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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 6891 | . . . . 5 ⊢ (𝑀 = 𝑁 → (𝐹‘𝑀) = (𝐹‘𝑁)) | |
2 | opeq12 4871 | . . . . 5 ⊢ ((𝑀 = 𝑁 ∧ (𝐹‘𝑀) = (𝐹‘𝑁)) → 〈𝑀, (𝐹‘𝑀)〉 = 〈𝑁, (𝐹‘𝑁)〉) | |
3 | 1, 2 | mpdan 686 | . . . 4 ⊢ (𝑀 = 𝑁 → 〈𝑀, (𝐹‘𝑀)〉 = 〈𝑁, (𝐹‘𝑁)〉) |
4 | rdgeq2 8426 | . . . 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 6056 | . 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 13993 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
8 | df-seq 13993 | . 2 ⊢ seq𝑁( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑁, (𝐹‘𝑁)〉) “ ω) | |
9 | 6, 7, 8 | 3eqtr4g 2792 | 1 ⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 Vcvv 3469 〈cop 4630 “ cima 5675 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 ωcom 7864 reccrdg 8423 1c1 11133 + caddc 11135 seqcseq 13992 |
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 2698 |
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 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-xp 5678 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-iota 6494 df-fv 6550 df-ov 7417 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-seq 13993 |
This theorem is referenced by: seqeq1d 13998 seqfn 14004 seq1 14005 seqp1 14007 seqf1olem2 14033 seqid 14038 seqz 14041 iserex 15629 summolem2 15688 summo 15689 zsum 15690 isumsplit 15812 ntrivcvg 15869 ntrivcvgn0 15870 ntrivcvgtail 15872 ntrivcvgmullem 15873 prodmolem2 15905 prodmo 15906 zprod 15907 fprodntriv 15912 ege2le3 16060 gsumval2a 18638 leibpi 26867 dvradcnv2 43756 binomcxplemnotnn0 43765 stirlinglem12 45445 |
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