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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 6102 | . . . . . . 7 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑥 + 1)) = (𝐺‘(𝑥 + 1))) | |
2 | 1 | oveq2d 6565 | . . . . . 6 ⊢ (𝐹 = 𝐺 → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐺‘(𝑥 + 1)))) |
3 | 2 | opeq2d 4347 | . . . . 5 ⊢ (𝐹 = 𝐺 → 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉 = 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉) |
4 | 3 | mpt2eq3dv 6619 | . . . 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 6102 | . . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑀) = (𝐺‘𝑀)) | |
6 | 5 | opeq2d 4347 | . . . 4 ⊢ (𝐹 = 𝐺 → 〈𝑀, (𝐹‘𝑀)〉 = 〈𝑀, (𝐺‘𝑀)〉) |
7 | rdgeq12 7396 | . . . 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 691 | . . 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 5384 | . 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 12664 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
11 | df-seq 12664 | . 2 ⊢ seq𝑀( + , 𝐺) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))〉), 〈𝑀, (𝐺‘𝑀)〉) “ ω) | |
12 | 9, 10, 11 | 3eqtr4g 2669 | 1 ⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 Vcvv 3173 〈cop 4131 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 reccrdg 7392 1c1 9816 + caddc 9818 seqcseq 12663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seq 12664 |
This theorem is referenced by: seqeq3d 12671 cbvprod 14484 iprodmul 14573 geolim3 23898 leibpilem2 24468 basel 24616 faclim 30885 ovoliunnfl 32621 voliunnfl 32623 heiborlem10 32789 binomcxplemnn0 37570 binomcxplemdvsum 37576 binomcxp 37578 fourierdlem112 39111 fouriersw 39124 voliunsge0lem 39365 |
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