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Mirrors > Home > MPE Home > Th. List > mulpqnq | 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, 26-Dec-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulpqnq | ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mq 10337 | . . . . 5 ⊢ ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q)) | |
2 | 1 | fveq1i 6671 | . . . 4 ⊢ ( ·Q ‘〈𝐴, 𝐵〉) = ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉) |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( ·Q ‘〈𝐴, 𝐵〉) = ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉)) |
4 | opelxpi 5592 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ (Q × Q)) | |
5 | 4 | fvresd 6690 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉) = (([Q] ∘ ·pQ )‘〈𝐴, 𝐵〉)) |
6 | df-mpq 10331 | . . . . 5 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | |
7 | opex 5356 | . . . . 5 ⊢ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ V | |
8 | 6, 7 | fnmpoi 7768 | . . . 4 ⊢ ·pQ Fn ((N × N) × (N × N)) |
9 | elpqn 10347 | . . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
10 | elpqn 10347 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
11 | opelxpi 5592 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) | |
12 | 9, 10, 11 | syl2an 597 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) |
13 | fvco2 6758 | . . . 4 ⊢ (( ·pQ Fn ((N × N) × (N × N)) ∧ 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) → (([Q] ∘ ·pQ )‘〈𝐴, 𝐵〉) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉))) | |
14 | 8, 12, 13 | sylancr 589 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (([Q] ∘ ·pQ )‘〈𝐴, 𝐵〉) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉))) |
15 | 3, 5, 14 | 3eqtrd 2860 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( ·Q ‘〈𝐴, 𝐵〉) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉))) |
16 | df-ov 7159 | . 2 ⊢ (𝐴 ·Q 𝐵) = ( ·Q ‘〈𝐴, 𝐵〉) | |
17 | df-ov 7159 | . . 3 ⊢ (𝐴 ·pQ 𝐵) = ( ·pQ ‘〈𝐴, 𝐵〉) | |
18 | 17 | fveq2i 6673 | . 2 ⊢ ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉)) |
19 | 15, 16, 18 | 3eqtr4g 2881 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 × cxp 5553 ↾ cres 5557 ∘ ccom 5559 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 2nd c2nd 7688 Ncnpi 10266 ·N cmi 10268 ·pQ cmpq 10271 Qcnq 10274 [Q]cerq 10276 ·Q cmq 10278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-mpq 10331 df-nq 10334 df-mq 10337 |
This theorem is referenced by: mulclnq 10369 mulcomnq 10375 mulerpq 10379 mulassnq 10381 distrnq 10383 mulidnq 10385 ltmnq 10394 |
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