Theorem seqeq1 10434

Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Assertion

Ref	Expression
seqeq1	⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))

Proof of Theorem seqeq1

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 5511	. . . . . 6 ⊢ (𝑀 = 𝑁 → (𝐹‘𝑀) = (𝐹‘𝑁))
3	1, 2	opeq12d 3784	. . . . 5 ⊢ (𝑀 = 𝑁 → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐹‘𝑁)⟩)
4		freceq2 6388	. . . . 5 ⊢ (⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐹‘𝑁)⟩ → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
5	3, 4	syl 14	. . . 4 ⊢ (𝑀 = 𝑁 → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
6		fveq2 5511	. . . . . 6 ⊢ (𝑀 = 𝑁 → (ℤ≥‘𝑀) = (ℤ≥‘𝑁))
7		eqid 2177	. . . . . 6 ⊢ V = V
8		mpoeq12 5929	. . . . . 6 ⊢ (((ℤ≥‘𝑀) = (ℤ≥‘𝑁) ∧ V = V) → (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩))
9	6, 7, 8	sylancl 413	. . . . 5 ⊢ (𝑀 = 𝑁 → (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩))
10		freceq1 6387	. . . . 5 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
11	9, 10	syl 14	. . . 4 ⊢ (𝑀 = 𝑁 → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
12	5, 11	eqtrd 2210	. . 3 ⊢ (𝑀 = 𝑁 → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
13	12	rneqd 4852	. 2 ⊢ (𝑀 = 𝑁 → ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = ran frec((𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
14		df-seqfrec 10432	. 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
15		df-seqfrec 10432	. 2 ⊢ seq𝑁( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩)
16	13, 14, 15	3eqtr4g 2235	1 ⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))

Syntax hints:   → wi 4   = wceq 1353  Vcvv 2737  ⟨cop 3594  ran crn 4624  ‘cfv 5212  (class class class)co 5869   ∈ cmpo 5871  freccfrec 6385  1c1 7803   + caddc 7805  ℤ_≥cuz 9517  seqcseq 10431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-cnv 4631  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fv 5220  df-oprab 5873  df-mpo 5874  df-recs 6300  df-frec 6386  df-seqfrec 10432

This theorem is referenced by:  seqeq1d  10437  seq3f1olemqsum  10486  seq3id  10494  seq3z  10497  iserex  11331  summodclem2  11374  summodc  11375  zsumdc  11376  isumsplit  11483  ntrivcvgap  11540  ntrivcvgap0  11541  prodmodclem2  11569  prodmodc  11570  zproddc  11571  fprodntrivap  11576  ege2le3  11663