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Mirrors > Home > ILE Home > Th. List > mulidnq | GIF version |
Description: Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) |
Ref | Expression |
---|---|
mulidnq | ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nqqs 6446 | . 2 ⊢ Q = ((N × N) / ~Q ) | |
2 | oveq1 5519 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~Q = 𝐴 → ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = (𝐴 ·Q 1Q)) | |
3 | id 19 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~Q = 𝐴 → [〈𝑥, 𝑦〉] ~Q = 𝐴) | |
4 | 2, 3 | eqeq12d 2054 | . 2 ⊢ ([〈𝑥, 𝑦〉] ~Q = 𝐴 → (([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = [〈𝑥, 𝑦〉] ~Q ↔ (𝐴 ·Q 1Q) = 𝐴)) |
5 | df-1nqqs 6449 | . . . . 5 ⊢ 1Q = [〈1𝑜, 1𝑜〉] ~Q | |
6 | 5 | oveq2i 5523 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = ([〈𝑥, 𝑦〉] ~Q ·Q [〈1𝑜, 1𝑜〉] ~Q ) |
7 | 1pi 6413 | . . . . 5 ⊢ 1𝑜 ∈ N | |
8 | mulpipqqs 6471 | . . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (1𝑜 ∈ N ∧ 1𝑜 ∈ N)) → ([〈𝑥, 𝑦〉] ~Q ·Q [〈1𝑜, 1𝑜〉] ~Q ) = [〈(𝑥 ·N 1𝑜), (𝑦 ·N 1𝑜)〉] ~Q ) | |
9 | 7, 7, 8 | mpanr12 415 | . . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([〈𝑥, 𝑦〉] ~Q ·Q [〈1𝑜, 1𝑜〉] ~Q ) = [〈(𝑥 ·N 1𝑜), (𝑦 ·N 1𝑜)〉] ~Q ) |
10 | 6, 9 | syl5eq 2084 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = [〈(𝑥 ·N 1𝑜), (𝑦 ·N 1𝑜)〉] ~Q ) |
11 | mulcompig 6429 | . . . . . . 7 ⊢ ((1𝑜 ∈ N ∧ 𝑥 ∈ N) → (1𝑜 ·N 𝑥) = (𝑥 ·N 1𝑜)) | |
12 | 7, 11 | mpan 400 | . . . . . 6 ⊢ (𝑥 ∈ N → (1𝑜 ·N 𝑥) = (𝑥 ·N 1𝑜)) |
13 | 12 | adantr 261 | . . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (1𝑜 ·N 𝑥) = (𝑥 ·N 1𝑜)) |
14 | mulcompig 6429 | . . . . . . 7 ⊢ ((1𝑜 ∈ N ∧ 𝑦 ∈ N) → (1𝑜 ·N 𝑦) = (𝑦 ·N 1𝑜)) | |
15 | 7, 14 | mpan 400 | . . . . . 6 ⊢ (𝑦 ∈ N → (1𝑜 ·N 𝑦) = (𝑦 ·N 1𝑜)) |
16 | 15 | adantl 262 | . . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (1𝑜 ·N 𝑦) = (𝑦 ·N 1𝑜)) |
17 | 13, 16 | opeq12d 3557 | . . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → 〈(1𝑜 ·N 𝑥), (1𝑜 ·N 𝑦)〉 = 〈(𝑥 ·N 1𝑜), (𝑦 ·N 1𝑜)〉) |
18 | 17 | eceq1d 6142 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [〈(1𝑜 ·N 𝑥), (1𝑜 ·N 𝑦)〉] ~Q = [〈(𝑥 ·N 1𝑜), (𝑦 ·N 1𝑜)〉] ~Q ) |
19 | mulcanenqec 6484 | . . . 4 ⊢ ((1𝑜 ∈ N ∧ 𝑥 ∈ N ∧ 𝑦 ∈ N) → [〈(1𝑜 ·N 𝑥), (1𝑜 ·N 𝑦)〉] ~Q = [〈𝑥, 𝑦〉] ~Q ) | |
20 | 7, 19 | mp3an1 1219 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [〈(1𝑜 ·N 𝑥), (1𝑜 ·N 𝑦)〉] ~Q = [〈𝑥, 𝑦〉] ~Q ) |
21 | 10, 18, 20 | 3eqtr2d 2078 | . 2 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = [〈𝑥, 𝑦〉] ~Q ) |
22 | 1, 4, 21 | ecoptocl 6193 | 1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 〈cop 3378 (class class class)co 5512 1𝑜c1o 5994 [cec 6104 Ncnpi 6370 ·N cmi 6372 ~Q ceq 6377 Qcnq 6378 1Qc1q 6379 ·Q cmq 6381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-mi 6404 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-mqqs 6448 df-1nqqs 6449 |
This theorem is referenced by: recmulnqg 6489 rec1nq 6493 ltaddnq 6505 halfnqq 6508 prarloclemarch 6516 ltrnqg 6518 addnqprllem 6625 addnqprulem 6626 addnqprl 6627 addnqpru 6628 appdivnq 6661 prmuloc2 6665 mulnqprl 6666 mulnqpru 6667 1idprl 6688 1idpru 6689 recexprlem1ssl 6731 recexprlem1ssu 6732 |
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