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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 5722 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
| 2 | opelxpi 5722 | . . 3 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → 〈𝐶, 𝐷〉 ∈ (N × N)) | |
| 3 | addpipq2 10976 | . . 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 8026 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) | |
| 6 | op2ndg 8027 | . . . . 5 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (2nd ‘〈𝐶, 𝐷〉) = 𝐷) | |
| 7 | 5, 6 | oveqan12d 7450 | . . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) = (𝐴 ·N 𝐷)) |
| 8 | op1stg 8026 | . . . . 5 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (1st ‘〈𝐶, 𝐷〉) = 𝐶) | |
| 9 | op2ndg 8027 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
| 10 | 8, 9 | oveqan12rd 7451 | . . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘〈𝐶, 𝐷〉) ·N (2nd ‘〈𝐴, 𝐵〉)) = (𝐶 ·N 𝐵)) |
| 11 | 7, 10 | oveq12d 7449 | . . 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 7450 | . . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((2nd ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) = (𝐵 ·N 𝐷)) |
| 13 | 11, 12 | opeq12d 4881 | . 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 2777 | 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 1540 ∈ wcel 2108 〈cop 4632 × cxp 5683 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 Ncnpi 10884 +N cpli 10885 ·N cmi 10886 +pQ cplpq 10888 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-plpq 10948 |
| This theorem is referenced by: addassnq 10998 distrnq 11001 1lt2nq 11013 ltexnq 11015 prlem934 11073 |
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