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Theorem mulidnq 11001
Description: Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulidnq (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)

Proof of Theorem mulidnq
StepHypRef Expression
1 1nq 10966 . . 3 1QQ
2 mulpqnq 10979 . . 3 ((𝐴Q ∧ 1QQ) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
31, 2mpan2 691 . 2 (𝐴Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
4 relxp 5707 . . . . . . 7 Rel (N × N)
5 elpqn 10963 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
6 1st2nd 8063 . . . . . . 7 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
74, 5, 6sylancr 587 . . . . . 6 (𝐴Q𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
8 df-1nq 10954 . . . . . . 7 1Q = ⟨1o, 1o
98a1i 11 . . . . . 6 (𝐴Q → 1Q = ⟨1o, 1o⟩)
107, 9oveq12d 7449 . . . . 5 (𝐴Q → (𝐴 ·pQ 1Q) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1o, 1o⟩))
11 xp1st 8045 . . . . . . 7 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
125, 11syl 17 . . . . . 6 (𝐴Q → (1st𝐴) ∈ N)
13 xp2nd 8046 . . . . . . 7 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
145, 13syl 17 . . . . . 6 (𝐴Q → (2nd𝐴) ∈ N)
15 1pi 10921 . . . . . . 7 1oN
1615a1i 11 . . . . . 6 (𝐴Q → 1oN)
17 mulpipq 10978 . . . . . 6 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ (1oN ∧ 1oN)) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1o, 1o⟩) = ⟨((1st𝐴) ·N 1o), ((2nd𝐴) ·N 1o)⟩)
1812, 14, 16, 16, 17syl22anc 839 . . . . 5 (𝐴Q → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1o, 1o⟩) = ⟨((1st𝐴) ·N 1o), ((2nd𝐴) ·N 1o)⟩)
19 mulidpi 10924 . . . . . . . 8 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1o) = (1st𝐴))
2011, 19syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((1st𝐴) ·N 1o) = (1st𝐴))
21 mulidpi 10924 . . . . . . . 8 ((2nd𝐴) ∈ N → ((2nd𝐴) ·N 1o) = (2nd𝐴))
2213, 21syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((2nd𝐴) ·N 1o) = (2nd𝐴))
2320, 22opeq12d 4886 . . . . . 6 (𝐴 ∈ (N × N) → ⟨((1st𝐴) ·N 1o), ((2nd𝐴) ·N 1o)⟩ = ⟨(1st𝐴), (2nd𝐴)⟩)
245, 23syl 17 . . . . 5 (𝐴Q → ⟨((1st𝐴) ·N 1o), ((2nd𝐴) ·N 1o)⟩ = ⟨(1st𝐴), (2nd𝐴)⟩)
2510, 18, 243eqtrd 2779 . . . 4 (𝐴Q → (𝐴 ·pQ 1Q) = ⟨(1st𝐴), (2nd𝐴)⟩)
2625, 7eqtr4d 2778 . . 3 (𝐴Q → (𝐴 ·pQ 1Q) = 𝐴)
2726fveq2d 6911 . 2 (𝐴Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴))
28 nqerid 10971 . 2 (𝐴Q → ([Q]‘𝐴) = 𝐴)
293, 27, 283eqtrd 2779 1 (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cop 4637   × cxp 5687  Rel wrel 5694  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  1oc1o 8498  Ncnpi 10882   ·N cmi 10884   ·pQ cmpq 10887  Qcnq 10890  1Qc1q 10891  [Q]cerq 10892   ·Q cmq 10894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-ni 10910  df-mi 10912  df-lti 10913  df-mpq 10947  df-enq 10949  df-nq 10950  df-erq 10951  df-mq 10953  df-1nq 10954
This theorem is referenced by:  recmulnq  11002  ltaddnq  11012  halfnq  11014  ltrnq  11017  addclprlem1  11054  addclprlem2  11055  mulclprlem  11057  1idpr  11067  prlem934  11071  prlem936  11085  reclem3pr  11087
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