Theorem seqeq3 13970

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

Assertion

Ref	Expression
seqeq3	⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))

Proof of Theorem seqeq3

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6890	. . . . . . 7 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑥 + 1)) = (𝐺‘(𝑥 + 1)))
2	1	oveq2d 7424	. . . . . 6 ⊢ (𝐹 = 𝐺 → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐺‘(𝑥 + 1))))
3	2	opeq2d 4880	. . . . 5 ⊢ (𝐹 = 𝐺 → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩)
4	3	mpoeq3dv 7487	. . . 4 ⊢ (𝐹 = 𝐺 → (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩))
5		fveq1 6890	. . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑀) = (𝐺‘𝑀))
6	5	opeq2d 4880	. . . 4 ⊢ (𝐹 = 𝐺 → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑀, (𝐺‘𝑀)⟩)
7		rdgeq12 8412	. . . 4 ⊢ (((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩) ∧ ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑀, (𝐺‘𝑀)⟩) → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺‘𝑀)⟩))
8	4, 6, 7	syl2anc 584	. . 3 ⊢ (𝐹 = 𝐺 → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺‘𝑀)⟩))
9	8	imaeq1d 6058	. 2 ⊢ (𝐹 = 𝐺 → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺‘𝑀)⟩) “ ω))
10		df-seq 13966	. 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
11		df-seq 13966	. 2 ⊢ seq𝑀( + , 𝐺) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺‘𝑀)⟩) “ ω)
12	9, 10, 11	3eqtr4g 2797	1 ⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))

Syntax hints:   → wi 4   = wceq 1541  Vcvv 3474  ⟨cop 4634   “ cima 5679  ‘cfv 6543  (class class class)co 7408   ∈ cmpo 7410  ωcom 7854  reccrdg 8408  1c1 11110   + caddc 11112  seqcseq 13965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-seq 13966

This theorem is referenced by:  seqeq3d  13973  cbvprod  15858  iprodmul  15946  geolim3  25851  leibpilem2  26443  basel  26591  faclim  34711  ovoliunnfl  36525  voliunnfl  36527  heiborlem10  36683  binomcxplemnn0  43098  binomcxplemdvsum  43104  binomcxp  43106  fourierdlem112  44924  fouriersw  44937  voliunsge0lem  45178