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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 10958 | . . . . 5 ⊢ ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q)) | |
2 | 1 | fveq1i 6902 | . . . 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 5719 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ (Q × Q)) | |
5 | 4 | fvresd 6921 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉) = (([Q] ∘ ·pQ )‘〈𝐴, 𝐵〉)) |
6 | df-mpq 10952 | . . . . 5 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | |
7 | opex 5470 | . . . . 5 ⊢ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ V | |
8 | 6, 7 | fnmpoi 8084 | . . . 4 ⊢ ·pQ Fn ((N × N) × (N × N)) |
9 | elpqn 10968 | . . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
10 | elpqn 10968 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
11 | opelxpi 5719 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) | |
12 | 9, 10, 11 | syl2an 594 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ ((N × N) × (N × N))) |
13 | fvco2 6999 | . . . 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 585 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (([Q] ∘ ·pQ )‘〈𝐴, 𝐵〉) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉))) |
15 | 3, 5, 14 | 3eqtrd 2770 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( ·Q ‘〈𝐴, 𝐵〉) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉))) |
16 | df-ov 7427 | . 2 ⊢ (𝐴 ·Q 𝐵) = ( ·Q ‘〈𝐴, 𝐵〉) | |
17 | df-ov 7427 | . . 3 ⊢ (𝐴 ·pQ 𝐵) = ( ·pQ ‘〈𝐴, 𝐵〉) | |
18 | 17 | fveq2i 6904 | . 2 ⊢ ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘( ·pQ ‘〈𝐴, 𝐵〉)) |
19 | 15, 16, 18 | 3eqtr4g 2791 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 〈cop 4639 × cxp 5680 ↾ cres 5684 ∘ ccom 5686 Fn wfn 6549 ‘cfv 6554 (class class class)co 7424 1st c1st 8001 2nd c2nd 8002 Ncnpi 10887 ·N cmi 10889 ·pQ cmpq 10892 Qcnq 10895 [Q]cerq 10897 ·Q cmq 10899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-mpq 10952 df-nq 10955 df-mq 10958 |
This theorem is referenced by: mulclnq 10990 mulcomnq 10996 mulerpq 11000 mulassnq 11002 distrnq 11004 mulidnq 11006 ltmnq 11015 |
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