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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 10339 | . . . . 5 ⊢ ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q)) | |
2 | 1 | fveq1i 6673 | . . . 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 5594 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ (Q × Q)) | |
5 | 4 | fvresd 6692 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉) = (([Q] ∘ ·pQ )‘〈𝐴, 𝐵〉)) |
6 | df-mpq 10333 | . . . . 5 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | |
7 | opex 5358 | . . . . 5 ⊢ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ V | |
8 | 6, 7 | fnmpoi 7770 | . . . 4 ⊢ ·pQ Fn ((N × N) × (N × N)) |
9 | elpqn 10349 | . . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
10 | elpqn 10349 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
11 | opelxpi 5594 | . . . . 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 6760 | . . . 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 2862 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( ·Q ‘〈𝐴, 𝐵〉) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉))) |
16 | df-ov 7161 | . 2 ⊢ (𝐴 ·Q 𝐵) = ( ·Q ‘〈𝐴, 𝐵〉) | |
17 | df-ov 7161 | . . 3 ⊢ (𝐴 ·pQ 𝐵) = ( ·pQ ‘〈𝐴, 𝐵〉) | |
18 | 17 | fveq2i 6675 | . 2 ⊢ ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉)) |
19 | 15, 16, 18 | 3eqtr4g 2883 | 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 4575 × cxp 5555 ↾ cres 5559 ∘ ccom 5561 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 Ncnpi 10268 ·N cmi 10270 ·pQ cmpq 10273 Qcnq 10276 [Q]cerq 10278 ·Q cmq 10280 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-mpq 10333 df-nq 10336 df-mq 10339 |
This theorem is referenced by: mulclnq 10371 mulcomnq 10377 mulerpq 10381 mulassnq 10383 distrnq 10385 mulidnq 10387 ltmnq 10396 |
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