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Theorem mulidnq 10181
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 10146 . . 3 1QQ
2 mulpqnq 10159 . . 3 ((𝐴Q ∧ 1QQ) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
31, 2mpan2 679 . 2 (𝐴Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q)))
4 relxp 5421 . . . . . . 7 Rel (N × N)
5 elpqn 10143 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
6 1st2nd 7548 . . . . . . 7 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
74, 5, 6sylancr 579 . . . . . 6 (𝐴Q𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
8 df-1nq 10134 . . . . . . 7 1Q = ⟨1o, 1o
98a1i 11 . . . . . 6 (𝐴Q → 1Q = ⟨1o, 1o⟩)
107, 9oveq12d 6992 . . . . 5 (𝐴Q → (𝐴 ·pQ 1Q) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1o, 1o⟩))
11 xp1st 7531 . . . . . . 7 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
125, 11syl 17 . . . . . 6 (𝐴Q → (1st𝐴) ∈ N)
13 xp2nd 7532 . . . . . . 7 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
145, 13syl 17 . . . . . 6 (𝐴Q → (2nd𝐴) ∈ N)
15 1pi 10101 . . . . . . 7 1oN
1615a1i 11 . . . . . 6 (𝐴Q → 1oN)
17 mulpipq 10158 . . . . . 6 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ (1oN ∧ 1oN)) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1o, 1o⟩) = ⟨((1st𝐴) ·N 1o), ((2nd𝐴) ·N 1o)⟩)
1812, 14, 16, 16, 17syl22anc 827 . . . . 5 (𝐴Q → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨1o, 1o⟩) = ⟨((1st𝐴) ·N 1o), ((2nd𝐴) ·N 1o)⟩)
19 mulidpi 10104 . . . . . . . 8 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1o) = (1st𝐴))
2011, 19syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((1st𝐴) ·N 1o) = (1st𝐴))
21 mulidpi 10104 . . . . . . . 8 ((2nd𝐴) ∈ N → ((2nd𝐴) ·N 1o) = (2nd𝐴))
2213, 21syl 17 . . . . . . 7 (𝐴 ∈ (N × N) → ((2nd𝐴) ·N 1o) = (2nd𝐴))
2320, 22opeq12d 4681 . . . . . 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 2811 . . . 4 (𝐴Q → (𝐴 ·pQ 1Q) = ⟨(1st𝐴), (2nd𝐴)⟩)
2625, 7eqtr4d 2810 . . 3 (𝐴Q → (𝐴 ·pQ 1Q) = 𝐴)
2726fveq2d 6500 . 2 (𝐴Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴))
28 nqerid 10151 . 2 (𝐴Q → ([Q]‘𝐴) = 𝐴)
293, 27, 283eqtrd 2811 1 (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1508  wcel 2051  cop 4441   × cxp 5401  Rel wrel 5408  cfv 6185  (class class class)co 6974  1st c1st 7497  2nd c2nd 7498  1oc1o 7896  Ncnpi 10062   ·N cmi 10064   ·pQ cmpq 10067  Qcnq 10070  1Qc1q 10071  [Q]cerq 10072   ·Q cmq 10074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-oadd 7907  df-omul 7908  df-er 8087  df-ni 10090  df-mi 10092  df-lti 10093  df-mpq 10127  df-enq 10129  df-nq 10130  df-erq 10131  df-mq 10133  df-1nq 10134
This theorem is referenced by:  recmulnq  10182  ltaddnq  10192  halfnq  10194  ltrnq  10197  addclprlem1  10234  addclprlem2  10235  mulclprlem  10237  1idpr  10247  prlem934  10251  prlem936  10265  reclem3pr  10267
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