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Theorem mulidnq 9737
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 9702 . . 3 1QQ
2 mulpqnq 9715 . . 3 ((𝐴Q ∧ 1QQ) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
31, 2mpan2 706 . 2 (𝐴Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
4 relxp 5193 . . . . . . 7 Rel (N × N)
5 elpqn 9699 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
6 1st2nd 7166 . . . . . . 7 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
74, 5, 6sylancr 694 . . . . . 6 (𝐴Q𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
8 df-1nq 9690 . . . . . . 7 1Q = ⟨1𝑜, 1𝑜
98a1i 11 . . . . . 6 (𝐴Q → 1Q = ⟨1𝑜, 1𝑜⟩)
107, 9oveq12d 6628 . . . . 5 (𝐴Q → (𝐴 ·pQ 1Q) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩))
11 xp1st 7150 . . . . . . 7 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
125, 11syl 17 . . . . . 6 (𝐴Q → (1st𝐴) ∈ N)
13 xp2nd 7151 . . . . . . 7 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
145, 13syl 17 . . . . . 6 (𝐴Q → (2nd𝐴) ∈ N)
15 1pi 9657 . . . . . . 7 1𝑜N
1615a1i 11 . . . . . 6 (𝐴Q → 1𝑜N)
17 mulpipq 9714 . . . . . 6 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ (1𝑜N ∧ 1𝑜N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1st𝐴) ·N 1𝑜), ((2nd𝐴) ·N 1𝑜)⟩)
1812, 14, 16, 16, 17syl22anc 1324 . . . . 5 (𝐴Q → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1st𝐴) ·N 1𝑜), ((2nd𝐴) ·N 1𝑜)⟩)
19 mulidpi 9660 . . . . . . . 8 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
2011, 19syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
21 mulidpi 9660 . . . . . . . 8 ((2nd𝐴) ∈ N → ((2nd𝐴) ·N 1𝑜) = (2nd𝐴))
2213, 21syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((2nd𝐴) ·N 1𝑜) = (2nd𝐴))
2320, 22opeq12d 4383 . . . . . 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 2659 . . . 4 (𝐴Q → (𝐴 ·pQ 1Q) = ⟨(1st𝐴), (2nd𝐴)⟩)
2625, 7eqtr4d 2658 . . 3 (𝐴Q → (𝐴 ·pQ 1Q) = 𝐴)
2726fveq2d 6157 . 2 (𝐴Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴))
28 nqerid 9707 . 2 (𝐴Q → ([Q]‘𝐴) = 𝐴)
293, 27, 283eqtrd 2659 1 (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cop 4159   × cxp 5077  Rel wrel 5084  cfv 5852  (class class class)co 6610  1st c1st 7118  2nd c2nd 7119  1𝑜c1o 7505  Ncnpi 9618   ·N cmi 9620   ·pQ cmpq 9623  Qcnq 9626  1Qc1q 9627  [Q]cerq 9628   ·Q cmq 9630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-omul 7517  df-er 7694  df-ni 9646  df-mi 9648  df-lti 9649  df-mpq 9683  df-enq 9685  df-nq 9686  df-erq 9687  df-mq 9689  df-1nq 9690
This theorem is referenced by:  recmulnq  9738  ltaddnq  9748  halfnq  9750  ltrnq  9753  addclprlem1  9790  addclprlem2  9791  mulclprlem  9793  1idpr  9803  prlem934  9807  prlem936  9821  reclem3pr  9823
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