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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 10953 | . . . . 5 ⊢ ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q)) | |
2 | 1 | fveq1i 6908 | . . . 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 5726 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ (Q × Q)) | |
5 | 4 | fvresd 6927 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉) = (([Q] ∘ ·pQ )‘〈𝐴, 𝐵〉)) |
6 | df-mpq 10947 | . . . . 5 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | |
7 | opex 5475 | . . . . 5 ⊢ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ V | |
8 | 6, 7 | fnmpoi 8094 | . . . 4 ⊢ ·pQ Fn ((N × N) × (N × N)) |
9 | elpqn 10963 | . . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
10 | elpqn 10963 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
11 | opelxpi 5726 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) | |
12 | 9, 10, 11 | syl2an 596 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) |
13 | fvco2 7006 | . . . 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 587 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (([Q] ∘ ·pQ )‘〈𝐴, 𝐵〉) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉))) |
15 | 3, 5, 14 | 3eqtrd 2779 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( ·Q ‘〈𝐴, 𝐵〉) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉))) |
16 | df-ov 7434 | . 2 ⊢ (𝐴 ·Q 𝐵) = ( ·Q ‘〈𝐴, 𝐵〉) | |
17 | df-ov 7434 | . . 3 ⊢ (𝐴 ·pQ 𝐵) = ( ·pQ ‘〈𝐴, 𝐵〉) | |
18 | 17 | fveq2i 6910 | . 2 ⊢ ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉)) |
19 | 15, 16, 18 | 3eqtr4g 2800 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 〈cop 4637 × cxp 5687 ↾ cres 5691 ∘ ccom 5693 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 Ncnpi 10882 ·N cmi 10884 ·pQ cmpq 10887 Qcnq 10890 [Q]cerq 10892 ·Q cmq 10894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-mpq 10947 df-nq 10950 df-mq 10953 |
This theorem is referenced by: mulclnq 10985 mulcomnq 10991 mulerpq 10995 mulassnq 10997 distrnq 10999 mulidnq 11001 ltmnq 11010 |
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