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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 6670 | . . . . 5 ⊢ (𝑀 = 𝑁 → (𝐹‘𝑀) = (𝐹‘𝑁)) | |
2 | opeq12 4805 | . . . . 5 ⊢ ((𝑀 = 𝑁 ∧ (𝐹‘𝑀) = (𝐹‘𝑁)) → 〈𝑀, (𝐹‘𝑀)〉 = 〈𝑁, (𝐹‘𝑁)〉) | |
3 | 1, 2 | mpdan 685 | . . . 4 ⊢ (𝑀 = 𝑁 → 〈𝑀, (𝐹‘𝑀)〉 = 〈𝑁, (𝐹‘𝑁)〉) |
4 | rdgeq2 8048 | . . . 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 5928 | . 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 13371 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
8 | df-seq 13371 | . 2 ⊢ seq𝑁( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑁, (𝐹‘𝑁)〉) “ ω) | |
9 | 6, 7, 8 | 3eqtr4g 2881 | 1 ⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Vcvv 3494 〈cop 4573 “ cima 5558 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ωcom 7580 reccrdg 8045 1c1 10538 + caddc 10540 seqcseq 13370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-iota 6314 df-fv 6363 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-seq 13371 |
This theorem is referenced by: seqeq1d 13376 seqfn 13382 seq1 13383 seqp1 13385 seqf1olem2 13411 seqid 13416 seqz 13419 iserex 15013 summolem2 15073 summo 15074 zsum 15075 isumsplit 15195 ntrivcvg 15253 ntrivcvgn0 15254 ntrivcvgtail 15256 ntrivcvgmullem 15257 prodmolem2 15289 prodmo 15290 zprod 15291 fprodntriv 15296 ege2le3 15443 gsumval2a 17895 leibpi 25520 dvradcnv2 40699 binomcxplemnotnn0 40708 stirlinglem12 42390 |
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