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Theorem mulidnq 9664
 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 9629 . . 3 1QQ
2 mulpqnq 9642 . . 3 ((𝐴Q ∧ 1QQ) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
31, 2mpan2 703 . 2 (𝐴Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
4 relxp 5150 . . . . . . 7 Rel (N × N)
5 elpqn 9626 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
6 1st2nd 7105 . . . . . . 7 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
74, 5, 6sylancr 694 . . . . . 6 (𝐴Q𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
8 df-1nq 9617 . . . . . . 7 1Q = ⟨1𝑜, 1𝑜
98a1i 11 . . . . . 6 (𝐴Q → 1Q = ⟨1𝑜, 1𝑜⟩)
107, 9oveq12d 6567 . . . . 5 (𝐴Q → (𝐴 ·pQ 1Q) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩))
11 xp1st 7089 . . . . . . 7 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
125, 11syl 17 . . . . . 6 (𝐴Q → (1st𝐴) ∈ N)
13 xp2nd 7090 . . . . . . 7 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
145, 13syl 17 . . . . . 6 (𝐴Q → (2nd𝐴) ∈ N)
15 1pi 9584 . . . . . . 7 1𝑜N
1615a1i 11 . . . . . 6 (𝐴Q → 1𝑜N)
17 mulpipq 9641 . . . . . 6 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ (1𝑜N ∧ 1𝑜N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1st𝐴) ·N 1𝑜), ((2nd𝐴) ·N 1𝑜)⟩)
1812, 14, 16, 16, 17syl22anc 1319 . . . . 5 (𝐴Q → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1st𝐴) ·N 1𝑜), ((2nd𝐴) ·N 1𝑜)⟩)
19 mulidpi 9587 . . . . . . . 8 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
2011, 19syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
21 mulidpi 9587 . . . . . . . 8 ((2nd𝐴) ∈ N → ((2nd𝐴) ·N 1𝑜) = (2nd𝐴))
2213, 21syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((2nd𝐴) ·N 1𝑜) = (2nd𝐴))
2320, 22opeq12d 4348 . . . . . 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 2648 . . . 4 (𝐴Q → (𝐴 ·pQ 1Q) = ⟨(1st𝐴), (2nd𝐴)⟩)
2625, 7eqtr4d 2647 . . 3 (𝐴Q → (𝐴 ·pQ 1Q) = 𝐴)
2726fveq2d 6107 . 2 (𝐴Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴))
28 nqerid 9634 . 2 (𝐴Q → ([Q]‘𝐴) = 𝐴)
293, 27, 283eqtrd 2648 1 (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  1𝑜c1o 7440  Ncnpi 9545   ·N cmi 9547   ·pQ cmpq 9550  Qcnq 9553  1Qc1q 9554  [Q]cerq 9555   ·Q cmq 9557 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ni 9573  df-mi 9575  df-lti 9576  df-mpq 9610  df-enq 9612  df-nq 9613  df-erq 9614  df-mq 9616  df-1nq 9617 This theorem is referenced by:  recmulnq  9665  ltaddnq  9675  halfnq  9677  ltrnq  9680  addclprlem1  9717  addclprlem2  9718  mulclprlem  9720  1idpr  9730  prlem934  9734  prlem936  9748  reclem3pr  9750
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