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| Mirrors > Home > MPE Home > Th. List > addpqnq | Structured version Visualization version GIF version | ||
| Description: Addition 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 |
|---|---|
| addpqnq | ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-plq 10954 | . . . . 5 ⊢ +Q = (([Q] ∘ +pQ ) ↾ (Q × Q)) | |
| 2 | 1 | fveq1i 6907 | . . . 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 5722 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 〈𝐴, 𝐵〉 ∈ (Q × Q)) | |
| 5 | 4 | fvresd 6926 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ +pQ ) ↾ (Q × Q))‘〈𝐴, 𝐵〉) = (([Q] ∘ +pQ )‘〈𝐴, 𝐵〉)) |
| 6 | df-plpq 10948 | . . . . 5 ⊢ +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | |
| 7 | opex 5469 | . . . . 5 ⊢ 〈(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ V | |
| 8 | 6, 7 | fnmpoi 8095 | . . . 4 ⊢ +pQ Fn ((N × N) × (N × N)) |
| 9 | elpqn 10965 | . . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
| 10 | elpqn 10965 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
| 11 | opelxpi 5722 | . . . . 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 2781 | . 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 6909 | . 2 ⊢ ([Q]‘(𝐴 +pQ 𝐵)) = ([Q]‘( +pQ ‘〈𝐴, 𝐵〉)) |
| 19 | 15, 16, 18 | 3eqtr4g 2802 | 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 1540 ∈ wcel 2108 〈cop 4632 × cxp 5683 ↾ cres 5687 ∘ ccom 5689 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 Ncnpi 10884 +N cpli 10885 ·N cmi 10886 +pQ cplpq 10888 Qcnq 10892 [Q]cerq 10894 +Q cplq 10895 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-plpq 10948 df-nq 10952 df-plq 10954 |
| This theorem is referenced by: addclnq 10985 addcomnq 10991 adderpq 10996 addassnq 10998 distrnq 11001 ltanq 11011 1lt2nq 11013 prlem934 11073 |
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