![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > seqeq3 | Structured version Visualization version GIF version |
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Ref | Expression |
---|---|
seqeq3 | ⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6896 | . . . . . . 7 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑥 + 1)) = (𝐺‘(𝑥 + 1))) | |
2 | 1 | oveq2d 7436 | . . . . . 6 ⊢ (𝐹 = 𝐺 → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐺‘(𝑥 + 1)))) |
3 | 2 | opeq2d 4881 | . . . . 5 ⊢ (𝐹 = 𝐺 → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩) |
4 | 3 | mpoeq3dv 7499 | . . . 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 6896 | . . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑀) = (𝐺‘𝑀)) | |
6 | 5 | opeq2d 4881 | . . . 4 ⊢ (𝐹 = 𝐺 → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑀, (𝐺‘𝑀)⟩) |
7 | rdgeq12 8433 | . . . 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 583 | . . 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 6062 | . 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 13999 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) | |
11 | df-seq 13999 | . 2 ⊢ seq𝑀( + , 𝐺) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺‘𝑀)⟩) “ ω) | |
12 | 9, 10, 11 | 3eqtr4g 2793 | 1 ⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 Vcvv 3471 ⟨cop 4635 “ cima 5681 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 ωcom 7870 reccrdg 8429 1c1 11139 + caddc 11141 seqcseq 13998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5684 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-iota 6500 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-seq 13999 |
This theorem is referenced by: seqeq3d 14006 cbvprod 15891 iprodmul 15979 geolim3 26273 leibpilem2 26872 basel 27021 faclim 35340 ovoliunnfl 37135 voliunnfl 37137 heiborlem10 37293 binomcxplemnn0 43786 binomcxplemdvsum 43792 binomcxp 43794 fourierdlem112 45606 fouriersw 45619 voliunsge0lem 45860 |
Copyright terms: Public domain | W3C validator |