Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5072 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | opelxpi 5072 | . . 3 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → 〈𝐶, 𝐷〉 ∈ (N × N)) | |
3 | addpipq2 9637 | . . 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 493 | . 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 7071 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) | |
6 | op2ndg 7072 | . . . . 5 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (2nd ‘〈𝐶, 𝐷〉) = 𝐷) | |
7 | 5, 6 | oveqan12d 6568 | . . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) = (𝐴 ·N 𝐷)) |
8 | op1stg 7071 | . . . . 5 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (1st ‘〈𝐶, 𝐷〉) = 𝐶) | |
9 | op2ndg 7072 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
10 | 8, 9 | oveqan12rd 6569 | . . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘〈𝐶, 𝐷〉) ·N (2nd ‘〈𝐴, 𝐵〉)) = (𝐶 ·N 𝐵)) |
11 | 7, 10 | oveq12d 6567 | . . 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 6568 | . . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((2nd ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) = (𝐵 ·N 𝐷)) |
13 | 11, 12 | opeq12d 4348 | . 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 2644 | 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 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 × cxp 5036 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 Ncnpi 9545 +N cpli 9546 ·N cmi 9547 +pQ cplpq 9549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-plpq 9609 |
This theorem is referenced by: addassnq 9659 distrnq 9662 1lt2nq 9674 ltexnq 9676 prlem934 9734 |
Copyright terms: Public domain | W3C validator |