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Theorem mulidnq 7418
Description: Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
Assertion
Ref Expression
mulidnq (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)

Proof of Theorem mulidnq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef 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)
42, 3eqeq12d 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
65oveq2i 5907 . . . 4 ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q )
7 1pi 7344 . . . . 5 1oN
8 mulpipqqs 7402 . . . . 5 (((𝑥N𝑦N) ∧ (1oN ∧ 1oN)) → ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q ) = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
97, 7, 8mpanr12 439 . . . 4 ((𝑥N𝑦N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q ) = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
106, 9eqtrid 2234 . . 3 ((𝑥N𝑦N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
11 mulcompig 7360 . . . . . . 7 ((1oN𝑥N) → (1o ·N 𝑥) = (𝑥 ·N 1o))
127, 11mpan 424 . . . . . 6 (𝑥N → (1o ·N 𝑥) = (𝑥 ·N 1o))
1312adantr 276 . . . . 5 ((𝑥N𝑦N) → (1o ·N 𝑥) = (𝑥 ·N 1o))
14 mulcompig 7360 . . . . . . 7 ((1oN𝑦N) → (1o ·N 𝑦) = (𝑦 ·N 1o))
157, 14mpan 424 . . . . . 6 (𝑦N → (1o ·N 𝑦) = (𝑦 ·N 1o))
1615adantl 277 . . . . 5 ((𝑥N𝑦N) → (1o ·N 𝑦) = (𝑦 ·N 1o))
1713, 16opeq12d 3801 . . . 4 ((𝑥N𝑦N) → ⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩ = ⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩)
1817eceq1d 6595 . . 3 ((𝑥N𝑦N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
19 mulcanenqec 7415 . . . 4 ((1oN𝑥N𝑦N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨𝑥, 𝑦⟩] ~Q )
207, 19mp3an1 1335 . . 3 ((𝑥N𝑦N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨𝑥, 𝑦⟩] ~Q )
2110, 18, 203eqtr2d 2228 . 2 ((𝑥N𝑦N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨𝑥, 𝑦⟩] ~Q )
221, 4, 21ecoptocl 6648 1 (𝐴Q → (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
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