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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 6645 | . . . . 5 ⊢ (𝑀 = 𝑁 → (𝐹‘𝑀) = (𝐹‘𝑁)) | |
2 | opeq12 4767 | . . . . 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 8031 | . . . 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 5895 | . 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 13365 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
8 | df-seq 13365 | . 2 ⊢ seq𝑁( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑁, (𝐹‘𝑁)〉) “ ω) | |
9 | 6, 7, 8 | 3eqtr4g 2858 | 1 ⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 Vcvv 3441 〈cop 4531 “ cima 5522 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ωcom 7560 reccrdg 8028 1c1 10527 + caddc 10529 seqcseq 13364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-iota 6283 df-fv 6332 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-seq 13365 |
This theorem is referenced by: seqeq1d 13370 seqfn 13376 seq1 13377 seqp1 13379 seqf1olem2 13406 seqid 13411 seqz 13414 iserex 15005 summolem2 15065 summo 15066 zsum 15067 isumsplit 15187 ntrivcvg 15245 ntrivcvgn0 15246 ntrivcvgtail 15248 ntrivcvgmullem 15249 prodmolem2 15281 prodmo 15282 zprod 15283 fprodntriv 15288 ege2le3 15435 gsumval2a 17887 leibpi 25528 dvradcnv2 41051 binomcxplemnotnn0 41060 stirlinglem12 42727 |
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