![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > seqeq2 | Structured version Visualization version GIF version |
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Ref | Expression |
---|---|
seqeq2 | ⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq 7412 | . . . . . 6 ⊢ ( + = 𝑄 → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦𝑄(𝐹‘(𝑥 + 1)))) | |
2 | 1 | opeq2d 4880 | . . . . 5 ⊢ ( + = 𝑄 → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩) |
3 | 2 | mpoeq3dv 7485 | . . . 4 ⊢ ( + = 𝑄 → (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩)) |
4 | rdgeq1 8408 | . . . 4 ⊢ ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩) → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ ( + = 𝑄 → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)) |
6 | 5 | imaeq1d 6057 | . 2 ⊢ ( + = 𝑄 → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)) |
7 | df-seq 13964 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) | |
8 | df-seq 13964 | . 2 ⊢ seq𝑀(𝑄, 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) | |
9 | 6, 7, 8 | 3eqtr4g 2798 | 1 ⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Vcvv 3475 ⟨cop 4634 “ cima 5679 ‘cfv 6541 (class class class)co 7406 ∈ cmpo 7408 ωcom 7852 reccrdg 8406 1c1 11108 + caddc 11110 seqcseq 13963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rab 3434 df-v 3477 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 6298 df-iota 6493 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-seq 13964 |
This theorem is referenced by: seqeq2d 13970 sadcom 16401 ressmulgnn 32172 cvmliftlem15 34278 |
Copyright terms: Public domain | W3C validator |