeq2d 3556	. . . 4 ⊢ (𝐹 = 𝐺 → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑀, (𝐺‘𝑀)⟩)
8		freceq1 5979	. . . . 5 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩) → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩))
9		freceq2 5980	. . . . 5 ⊢ (⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑀, (𝐺‘𝑀)⟩ → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺‘𝑀)⟩))
10	8, 9	sylan9eq 2092	. . . 4 ⊢ (((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩) ∧ ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑀, (𝐺‘𝑀)⟩) → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺‘𝑀)⟩))
11	5, 7, 10	syl2anc 391	. . 3 ⊢ (𝐹 = 𝐺 → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺‘𝑀)⟩))
12	11	rneqd 4563	. 2 ⊢ (𝐹 = 𝐺 → ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺‘𝑀)⟩))
13		df-iseq 9212	. 2 ⊢ seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
14		df-iseq 9212	. 2 ⊢ seq𝑀( + , 𝐺, 𝑆) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺‘𝑀)⟩)
15	12, 13, 14	3eqtr4g 2097	1 ⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))

Syntax hints:   → wi 4   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ⟨cop 3378  ran crn 4346  ‘cfv 4902  (class class class)co 5512   ↦ cmpt2 5514  freccfrec 5977  1c1 6890   + caddc 6892  ℤ_≥cuz 8473  seqcseq 9211

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-recs 5920  df-frec 5978  df-iseq 9212

This theorem is referenced by:  expival  9257  sumeq1  9874