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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 6837 | . . . . . . 7 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑥 + 1)) = (𝐺‘(𝑥 + 1))) | |
2 | 1 | oveq2d 7366 | . . . . . 6 ⊢ (𝐹 = 𝐺 → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐺‘(𝑥 + 1)))) |
3 | 2 | opeq2d 4836 | . . . . 5 ⊢ (𝐹 = 𝐺 → 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉 = 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉) |
4 | 3 | mpoeq3dv 7429 | . . . 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 6837 | . . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑀) = (𝐺‘𝑀)) | |
6 | 5 | opeq2d 4836 | . . . 4 ⊢ (𝐹 = 𝐺 → 〈𝑀, (𝐹‘𝑀)〉 = 〈𝑀, (𝐺‘𝑀)〉) |
7 | rdgeq12 8327 | . . . 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 585 | . . 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 6009 | . 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 13837 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
11 | df-seq 13837 | . 2 ⊢ seq𝑀( + , 𝐺) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉), 〈𝑀, (𝐺‘𝑀)〉) “ ω) | |
12 | 9, 10, 11 | 3eqtr4g 2803 | 1 ⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Vcvv 3444 〈cop 4591 “ cima 5634 ‘cfv 6492 (class class class)co 7350 ∈ cmpo 7352 ωcom 7793 reccrdg 8323 1c1 10986 + caddc 10988 seqcseq 13836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-xp 5637 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-iota 6444 df-fv 6500 df-ov 7353 df-oprab 7354 df-mpo 7355 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-seq 13837 |
This theorem is referenced by: seqeq3d 13844 cbvprod 15734 iprodmul 15822 geolim3 25627 leibpilem2 26219 basel 26367 faclim 34113 ovoliunnfl 36051 voliunnfl 36053 heiborlem10 36210 binomcxplemnn0 42430 binomcxplemdvsum 42436 binomcxp 42438 fourierdlem112 44250 fouriersw 44263 voliunsge0lem 44504 |
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