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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 6669 | . . . . . . 7 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑥 + 1)) = (𝐺‘(𝑥 + 1))) | |
2 | 1 | oveq2d 7172 | . . . . . 6 ⊢ (𝐹 = 𝐺 → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐺‘(𝑥 + 1)))) |
3 | 2 | opeq2d 4810 | . . . . 5 ⊢ (𝐹 = 𝐺 → 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉 = 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉) |
4 | 3 | mpoeq3dv 7233 | . . . 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 6669 | . . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑀) = (𝐺‘𝑀)) | |
6 | 5 | opeq2d 4810 | . . . 4 ⊢ (𝐹 = 𝐺 → 〈𝑀, (𝐹‘𝑀)〉 = 〈𝑀, (𝐺‘𝑀)〉) |
7 | rdgeq12 8049 | . . . 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 586 | . . 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 5928 | . 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 13371 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
11 | df-seq 13371 | . 2 ⊢ seq𝑀( + , 𝐺) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉), 〈𝑀, (𝐺‘𝑀)〉) “ ω) | |
12 | 9, 10, 11 | 3eqtr4g 2881 | 1 ⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺)) |
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-ov 7159 df-oprab 7160 df-mpo 7161 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-seq 13371 |
This theorem is referenced by: seqeq3d 13378 cbvprod 15269 iprodmul 15357 geolim3 24928 leibpilem2 25519 basel 25667 faclim 32978 ovoliunnfl 34949 voliunnfl 34951 heiborlem10 35113 binomcxplemnn0 40701 binomcxplemdvsum 40707 binomcxp 40709 fourierdlem112 42523 fouriersw 42536 voliunsge0lem 42774 |
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