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Theorem mulerpq 10711
Description: Multiplication is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulerpq (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵))

Proof of Theorem mulerpq
StepHypRef Expression
1 nqercl 10685 . . . 4 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
2 nqercl 10685 . . . 4 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
3 mulpqnq 10695 . . . 4 ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵))))
41, 2, 3syl2an 596 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵))))
5 enqer 10675 . . . . . 6 ~Q Er (N × N)
65a1i 11 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ~Q Er (N × N))
7 nqerrel 10686 . . . . . . 7 (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
87adantr 481 . . . . . 6 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐴 ~Q ([Q]‘𝐴))
9 elpqn 10679 . . . . . . . . 9 (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
101, 9syl 17 . . . . . . . 8 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
11 mulerpqlem 10709 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ ([Q]‘𝐴) ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵)))
12113exp 1118 . . . . . . . 8 (𝐴 ∈ (N × N) → (([Q]‘𝐴) ∈ (N × N) → (𝐵 ∈ (N × N) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵)))))
1310, 12mpd 15 . . . . . . 7 (𝐴 ∈ (N × N) → (𝐵 ∈ (N × N) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵))))
1413imp 407 . . . . . 6 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵)))
158, 14mpbid 231 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵))
16 nqerrel 10686 . . . . . . . 8 (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
1716adantl 482 . . . . . . 7 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 ~Q ([Q]‘𝐵))
18 elpqn 10679 . . . . . . . . . 10 (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
192, 18syl 17 . . . . . . . . 9 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
20 mulerpqlem 10709 . . . . . . . . . 10 ((𝐵 ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N) ∧ ([Q]‘𝐴) ∈ (N × N)) → (𝐵 ~Q ([Q]‘𝐵) ↔ (𝐵 ·pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) ·pQ ([Q]‘𝐴))))
21203exp 1118 . . . . . . . . 9 (𝐵 ∈ (N × N) → (([Q]‘𝐵) ∈ (N × N) → (([Q]‘𝐴) ∈ (N × N) → (𝐵 ~Q ([Q]‘𝐵) ↔ (𝐵 ·pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) ·pQ ([Q]‘𝐴))))))
2219, 21mpd 15 . . . . . . . 8 (𝐵 ∈ (N × N) → (([Q]‘𝐴) ∈ (N × N) → (𝐵 ~Q ([Q]‘𝐵) ↔ (𝐵 ·pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) ·pQ ([Q]‘𝐴)))))
2310, 22mpan9 507 . . . . . . 7 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 ~Q ([Q]‘𝐵) ↔ (𝐵 ·pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) ·pQ ([Q]‘𝐴))))
2417, 23mpbid 231 . . . . . 6 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 ·pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) ·pQ ([Q]‘𝐴)))
25 mulcompq 10706 . . . . . 6 (𝐵 ·pQ ([Q]‘𝐴)) = (([Q]‘𝐴) ·pQ 𝐵)
26 mulcompq 10706 . . . . . 6 (([Q]‘𝐵) ·pQ ([Q]‘𝐴)) = (([Q]‘𝐴) ·pQ ([Q]‘𝐵))
2724, 25, 263brtr3g 5109 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)))
286, 15, 27ertrd 8512 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)))
29 mulpqf 10700 . . . . . 6 ·pQ :((N × N) × (N × N))⟶(N × N)
3029fovcl 7402 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ∈ (N × N))
3129fovcl 7402 . . . . . 6 ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (([Q]‘𝐴) ·pQ ([Q]‘𝐵)) ∈ (N × N))
3210, 19, 31syl2an 596 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·pQ ([Q]‘𝐵)) ∈ (N × N))
33 nqereq 10689 . . . . 5 (((𝐴 ·pQ 𝐵) ∈ (N × N) ∧ (([Q]‘𝐴) ·pQ ([Q]‘𝐵)) ∈ (N × N)) → ((𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)) ↔ ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵)))))
3430, 32, 33syl2anc 584 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)) ↔ ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵)))))
3528, 34mpbid 231 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵))))
364, 35eqtr4d 2781 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵)))
37 0nnq 10678 . . . . . . 7 ¬ ∅ ∈ Q
38 nqerf 10684 . . . . . . . . . . 11 [Q]:(N × N)⟶Q
3938fdmi 6614 . . . . . . . . . 10 dom [Q] = (N × N)
4039eleq2i 2830 . . . . . . . . 9 (𝐴 ∈ dom [Q] ↔ 𝐴 ∈ (N × N))
41 ndmfv 6806 . . . . . . . . 9 𝐴 ∈ dom [Q] → ([Q]‘𝐴) = ∅)
4240, 41sylnbir 331 . . . . . . . 8 𝐴 ∈ (N × N) → ([Q]‘𝐴) = ∅)
4342eleq1d 2823 . . . . . . 7 𝐴 ∈ (N × N) → (([Q]‘𝐴) ∈ Q ↔ ∅ ∈ Q))
4437, 43mtbiri 327 . . . . . 6 𝐴 ∈ (N × N) → ¬ ([Q]‘𝐴) ∈ Q)
4544con4i 114 . . . . 5 (([Q]‘𝐴) ∈ Q𝐴 ∈ (N × N))
4639eleq2i 2830 . . . . . . . . 9 (𝐵 ∈ dom [Q] ↔ 𝐵 ∈ (N × N))
47 ndmfv 6806 . . . . . . . . 9 𝐵 ∈ dom [Q] → ([Q]‘𝐵) = ∅)
4846, 47sylnbir 331 . . . . . . . 8 𝐵 ∈ (N × N) → ([Q]‘𝐵) = ∅)
4948eleq1d 2823 . . . . . . 7 𝐵 ∈ (N × N) → (([Q]‘𝐵) ∈ Q ↔ ∅ ∈ Q))
5037, 49mtbiri 327 . . . . . 6 𝐵 ∈ (N × N) → ¬ ([Q]‘𝐵) ∈ Q)
5150con4i 114 . . . . 5 (([Q]‘𝐵) ∈ Q𝐵 ∈ (N × N))
5245, 51anim12i 613 . . . 4 ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
53 mulnqf 10703 . . . . . 6 ·Q :(Q × Q)⟶Q
5453fdmi 6614 . . . . 5 dom ·Q = (Q × Q)
5554ndmov 7456 . . . 4 (¬ (([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ∅)
5652, 55nsyl5 159 . . 3 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ∅)
57 0nelxp 5625 . . . . . 6 ¬ ∅ ∈ (N × N)
5839eleq2i 2830 . . . . . 6 (∅ ∈ dom [Q] ↔ ∅ ∈ (N × N))
5957, 58mtbir 323 . . . . 5 ¬ ∅ ∈ dom [Q]
6029fdmi 6614 . . . . . . 7 dom ·pQ = ((N × N) × (N × N))
6160ndmov 7456 . . . . . 6 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ∅)
6261eleq1d 2823 . . . . 5 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((𝐴 ·pQ 𝐵) ∈ dom [Q] ↔ ∅ ∈ dom [Q]))
6359, 62mtbiri 327 . . . 4 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ¬ (𝐴 ·pQ 𝐵) ∈ dom [Q])
64 ndmfv 6806 . . . 4 (¬ (𝐴 ·pQ 𝐵) ∈ dom [Q] → ([Q]‘(𝐴 ·pQ 𝐵)) = ∅)
6563, 64syl 17 . . 3 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ([Q]‘(𝐴 ·pQ 𝐵)) = ∅)
6656, 65eqtr4d 2781 . 2 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵)))
6736, 66pm2.61i 182 1 (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  c0 4258   class class class wbr 5076   × cxp 5589  dom cdm 5591  cfv 6435  (class class class)co 7277   Er wer 8493  Ncnpi 10598   ·pQ cmpq 10603   ~Q ceq 10605  Qcnq 10606  [Q]cerq 10608   ·Q cmq 10610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pr 5354  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5077  df-opab 5139  df-mpt 5160  df-tr 5194  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-pred 6204  df-ord 6271  df-on 6272  df-lim 6273  df-suc 6274  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-f1 6440  df-fo 6441  df-f1o 6442  df-fv 6443  df-ov 7280  df-oprab 7281  df-mpo 7282  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8095  df-wrecs 8126  df-recs 8200  df-rdg 8239  df-1o 8295  df-oadd 8299  df-omul 8300  df-er 8496  df-ni 10626  df-mi 10628  df-lti 10629  df-mpq 10663  df-enq 10665  df-nq 10666  df-erq 10667  df-mq 10669  df-1nq 10670
This theorem is referenced by:  mulassnq  10713  distrnq  10715  recmulnq  10718
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