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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 10494 | . . . . 5 ⊢ ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q)) | |
2 | 1 | fveq1i 6696 | . . . 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 5573 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ (Q × Q)) | |
5 | 4 | fvresd 6715 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉) = (([Q] ∘ ·pQ )‘〈𝐴, 𝐵〉)) |
6 | df-mpq 10488 | . . . . 5 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | |
7 | opex 5333 | . . . . 5 ⊢ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ V | |
8 | 6, 7 | fnmpoi 7818 | . . . 4 ⊢ ·pQ Fn ((N × N) × (N × N)) |
9 | elpqn 10504 | . . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
10 | elpqn 10504 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
11 | opelxpi 5573 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) | |
12 | 9, 10, 11 | syl2an 599 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) |
13 | fvco2 6786 | . . . 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 590 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (([Q] ∘ ·pQ )‘〈𝐴, 𝐵〉) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉))) |
15 | 3, 5, 14 | 3eqtrd 2775 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( ·Q ‘〈𝐴, 𝐵〉) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉))) |
16 | df-ov 7194 | . 2 ⊢ (𝐴 ·Q 𝐵) = ( ·Q ‘〈𝐴, 𝐵〉) | |
17 | df-ov 7194 | . . 3 ⊢ (𝐴 ·pQ 𝐵) = ( ·pQ ‘〈𝐴, 𝐵〉) | |
18 | 17 | fveq2i 6698 | . 2 ⊢ ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉)) |
19 | 15, 16, 18 | 3eqtr4g 2796 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 〈cop 4533 × cxp 5534 ↾ cres 5538 ∘ ccom 5540 Fn wfn 6353 ‘cfv 6358 (class class class)co 7191 1st c1st 7737 2nd c2nd 7738 Ncnpi 10423 ·N cmi 10425 ·pQ cmpq 10428 Qcnq 10431 [Q]cerq 10433 ·Q cmq 10435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-mpq 10488 df-nq 10491 df-mq 10494 |
This theorem is referenced by: mulclnq 10526 mulcomnq 10532 mulerpq 10536 mulassnq 10538 distrnq 10540 mulidnq 10542 ltmnq 10551 |
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