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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 10553 | . . . . 5 ⊢ +Q = (([Q] ∘ +pQ ) ↾ (Q × Q)) | |
2 | 1 | fveq1i 6737 | . . . 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 5603 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ (Q × Q)) | |
5 | 4 | fvresd 6756 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ +pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉) = (([Q] ∘ +pQ )‘〈𝐴, 𝐵〉)) |
6 | df-plpq 10547 | . . . . 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 5363 | . . . . 5 ⊢ 〈(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ V | |
8 | 6, 7 | fnmpoi 7859 | . . . 4 ⊢ +pQ Fn ((N × N) × (N × N)) |
9 | elpqn 10564 | . . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
10 | elpqn 10564 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
11 | opelxpi 5603 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) | |
12 | 9, 10, 11 | syl2an 599 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) |
13 | fvco2 6827 | . . . 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 590 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (([Q] ∘ +pQ )‘〈𝐴, 𝐵〉) = ([Q]‘( +pQ ‘〈𝐴, 𝐵〉))) |
15 | 3, 5, 14 | 3eqtrd 2782 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( +Q ‘〈𝐴, 𝐵〉) = ([Q]‘( +pQ ‘〈𝐴, 𝐵〉))) |
16 | df-ov 7235 | . 2 ⊢ (𝐴 +Q 𝐵) = ( +Q ‘〈𝐴, 𝐵〉) | |
17 | df-ov 7235 | . . 3 ⊢ (𝐴 +pQ 𝐵) = ( +pQ ‘〈𝐴, 𝐵〉) | |
18 | 17 | fveq2i 6739 | . 2 ⊢ ([Q]‘(𝐴 +pQ 𝐵)) = ([Q]‘( +pQ ‘〈𝐴, 𝐵〉)) |
19 | 15, 16, 18 | 3eqtr4g 2804 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 〈cop 4562 × cxp 5564 ↾ cres 5568 ∘ ccom 5570 Fn wfn 6393 ‘cfv 6398 (class class class)co 7232 1st c1st 7778 2nd c2nd 7779 Ncnpi 10483 +N cpli 10484 ·N cmi 10485 +pQ cplpq 10487 Qcnq 10491 [Q]cerq 10493 +Q cplq 10494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 ax-un 7542 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-fv 6406 df-ov 7235 df-oprab 7236 df-mpo 7237 df-1st 7780 df-2nd 7781 df-plpq 10547 df-nq 10551 df-plq 10553 |
This theorem is referenced by: addclnq 10584 addcomnq 10590 adderpq 10595 addassnq 10597 distrnq 10600 ltanq 10610 1lt2nq 10612 prlem934 10672 |
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