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Theorem mulidnq 7209
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 7168 . 2 Q = ((N × N) / ~Q )
2 oveq1 5781 . . 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 2154 . 2 ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨𝑥, 𝑦⟩] ~Q ↔ (𝐴 ·Q 1Q) = 𝐴))
5 df-1nqqs 7171 . . . . 5 1Q = [⟨1o, 1o⟩] ~Q
65oveq2i 5785 . . . 4 ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q )
7 1pi 7135 . . . . 5 1oN
8 mulpipqqs 7193 . . . . 5 (((𝑥N𝑦N) ∧ (1oN ∧ 1oN)) → ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q ) = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
97, 7, 8mpanr12 435 . . . 4 ((𝑥N𝑦N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q [⟨1o, 1o⟩] ~Q ) = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
106, 9syl5eq 2184 . . 3 ((𝑥N𝑦N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
11 mulcompig 7151 . . . . . . 7 ((1oN𝑥N) → (1o ·N 𝑥) = (𝑥 ·N 1o))
127, 11mpan 420 . . . . . 6 (𝑥N → (1o ·N 𝑥) = (𝑥 ·N 1o))
1312adantr 274 . . . . 5 ((𝑥N𝑦N) → (1o ·N 𝑥) = (𝑥 ·N 1o))
14 mulcompig 7151 . . . . . . 7 ((1oN𝑦N) → (1o ·N 𝑦) = (𝑦 ·N 1o))
157, 14mpan 420 . . . . . 6 (𝑦N → (1o ·N 𝑦) = (𝑦 ·N 1o))
1615adantl 275 . . . . 5 ((𝑥N𝑦N) → (1o ·N 𝑦) = (𝑦 ·N 1o))
1713, 16opeq12d 3713 . . . 4 ((𝑥N𝑦N) → ⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩ = ⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩)
1817eceq1d 6465 . . 3 ((𝑥N𝑦N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨(𝑥 ·N 1o), (𝑦 ·N 1o)⟩] ~Q )
19 mulcanenqec 7206 . . . 4 ((1oN𝑥N𝑦N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨𝑥, 𝑦⟩] ~Q )
207, 19mp3an1 1302 . . 3 ((𝑥N𝑦N) → [⟨(1o ·N 𝑥), (1o ·N 𝑦)⟩] ~Q = [⟨𝑥, 𝑦⟩] ~Q )
2110, 18, 203eqtr2d 2178 . 2 ((𝑥N𝑦N) → ([⟨𝑥, 𝑦⟩] ~Q ·Q 1Q) = [⟨𝑥, 𝑦⟩] ~Q )
221, 4, 21ecoptocl 6516 1 (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  cop 3530  (class class class)co 5774  1oc1o 6306  [cec 6427  Ncnpi 7092   ·N cmi 7094   ~Q ceq 7099  Qcnq 7100  1Qc1q 7101   ·Q cmq 7103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-mi 7126  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-mqqs 7170  df-1nqqs 7171
This theorem is referenced by:  recmulnqg  7211  rec1nq  7215  ltaddnq  7227  halfnqq  7230  prarloclemarch  7238  ltrnqg  7240  addnqprllem  7347  addnqprulem  7348  addnqprl  7349  addnqpru  7350  appdivnq  7383  prmuloc2  7387  mulnqprl  7388  mulnqpru  7389  1idprl  7410  1idpru  7411  recexprlem1ssl  7453  recexprlem1ssu  7454
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