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Mirrors > Home > MPE Home > Th. List > seqeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Ref | Expression |
---|---|
seqeq1 | ⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6882 | . . . . 5 ⊢ (𝑀 = 𝑁 → (𝐹‘𝑀) = (𝐹‘𝑁)) | |
2 | opeq12 4868 | . . . . 5 ⊢ ((𝑀 = 𝑁 ∧ (𝐹‘𝑀) = (𝐹‘𝑁)) → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐹‘𝑁)⟩) | |
3 | 1, 2 | mpdan 684 | . . . 4 ⊢ (𝑀 = 𝑁 → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐹‘𝑁)⟩) |
4 | rdgeq2 8408 | . . . 4 ⊢ (⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐹‘𝑁)⟩ → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑀 = 𝑁 → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩)) |
6 | 5 | imaeq1d 6049 | . 2 ⊢ (𝑀 = 𝑁 → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩) “ ω)) |
7 | df-seq 13968 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) | |
8 | df-seq 13968 | . 2 ⊢ seq𝑁( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩) “ ω) | |
9 | 6, 7, 8 | 3eqtr4g 2789 | 1 ⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Vcvv 3466 ⟨cop 4627 “ cima 5670 ‘cfv 6534 (class class class)co 7402 ∈ cmpo 7404 ωcom 7849 reccrdg 8405 1c1 11108 + caddc 11110 seqcseq 13967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-xp 5673 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-iota 6486 df-fv 6542 df-ov 7405 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-seq 13968 |
This theorem is referenced by: seqeq1d 13973 seqfn 13979 seq1 13980 seqp1 13982 seqf1olem2 14009 seqid 14014 seqz 14017 iserex 15605 summolem2 15664 summo 15665 zsum 15666 isumsplit 15788 ntrivcvg 15845 ntrivcvgn0 15846 ntrivcvgtail 15848 ntrivcvgmullem 15849 prodmolem2 15881 prodmo 15882 zprod 15883 fprodntriv 15888 ege2le3 16036 gsumval2a 18614 leibpi 26815 dvradcnv2 43656 binomcxplemnotnn0 43665 stirlinglem12 45347 |
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