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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 5585 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | opelxpi 5585 | . . 3 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → 〈𝐶, 𝐷〉 ∈ (N × N)) | |
3 | mulpipq2 10349 | . . 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 595 | . 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 7690 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) | |
6 | op1stg 7690 | . . . 4 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (1st ‘〈𝐶, 𝐷〉) = 𝐶) | |
7 | 5, 6 | oveqan12d 7164 | . . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘〈𝐴, 𝐵〉) ·N (1st ‘〈𝐶, 𝐷〉)) = (𝐴 ·N 𝐶)) |
8 | op2ndg 7691 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
9 | op2ndg 7691 | . . . 4 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (2nd ‘〈𝐶, 𝐷〉) = 𝐷) | |
10 | 8, 9 | oveqan12d 7164 | . . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((2nd ‘〈𝐴, 𝐵〉) ·N (2nd ‘〈𝐶, 𝐷〉)) = (𝐵 ·N 𝐷)) |
11 | 7, 10 | opeq12d 4803 | . 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 2853 | 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 396 = wceq 1528 ∈ wcel 2105 〈cop 4563 × cxp 5546 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 Ncnpi 10254 ·N cmi 10256 ·pQ cmpq 10259 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-mpq 10319 |
This theorem is referenced by: mulassnq 10369 distrnq 10371 mulidnq 10373 recmulnq 10374 |
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