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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 10330 | . . . . 5 ⊢ +Q = (([Q] ∘ +pQ ) ↾ (Q × Q)) | |
2 | 1 | fveq1i 6665 | . . . 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 5586 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ (Q × Q)) | |
5 | 4 | fvresd 6684 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ +pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉) = (([Q] ∘ +pQ )‘〈𝐴, 𝐵〉)) |
6 | df-plpq 10324 | . . . . 5 ⊢ +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | |
7 | opex 5348 | . . . . 5 ⊢ 〈(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ V | |
8 | 6, 7 | fnmpoi 7762 | . . . 4 ⊢ +pQ Fn ((N × N) × (N × N)) |
9 | elpqn 10341 | . . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
10 | elpqn 10341 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
11 | opelxpi 5586 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) | |
12 | 9, 10, 11 | syl2an 597 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) |
13 | fvco2 6752 | . . . 4 ⊢ (( +pQ Fn ((N × N) × (N × N)) ∧ 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) → (([Q] ∘ +pQ )‘〈𝐴, 𝐵〉) = ([Q]‘( +pQ ‘〈𝐴, 𝐵〉))) | |
14 | 8, 12, 13 | sylancr 589 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (([Q] ∘ +pQ )‘〈𝐴, 𝐵〉) = ([Q]‘( +pQ ‘〈𝐴, 𝐵〉))) |
15 | 3, 5, 14 | 3eqtrd 2860 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( +Q ‘〈𝐴, 𝐵〉) = ([Q]‘( +pQ ‘〈𝐴, 𝐵〉))) |
16 | df-ov 7153 | . 2 ⊢ (𝐴 +Q 𝐵) = ( +Q ‘〈𝐴, 𝐵〉) | |
17 | df-ov 7153 | . . 3 ⊢ (𝐴 +pQ 𝐵) = ( +pQ ‘〈𝐴, 𝐵〉) | |
18 | 17 | fveq2i 6667 | . 2 ⊢ ([Q]‘(𝐴 +pQ 𝐵)) = ([Q]‘( +pQ ‘〈𝐴, 𝐵〉)) |
19 | 15, 16, 18 | 3eqtr4g 2881 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 〈cop 4566 × cxp 5547 ↾ cres 5551 ∘ ccom 5553 Fn wfn 6344 ‘cfv 6349 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 Ncnpi 10260 +N cpli 10261 ·N cmi 10262 +pQ cplpq 10264 Qcnq 10268 [Q]cerq 10270 +Q cplq 10271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-plpq 10324 df-nq 10328 df-plq 10330 |
This theorem is referenced by: addclnq 10361 addcomnq 10367 adderpq 10372 addassnq 10374 distrnq 10377 ltanq 10387 1lt2nq 10389 prlem934 10449 |
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