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Mirrors > Home > MPE Home > Th. List > addpipq | Structured version Visualization version GIF version |
Description: Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addpipq | ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 +pQ 〈𝐶, 𝐷〉) = 〈((𝐴 ·N 𝐷) +N (𝐶 ·N 𝐵)), (𝐵 ·N 𝐷)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5712 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | opelxpi 5712 | . . 3 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → 〈𝐶, 𝐷〉 ∈ (N × N)) | |
3 | addpipq2 10927 | . . 3 ⊢ ((〈𝐴, 𝐵〉 ∈ (N × N) ∧ 〈𝐶, 𝐷〉 ∈ (N × N)) → (〈𝐴, 𝐵〉 +pQ 〈𝐶, 𝐷〉) = 〈(((1st ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) +N ((1st ‘〈𝐶, 𝐷〉) ·N (2nd ‘〈𝐴, 𝐵〉))), ((2nd ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉))〉) | |
4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 +pQ 〈𝐶, 𝐷〉) = 〈(((1st ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) +N ((1st ‘〈𝐶, 𝐷〉) ·N (2nd ‘〈𝐴, 𝐵〉))), ((2nd ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉))〉) |
5 | op1stg 7983 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) | |
6 | op2ndg 7984 | . . . . 5 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (2nd ‘〈𝐶, 𝐷〉) = 𝐷) | |
7 | 5, 6 | oveqan12d 7424 | . . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) = (𝐴 ·N 𝐷)) |
8 | op1stg 7983 | . . . . 5 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (1st ‘〈𝐶, 𝐷〉) = 𝐶) | |
9 | op2ndg 7984 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
10 | 8, 9 | oveqan12rd 7425 | . . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘〈𝐶, 𝐷〉) ·N (2nd ‘〈𝐴, 𝐵〉)) = (𝐶 ·N 𝐵)) |
11 | 7, 10 | oveq12d 7423 | . . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (((1st ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) +N ((1st ‘〈𝐶, 𝐷〉) ·N (2nd ‘〈𝐴, 𝐵〉))) = ((𝐴 ·N 𝐷) +N (𝐶 ·N 𝐵))) |
12 | 9, 6 | oveqan12d 7424 | . . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((2nd ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) = (𝐵 ·N 𝐷)) |
13 | 11, 12 | opeq12d 4880 | . 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → 〈(((1st ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) +N ((1st ‘〈𝐶, 𝐷〉) ·N (2nd ‘〈𝐴, 𝐵〉))), ((2nd ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉))〉 = 〈((𝐴 ·N 𝐷) +N (𝐶 ·N 𝐵)), (𝐵 ·N 𝐷)〉) |
14 | 4, 13 | eqtrd 2772 | 1 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 +pQ 〈𝐶, 𝐷〉) = 〈((𝐴 ·N 𝐷) +N (𝐶 ·N 𝐵)), (𝐵 ·N 𝐷)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 〈cop 4633 × cxp 5673 ‘cfv 6540 (class class class)co 7405 1st c1st 7969 2nd c2nd 7970 Ncnpi 10835 +N cpli 10836 ·N cmi 10837 +pQ cplpq 10839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-plpq 10899 |
This theorem is referenced by: addassnq 10949 distrnq 10952 1lt2nq 10964 ltexnq 10966 prlem934 11024 |
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