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Theorem mulidnq 10374
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 10339 . . 3 1QQ
2 mulpqnq 10352 . . 3 ((𝐴Q ∧ 1QQ) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
31, 2mpan2 690 . 2 (𝐴Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
4 relxp 5537 . . . . . . 7 Rel (N × N)
5 elpqn 10336 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
6 1st2nd 7720 . . . . . . 7 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
74, 5, 6sylancr 590 . . . . . 6 (𝐴Q𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
8 df-1nq 10327 . . . . . . 7 1Q = ⟨1o, 1o
98a1i 11 . . . . . 6 (𝐴Q → 1Q = ⟨1o, 1o⟩)
107, 9oveq12d 7153 . . . . 5 (𝐴Q → (𝐴 ·pQ 1Q) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1o, 1o⟩))
11 xp1st 7703 . . . . . . 7 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
125, 11syl 17 . . . . . 6 (𝐴Q → (1st𝐴) ∈ N)
13 xp2nd 7704 . . . . . . 7 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
145, 13syl 17 . . . . . 6 (𝐴Q → (2nd𝐴) ∈ N)
15 1pi 10294 . . . . . . 7 1oN
1615a1i 11 . . . . . 6 (𝐴Q → 1oN)
17 mulpipq 10351 . . . . . 6 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ (1oN ∧ 1oN)) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1o, 1o⟩) = ⟨((1st𝐴) ·N 1o), ((2nd𝐴) ·N 1o)⟩)
1812, 14, 16, 16, 17syl22anc 837 . . . . 5 (𝐴Q → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1o, 1o⟩) = ⟨((1st𝐴) ·N 1o), ((2nd𝐴) ·N 1o)⟩)
19 mulidpi 10297 . . . . . . . 8 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1o) = (1st𝐴))
2011, 19syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((1st𝐴) ·N 1o) = (1st𝐴))
21 mulidpi 10297 . . . . . . . 8 ((2nd𝐴) ∈ N → ((2nd𝐴) ·N 1o) = (2nd𝐴))
2213, 21syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((2nd𝐴) ·N 1o) = (2nd𝐴))
2320, 22opeq12d 4773 . . . . . 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 2837 . . . 4 (𝐴Q → (𝐴 ·pQ 1Q) = ⟨(1st𝐴), (2nd𝐴)⟩)
2625, 7eqtr4d 2836 . . 3 (𝐴Q → (𝐴 ·pQ 1Q) = 𝐴)
2726fveq2d 6649 . 2 (𝐴Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴))
28 nqerid 10344 . 2 (𝐴Q → ([Q]‘𝐴) = 𝐴)
293, 27, 283eqtrd 2837 1 (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  cop 4531   × cxp 5517  Rel wrel 5524  cfv 6324  (class class class)co 7135  1st c1st 7669  2nd c2nd 7670  1oc1o 8078  Ncnpi 10255   ·N cmi 10257   ·pQ cmpq 10260  Qcnq 10263  1Qc1q 10264  [Q]cerq 10265   ·Q cmq 10267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-omul 8090  df-er 8272  df-ni 10283  df-mi 10285  df-lti 10286  df-mpq 10320  df-enq 10322  df-nq 10323  df-erq 10324  df-mq 10326  df-1nq 10327
This theorem is referenced by:  recmulnq  10375  ltaddnq  10385  halfnq  10387  ltrnq  10390  addclprlem1  10427  addclprlem2  10428  mulclprlem  10430  1idpr  10440  prlem934  10444  prlem936  10458  reclem3pr  10460
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