Theorem iseqeq1 9434

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.

Step	Hyp	Ref	Expression
1		id 19	. . . . . 6 ⊢ (𝑀 = 𝑁 → 𝑀 = 𝑁)
2		fveq2 5198	. . . . . 6 ⊢ (𝑀 = 𝑁 → (𝐹‘𝑀) = (𝐹‘𝑁))
3	1, 2	opeq12d 3578	. . . . 5 ⊢ (𝑀 = 𝑁 → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐹‘𝑁)⟩)
4		freceq2 6003	. . . . 5 ⊢ (⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐹‘𝑁)⟩ → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
5	3, 4	syl 14	. . . 4 ⊢ (𝑀 = 𝑁 → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
6		fveq2 5198	. . . . . 6 ⊢ (𝑀 = 𝑁 → (ℤ≥‘𝑀) = (ℤ≥‘𝑁))
7		eqid 2081	. . . . . 6 ⊢ 𝑆 = 𝑆
8		mpt2eq12 5585	. . . . . 6 ⊢ (((ℤ≥‘𝑀) = (ℤ≥‘𝑁) ∧ 𝑆 = 𝑆) → (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩))
9	6, 7, 8	sylancl 404	. . . . 5 ⊢ (𝑀 = 𝑁 → (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩))
10		freceq1 6002	. . . . 5 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
11	9, 10	syl 14	. . . 4 ⊢ (𝑀 = 𝑁 → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
12	5, 11	eqtrd 2113	. . 3 ⊢ (𝑀 = 𝑁 → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
13	12	rneqd 4581	. 2 ⊢ (𝑀 = 𝑁 → ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = ran frec((𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
14		df-iseq 9432	. 2 ⊢ seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
15		df-iseq 9432	. 2 ⊢ seq𝑁( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩)
16	13, 14, 15	3eqtr4g 2138	1 ⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹, 𝑆) = seq𝑁( + , 𝐹, 𝑆))

Syntax hints:   → wi 4   = wceq 1284  ⟨cop 3401  ran crn 4364  ‘cfv 4922  (class class class)co 5532   ↦ cmpt2 5534  freccfrec 6000  1c1 6982   + caddc 6984  ℤ_≥cuz 8619  seqcseq 9431

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-iota 4887  df-fv 4930  df-oprab 5536  df-mpt2 5537  df-recs 5943  df-frec 6001  df-iseq 9432

This theorem is referenced by:  iseqid  9467  iseqz  9469  ibcval5  9690  bcn2  9691  iiserex  10177