![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 7377 | . 2 ⊢ Q = ((N × N) / ~Q ) | |
2 | oveq1 5903 | . . 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 2204 | . 2 ⊢ ([〈𝑥, 𝑦〉] ~Q = 𝐴 → (([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = [〈𝑥, 𝑦〉] ~Q ↔ (𝐴 ·Q 1Q) = 𝐴)) |
5 | df-1nqqs 7380 | . . . . 5 ⊢ 1Q = [〈1o, 1o〉] ~Q | |
6 | 5 | oveq2i 5907 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = ([〈𝑥, 𝑦〉] ~Q ·Q [〈1o, 1o〉] ~Q ) |
7 | 1pi 7344 | . . . . 5 ⊢ 1o ∈ N | |
8 | mulpipqqs 7402 | . . . . 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 439 | . . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([〈𝑥, 𝑦〉] ~Q ·Q [〈1o, 1o〉] ~Q ) = [〈(𝑥 ·N 1o), (𝑦 ·N 1o)〉] ~Q ) |
10 | 6, 9 | eqtrid 2234 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = [〈(𝑥 ·N 1o), (𝑦 ·N 1o)〉] ~Q ) |
11 | mulcompig 7360 | . . . . . . 7 ⊢ ((1o ∈ N ∧ 𝑥 ∈ N) → (1o ·N 𝑥) = (𝑥 ·N 1o)) | |
12 | 7, 11 | mpan 424 | . . . . . 6 ⊢ (𝑥 ∈ N → (1o ·N 𝑥) = (𝑥 ·N 1o)) |
13 | 12 | adantr 276 | . . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑥) = (𝑥 ·N 1o)) |
14 | mulcompig 7360 | . . . . . . 7 ⊢ ((1o ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑦) = (𝑦 ·N 1o)) | |
15 | 7, 14 | mpan 424 | . . . . . 6 ⊢ (𝑦 ∈ N → (1o ·N 𝑦) = (𝑦 ·N 1o)) |
16 | 15 | adantl 277 | . . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑦) = (𝑦 ·N 1o)) |
17 | 13, 16 | opeq12d 3801 | . . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → 〈(1o ·N 𝑥), (1o ·N 𝑦)〉 = 〈(𝑥 ·N 1o), (𝑦 ·N 1o)〉) |
18 | 17 | eceq1d 6595 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [〈(1o ·N 𝑥), (1o ·N 𝑦)〉] ~Q = [〈(𝑥 ·N 1o), (𝑦 ·N 1o)〉] ~Q ) |
19 | mulcanenqec 7415 | . . . 4 ⊢ ((1o ∈ N ∧ 𝑥 ∈ N ∧ 𝑦 ∈ N) → [〈(1o ·N 𝑥), (1o ·N 𝑦)〉] ~Q = [〈𝑥, 𝑦〉] ~Q ) | |
20 | 7, 19 | mp3an1 1335 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [〈(1o ·N 𝑥), (1o ·N 𝑦)〉] ~Q = [〈𝑥, 𝑦〉] ~Q ) |
21 | 10, 18, 20 | 3eqtr2d 2228 | . 2 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = [〈𝑥, 𝑦〉] ~Q ) |
22 | 1, 4, 21 | ecoptocl 6648 | 1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 〈cop 3610 (class class class)co 5896 1oc1o 6434 [cec 6557 Ncnpi 7301 ·N cmi 7303 ~Q ceq 7308 Qcnq 7309 1Qc1q 7310 ·Q cmq 7312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-irdg 6395 df-1o 6441 df-oadd 6445 df-omul 6446 df-er 6559 df-ec 6561 df-qs 6565 df-ni 7333 df-mi 7335 df-mpq 7374 df-enq 7376 df-nqqs 7377 df-mqqs 7379 df-1nqqs 7380 |
This theorem is referenced by: recmulnqg 7420 rec1nq 7424 ltaddnq 7436 halfnqq 7439 prarloclemarch 7447 ltrnqg 7449 addnqprllem 7556 addnqprulem 7557 addnqprl 7558 addnqpru 7559 appdivnq 7592 prmuloc2 7596 mulnqprl 7597 mulnqpru 7598 1idprl 7619 1idpru 7620 recexprlem1ssl 7662 recexprlem1ssu 7663 |
Copyright terms: Public domain | W3C validator |