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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 5737 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | opelxpi 5737 | . . 3 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → 〈𝐶, 𝐷〉 ∈ (N × N)) | |
3 | addpipq2 11005 | . . 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 595 | . 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 8042 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) | |
6 | op2ndg 8043 | . . . . 5 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (2nd ‘〈𝐶, 𝐷〉) = 𝐷) | |
7 | 5, 6 | oveqan12d 7467 | . . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) = (𝐴 ·N 𝐷)) |
8 | op1stg 8042 | . . . . 5 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (1st ‘〈𝐶, 𝐷〉) = 𝐶) | |
9 | op2ndg 8043 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
10 | 8, 9 | oveqan12rd 7468 | . . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘〈𝐶, 𝐷〉) ·N (2nd ‘〈𝐴, 𝐵〉)) = (𝐶 ·N 𝐵)) |
11 | 7, 10 | oveq12d 7466 | . . 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 7467 | . . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((2nd ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) = (𝐵 ·N 𝐷)) |
13 | 11, 12 | opeq12d 4905 | . 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 2780 | 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 395 = wceq 1537 ∈ wcel 2108 〈cop 4654 × cxp 5698 ‘cfv 6573 (class class class)co 7448 1st c1st 8028 2nd c2nd 8029 Ncnpi 10913 +N cpli 10914 ·N cmi 10915 +pQ cplpq 10917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-plpq 10977 |
This theorem is referenced by: addassnq 11027 distrnq 11030 1lt2nq 11042 ltexnq 11044 prlem934 11102 |
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