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Mirrors > Home > MPE Home > Th. List > seqeq3 | Structured version Visualization version GIF version |
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Ref | Expression |
---|---|
seqeq3 | ⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6347 | . . . . . . 7 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑥 + 1)) = (𝐺‘(𝑥 + 1))) | |
2 | 1 | oveq2d 6825 | . . . . . 6 ⊢ (𝐹 = 𝐺 → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐺‘(𝑥 + 1)))) |
3 | 2 | opeq2d 4556 | . . . . 5 ⊢ (𝐹 = 𝐺 → 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉 = 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉) |
4 | 3 | mpt2eq3dv 6882 | . . . 4 ⊢ (𝐹 = 𝐺 → (𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉) = (𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉)) |
5 | fveq1 6347 | . . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑀) = (𝐺‘𝑀)) | |
6 | 5 | opeq2d 4556 | . . . 4 ⊢ (𝐹 = 𝐺 → 〈𝑀, (𝐹‘𝑀)〉 = 〈𝑀, (𝐺‘𝑀)〉) |
7 | rdgeq12 7674 | . . . 4 ⊢ (((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉) = (𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉) ∧ 〈𝑀, (𝐹‘𝑀)〉 = 〈𝑀, (𝐺‘𝑀)〉) → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉), 〈𝑀, (𝐺‘𝑀)〉)) | |
8 | 4, 6, 7 | syl2anc 696 | . . 3 ⊢ (𝐹 = 𝐺 → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉), 〈𝑀, (𝐺‘𝑀)〉)) |
9 | 8 | imaeq1d 5619 | . 2 ⊢ (𝐹 = 𝐺 → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉), 〈𝑀, (𝐺‘𝑀)〉) “ ω)) |
10 | df-seq 12992 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
11 | df-seq 12992 | . 2 ⊢ seq𝑀( + , 𝐺) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉), 〈𝑀, (𝐺‘𝑀)〉) “ ω) | |
12 | 9, 10, 11 | 3eqtr4g 2815 | 1 ⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1628 Vcvv 3336 〈cop 4323 “ cima 5265 ‘cfv 6045 (class class class)co 6809 ↦ cmpt2 6811 ωcom 7226 reccrdg 7670 1c1 10125 + caddc 10127 seqcseq 12991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ral 3051 df-rex 3052 df-rab 3055 df-v 3338 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-nul 4055 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4585 df-br 4801 df-opab 4861 df-mpt 4878 df-xp 5268 df-cnv 5270 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-pred 5837 df-iota 6008 df-fv 6053 df-ov 6812 df-oprab 6813 df-mpt2 6814 df-wrecs 7572 df-recs 7633 df-rdg 7671 df-seq 12992 |
This theorem is referenced by: seqeq3d 12999 cbvprod 14840 iprodmul 14929 geolim3 24289 leibpilem2 24863 basel 25011 faclim 31935 ovoliunnfl 33760 voliunnfl 33762 heiborlem10 33928 binomcxplemnn0 39046 binomcxplemdvsum 39052 binomcxp 39054 fourierdlem112 40934 fouriersw 40947 voliunsge0lem 41188 |
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