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Theorem mulerpq 9724
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 9698 . . . 4 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
2 nqercl 9698 . . . 4 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
3 mulpqnq 9708 . . . 4 ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵))))
41, 2, 3syl2an 494 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵))))
5 enqer 9688 . . . . . 6 ~Q Er (N × N)
65a1i 11 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ~Q Er (N × N))
7 nqerrel 9699 . . . . . . 7 (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
87adantr 481 . . . . . 6 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐴 ~Q ([Q]‘𝐴))
9 elpqn 9692 . . . . . . . . 9 (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
101, 9syl 17 . . . . . . . 8 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
11 mulerpqlem 9722 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ ([Q]‘𝐴) ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵)))
12113exp 1261 . . . . . . . 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 445 . . . . . 6 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵)))
158, 14mpbid 222 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵))
16 nqerrel 9699 . . . . . . . 8 (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
1716adantl 482 . . . . . . 7 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 ~Q ([Q]‘𝐵))
18 elpqn 9692 . . . . . . . . . 10 (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
192, 18syl 17 . . . . . . . . 9 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
20 mulerpqlem 9722 . . . . . . . . . 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 1261 . . . . . . . . 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 486 . . . . . . 7 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 ~Q ([Q]‘𝐵) ↔ (𝐵 ·pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) ·pQ ([Q]‘𝐴))))
2417, 23mpbid 222 . . . . . 6 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 ·pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) ·pQ ([Q]‘𝐴)))
25 mulcompq 9719 . . . . . 6 (𝐵 ·pQ ([Q]‘𝐴)) = (([Q]‘𝐴) ·pQ 𝐵)
26 mulcompq 9719 . . . . . 6 (([Q]‘𝐵) ·pQ ([Q]‘𝐴)) = (([Q]‘𝐴) ·pQ ([Q]‘𝐵))
2724, 25, 263brtr3g 4651 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)))
286, 15, 27ertrd 7704 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)))
29 mulpqf 9713 . . . . . 6 ·pQ :((N × N) × (N × N))⟶(N × N)
3029fovcl 6719 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ∈ (N × N))
3129fovcl 6719 . . . . . 6 ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (([Q]‘𝐴) ·pQ ([Q]‘𝐵)) ∈ (N × N))
3210, 19, 31syl2an 494 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·pQ ([Q]‘𝐵)) ∈ (N × N))
33 nqereq 9702 . . . . 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 692 . . . 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 222 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵))))
364, 35eqtr4d 2663 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵)))
37 0nnq 9691 . . . . . . . 8 ¬ ∅ ∈ Q
38 nqerf 9697 . . . . . . . . . . . 12 [Q]:(N × N)⟶Q
3938fdmi 6011 . . . . . . . . . . 11 dom [Q] = (N × N)
4039eleq2i 2696 . . . . . . . . . 10 (𝐴 ∈ dom [Q] ↔ 𝐴 ∈ (N × N))
41 ndmfv 6176 . . . . . . . . . 10 𝐴 ∈ dom [Q] → ([Q]‘𝐴) = ∅)
4240, 41sylnbir 321 . . . . . . . . 9 𝐴 ∈ (N × N) → ([Q]‘𝐴) = ∅)
4342eleq1d 2688 . . . . . . . 8 𝐴 ∈ (N × N) → (([Q]‘𝐴) ∈ Q ↔ ∅ ∈ Q))
4437, 43mtbiri 317 . . . . . . 7 𝐴 ∈ (N × N) → ¬ ([Q]‘𝐴) ∈ Q)
4544con4i 113 . . . . . 6 (([Q]‘𝐴) ∈ Q𝐴 ∈ (N × N))
4639eleq2i 2696 . . . . . . . . . 10 (𝐵 ∈ dom [Q] ↔ 𝐵 ∈ (N × N))
47 ndmfv 6176 . . . . . . . . . 10 𝐵 ∈ dom [Q] → ([Q]‘𝐵) = ∅)
4846, 47sylnbir 321 . . . . . . . . 9 𝐵 ∈ (N × N) → ([Q]‘𝐵) = ∅)
4948eleq1d 2688 . . . . . . . 8 𝐵 ∈ (N × N) → (([Q]‘𝐵) ∈ Q ↔ ∅ ∈ Q))
5037, 49mtbiri 317 . . . . . . 7 𝐵 ∈ (N × N) → ¬ ([Q]‘𝐵) ∈ Q)
5150con4i 113 . . . . . 6 (([Q]‘𝐵) ∈ Q𝐵 ∈ (N × N))
5245, 51anim12i 589 . . . . 5 ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
5352con3i 150 . . . 4 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ¬ (([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q))
54 mulnqf 9716 . . . . . 6 ·Q :(Q × Q)⟶Q
5554fdmi 6011 . . . . 5 dom ·Q = (Q × Q)
5655ndmov 6772 . . . 4 (¬ (([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ∅)
5753, 56syl 17 . . 3 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ∅)
58 0nelxp 5108 . . . . . 6 ¬ ∅ ∈ (N × N)
5939eleq2i 2696 . . . . . 6 (∅ ∈ dom [Q] ↔ ∅ ∈ (N × N))
6058, 59mtbir 313 . . . . 5 ¬ ∅ ∈ dom [Q]
6129fdmi 6011 . . . . . . 7 dom ·pQ = ((N × N) × (N × N))
6261ndmov 6772 . . . . . 6 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ∅)
6362eleq1d 2688 . . . . 5 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((𝐴 ·pQ 𝐵) ∈ dom [Q] ↔ ∅ ∈ dom [Q]))
6460, 63mtbiri 317 . . . 4 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ¬ (𝐴 ·pQ 𝐵) ∈ dom [Q])
65 ndmfv 6176 . . . 4 (¬ (𝐴 ·pQ 𝐵) ∈ dom [Q] → ([Q]‘(𝐴 ·pQ 𝐵)) = ∅)
6664, 65syl 17 . . 3 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ([Q]‘(𝐴 ·pQ 𝐵)) = ∅)
6757, 66eqtr4d 2663 . 2 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵)))
6836, 67pm2.61i 176 1 (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  c0 3896   class class class wbr 4618   × cxp 5077  dom cdm 5079  cfv 5850  (class class class)co 6605   Er wer 7685  Ncnpi 9611   ·pQ cmpq 9616   ~Q ceq 9618  Qcnq 9619  [Q]cerq 9621   ·Q cmq 9623
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  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 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-omul 7511  df-er 7688  df-ni 9639  df-mi 9641  df-lti 9642  df-mpq 9676  df-enq 9678  df-nq 9679  df-erq 9680  df-mq 9682  df-1nq 9683
This theorem is referenced by:  mulassnq  9726  distrnq  9728  recmulnq  9731
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