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Theorem mulidnq 9986
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 9951 . . 3 1QQ
2 mulpqnq 9964 . . 3 ((𝐴Q ∧ 1QQ) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
31, 2mpan2 663 . 2 (𝐴Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
4 relxp 5266 . . . . . . 7 Rel (N × N)
5 elpqn 9948 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
6 1st2nd 7362 . . . . . . 7 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
74, 5, 6sylancr 567 . . . . . 6 (𝐴Q𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
8 df-1nq 9939 . . . . . . 7 1Q = ⟨1𝑜, 1𝑜
98a1i 11 . . . . . 6 (𝐴Q → 1Q = ⟨1𝑜, 1𝑜⟩)
107, 9oveq12d 6810 . . . . 5 (𝐴Q → (𝐴 ·pQ 1Q) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩))
11 xp1st 7346 . . . . . . 7 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
125, 11syl 17 . . . . . 6 (𝐴Q → (1st𝐴) ∈ N)
13 xp2nd 7347 . . . . . . 7 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
145, 13syl 17 . . . . . 6 (𝐴Q → (2nd𝐴) ∈ N)
15 1pi 9906 . . . . . . 7 1𝑜N
1615a1i 11 . . . . . 6 (𝐴Q → 1𝑜N)
17 mulpipq 9963 . . . . . 6 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ (1𝑜N ∧ 1𝑜N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1st𝐴) ·N 1𝑜), ((2nd𝐴) ·N 1𝑜)⟩)
1812, 14, 16, 16, 17syl22anc 1476 . . . . 5 (𝐴Q → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1st𝐴) ·N 1𝑜), ((2nd𝐴) ·N 1𝑜)⟩)
19 mulidpi 9909 . . . . . . . 8 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
2011, 19syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
21 mulidpi 9909 . . . . . . . 8 ((2nd𝐴) ∈ N → ((2nd𝐴) ·N 1𝑜) = (2nd𝐴))
2213, 21syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((2nd𝐴) ·N 1𝑜) = (2nd𝐴))
2320, 22opeq12d 4545 . . . . . 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 2808 . . . 4 (𝐴Q → (𝐴 ·pQ 1Q) = ⟨(1st𝐴), (2nd𝐴)⟩)
2625, 7eqtr4d 2807 . . 3 (𝐴Q → (𝐴 ·pQ 1Q) = 𝐴)
2726fveq2d 6336 . 2 (𝐴Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴))
28 nqerid 9956 . 2 (𝐴Q → ([Q]‘𝐴) = 𝐴)
293, 27, 283eqtrd 2808 1 (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  cop 4320   × cxp 5247  Rel wrel 5254  cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  1𝑜c1o 7705  Ncnpi 9867   ·N cmi 9869   ·pQ cmpq 9872  Qcnq 9875  1Qc1q 9876  [Q]cerq 9877   ·Q cmq 9879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-omul 7717  df-er 7895  df-ni 9895  df-mi 9897  df-lti 9898  df-mpq 9932  df-enq 9934  df-nq 9935  df-erq 9936  df-mq 9938  df-1nq 9939
This theorem is referenced by:  recmulnq  9987  ltaddnq  9997  halfnq  9999  ltrnq  10002  addclprlem1  10039  addclprlem2  10040  mulclprlem  10042  1idpr  10052  prlem934  10056  prlem936  10070  reclem3pr  10072
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