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Mirrors > Home > MPE Home > Th. List > mulpipq | Structured version Visualization version GIF version |
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulpipq | ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 ·pQ 〈𝐶, 𝐷〉) = 〈(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5556 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | opelxpi 5556 | . . 3 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → 〈𝐶, 𝐷〉 ∈ (N × N)) | |
3 | mulpipq2 10350 | . . 3 ⊢ ((〈𝐴, 𝐵〉 ∈ (N × N) ∧ 〈𝐶, 𝐷〉 ∈ (N × N)) → (〈𝐴, 𝐵〉 ·pQ 〈𝐶, 𝐷〉) = 〈((1st ‘〈𝐴, 𝐵〉) ·N (1st ‘〈𝐶, 𝐷〉)), ((2nd ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉))〉) | |
4 | 1, 2, 3 | syl2an 598 | . 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 ·pQ 〈𝐶, 𝐷〉) = 〈((1st ‘〈𝐴, 𝐵〉) ·N (1st ‘〈𝐶, 𝐷〉)), ((2nd ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉))〉) |
5 | op1stg 7683 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) | |
6 | op1stg 7683 | . . . 4 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (1st ‘〈𝐶, 𝐷〉) = 𝐶) | |
7 | 5, 6 | oveqan12d 7154 | . . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘〈𝐴, 𝐵〉) ·N (1st ‘〈𝐶, 𝐷〉)) = (𝐴 ·N 𝐶)) |
8 | op2ndg 7684 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
9 | op2ndg 7684 | . . . 4 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (2nd ‘〈𝐶, 𝐷〉) = 𝐷) | |
10 | 8, 9 | oveqan12d 7154 | . . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((2nd ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) = (𝐵 ·N 𝐷)) |
11 | 7, 10 | opeq12d 4773 | . 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → 〈((1st ‘〈𝐴, 𝐵〉) ·N (1st ‘〈𝐶, 𝐷〉)), ((2nd ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉))〉 = 〈(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)〉) |
12 | 4, 11 | eqtrd 2833 | 1 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 ·pQ 〈𝐶, 𝐷〉) = 〈(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 〈cop 4531 × cxp 5517 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 Ncnpi 10255 ·N cmi 10257 ·pQ cmpq 10260 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-mpq 10320 |
This theorem is referenced by: mulassnq 10370 distrnq 10372 mulidnq 10374 recmulnq 10375 |
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