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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 6672 | . . . . 5 ⊢ (𝑀 = 𝑁 → (𝐹‘𝑀) = (𝐹‘𝑁)) | |
2 | opeq12 4807 | . . . . 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 8050 | . . . 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 5930 | . 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 13373 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
8 | df-seq 13373 | . 2 ⊢ seq𝑁( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑁, (𝐹‘𝑁)〉) “ ω) | |
9 | 6, 7, 8 | 3eqtr4g 2883 | 1 ⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Vcvv 3496 〈cop 4575 “ cima 5560 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 ωcom 7582 reccrdg 8047 1c1 10540 + caddc 10542 seqcseq 13372 |
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 2795 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-iota 6316 df-fv 6365 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-seq 13373 |
This theorem is referenced by: seqeq1d 13378 seqfn 13384 seq1 13385 seqp1 13387 seqf1olem2 13413 seqid 13418 seqz 13421 iserex 15015 summolem2 15075 summo 15076 zsum 15077 isumsplit 15197 ntrivcvg 15255 ntrivcvgn0 15256 ntrivcvgtail 15258 ntrivcvgmullem 15259 prodmolem2 15291 prodmo 15292 zprod 15293 fprodntriv 15298 ege2le3 15445 gsumval2a 17897 leibpi 25522 dvradcnv2 40686 binomcxplemnotnn0 40695 stirlinglem12 42377 |
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