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| 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 6904 | . . . . . . 7 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑥 + 1)) = (𝐺‘(𝑥 + 1))) | |
| 2 | 1 | oveq2d 7448 | . . . . . 6 ⊢ (𝐹 = 𝐺 → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐺‘(𝑥 + 1)))) | 
| 3 | 2 | opeq2d 4879 | . . . . 5 ⊢ (𝐹 = 𝐺 → 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉 = 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉) | 
| 4 | 3 | mpoeq3dv 7513 | . . . 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 6904 | . . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑀) = (𝐺‘𝑀)) | |
| 6 | 5 | opeq2d 4879 | . . . 4 ⊢ (𝐹 = 𝐺 → 〈𝑀, (𝐹‘𝑀)〉 = 〈𝑀, (𝐺‘𝑀)〉) | 
| 7 | rdgeq12 8454 | . . . 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 6076 | . 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 14044 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
| 11 | df-seq 14044 | . 2 ⊢ seq𝑀( + , 𝐺) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉), 〈𝑀, (𝐺‘𝑀)〉) “ ω) | |
| 12 | 9, 10, 11 | 3eqtr4g 2801 | 1 ⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 Vcvv 3479 〈cop 4631 “ cima 5687 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 ωcom 7888 reccrdg 8450 1c1 11157 + caddc 11159 seqcseq 14043 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-xp 5690 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-iota 6513 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-seq 14044 | 
| This theorem is referenced by: seqeq3d 14051 cbvprod 15950 cbvprodv 15951 prodeq1i 15953 iprodmul 16040 geolim3 26382 leibpilem2 26985 basel 27134 faclim 35747 sumeq2si 36204 prodeq2si 36206 cbvprodvw2 36249 ovoliunnfl 37670 voliunnfl 37672 heiborlem10 37828 binomcxplemnn0 44373 binomcxplemdvsum 44379 binomcxp 44381 fourierdlem112 46238 fouriersw 46251 voliunsge0lem 46492 | 
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