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Mirrors > Home > MPE Home > Th. List > addpqnq | Structured version Visualization version GIF version |
Description: Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addpqnq | ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-plq 9615 | . . . . 5 ⊢ +Q = (([Q] ∘ +pQ ) ↾ (Q × Q)) | |
2 | 1 | fveq1i 6104 | . . . 4 ⊢ ( +Q ‘〈𝐴, 𝐵〉) = ((([Q] ∘ +pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉) |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( +Q ‘〈𝐴, 𝐵〉) = ((([Q] ∘ +pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉)) |
4 | opelxpi 5072 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ (Q × Q)) | |
5 | fvres 6117 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (Q × Q) → ((([Q] ∘ +pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉) = (([Q] ∘ +pQ )‘〈𝐴, 𝐵〉)) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ +pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉) = (([Q] ∘ +pQ )‘〈𝐴, 𝐵〉)) |
7 | df-plpq 9609 | . . . . 5 ⊢ +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | |
8 | opex 4859 | . . . . 5 ⊢ 〈(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ V | |
9 | 7, 8 | fnmpt2i 7128 | . . . 4 ⊢ +pQ Fn ((N × N) × (N × N)) |
10 | elpqn 9626 | . . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
11 | elpqn 9626 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
12 | opelxpi 5072 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) | |
13 | 10, 11, 12 | syl2an 493 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) |
14 | fvco2 6183 | . . . 4 ⊢ (( +pQ Fn ((N × N) × (N × N)) ∧ 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) → (([Q] ∘ +pQ )‘〈𝐴, 𝐵〉) = ([Q]‘( +pQ ‘〈𝐴, 𝐵〉))) | |
15 | 9, 13, 14 | sylancr 694 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (([Q] ∘ +pQ )‘〈𝐴, 𝐵〉) = ([Q]‘( +pQ ‘〈𝐴, 𝐵〉))) |
16 | 3, 6, 15 | 3eqtrd 2648 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( +Q ‘〈𝐴, 𝐵〉) = ([Q]‘( +pQ ‘〈𝐴, 𝐵〉))) |
17 | df-ov 6552 | . 2 ⊢ (𝐴 +Q 𝐵) = ( +Q ‘〈𝐴, 𝐵〉) | |
18 | df-ov 6552 | . . 3 ⊢ (𝐴 +pQ 𝐵) = ( +pQ ‘〈𝐴, 𝐵〉) | |
19 | 18 | fveq2i 6106 | . 2 ⊢ ([Q]‘(𝐴 +pQ 𝐵)) = ([Q]‘( +pQ ‘〈𝐴, 𝐵〉)) |
20 | 16, 17, 19 | 3eqtr4g 2669 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 × cxp 5036 ↾ cres 5040 ∘ ccom 5042 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 Ncnpi 9545 +N cpli 9546 ·N cmi 9547 +pQ cplpq 9549 Qcnq 9553 [Q]cerq 9555 +Q cplq 9556 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-plpq 9609 df-nq 9613 df-plq 9615 |
This theorem is referenced by: addclnq 9646 addcomnq 9652 adderpq 9657 addassnq 9659 distrnq 9662 ltanq 9672 1lt2nq 9674 prlem934 9734 |
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