![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mulpipq2 | Structured version Visualization version GIF version |
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulpipq2 | ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = 〈((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6896 | . . . 4 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴)) | |
2 | 1 | oveq1d 7434 | . . 3 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑥) ·N (1st ‘𝑦)) = ((1st ‘𝐴) ·N (1st ‘𝑦))) |
3 | fveq2 6896 | . . . 4 ⊢ (𝑥 = 𝐴 → (2nd ‘𝑥) = (2nd ‘𝐴)) | |
4 | 3 | oveq1d 7434 | . . 3 ⊢ (𝑥 = 𝐴 → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝑦))) |
5 | 2, 4 | opeq12d 4883 | . 2 ⊢ (𝑥 = 𝐴 → 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 = 〈((1st ‘𝐴) ·N (1st ‘𝑦)), ((2nd ‘𝐴) ·N (2nd ‘𝑦))〉) |
6 | fveq2 6896 | . . . 4 ⊢ (𝑦 = 𝐵 → (1st ‘𝑦) = (1st ‘𝐵)) | |
7 | 6 | oveq2d 7435 | . . 3 ⊢ (𝑦 = 𝐵 → ((1st ‘𝐴) ·N (1st ‘𝑦)) = ((1st ‘𝐴) ·N (1st ‘𝐵))) |
8 | fveq2 6896 | . . . 4 ⊢ (𝑦 = 𝐵 → (2nd ‘𝑦) = (2nd ‘𝐵)) | |
9 | 8 | oveq2d 7435 | . . 3 ⊢ (𝑦 = 𝐵 → ((2nd ‘𝐴) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝐵))) |
10 | 7, 9 | opeq12d 4883 | . 2 ⊢ (𝑦 = 𝐵 → 〈((1st ‘𝐴) ·N (1st ‘𝑦)), ((2nd ‘𝐴) ·N (2nd ‘𝑦))〉 = 〈((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))〉) |
11 | df-mpq 10934 | . 2 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | |
12 | opex 5466 | . 2 ⊢ 〈((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))〉 ∈ V | |
13 | 5, 10, 11, 12 | ovmpo 7581 | 1 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = 〈((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 〈cop 4636 × cxp 5676 ‘cfv 6549 (class class class)co 7419 1st c1st 7992 2nd c2nd 7993 Ncnpi 10869 ·N cmi 10871 ·pQ cmpq 10874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-mpq 10934 |
This theorem is referenced by: mulpipq 10965 mulcompq 10977 mulerpqlem 10980 mulassnq 10984 distrnq 10986 ltmnq 10997 |
Copyright terms: Public domain | W3C validator |