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Theorem mulerpq 9817
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 9791 . . . 4 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
2 nqercl 9791 . . . 4 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
3 mulpqnq 9801 . . . 4 ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵))))
41, 2, 3syl2an 493 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵))))
5 enqer 9781 . . . . . 6 ~Q Er (N × N)
65a1i 11 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ~Q Er (N × N))
7 nqerrel 9792 . . . . . . 7 (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
87adantr 480 . . . . . 6 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐴 ~Q ([Q]‘𝐴))
9 elpqn 9785 . . . . . . . . 9 (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
101, 9syl 17 . . . . . . . 8 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
11 mulerpqlem 9815 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ ([Q]‘𝐴) ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵)))
12113exp 1283 . . . . . . . 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 444 . . . . . 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 9792 . . . . . . . 8 (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
1716adantl 481 . . . . . . 7 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 ~Q ([Q]‘𝐵))
18 elpqn 9785 . . . . . . . . . 10 (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
192, 18syl 17 . . . . . . . . 9 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
20 mulerpqlem 9815 . . . . . . . . . 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 1283 . . . . . . . . 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 485 . . . . . . 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 9812 . . . . . 6 (𝐵 ·pQ ([Q]‘𝐴)) = (([Q]‘𝐴) ·pQ 𝐵)
26 mulcompq 9812 . . . . . 6 (([Q]‘𝐵) ·pQ ([Q]‘𝐴)) = (([Q]‘𝐴) ·pQ ([Q]‘𝐵))
2724, 25, 263brtr3g 4718 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)))
286, 15, 27ertrd 7803 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)))
29 mulpqf 9806 . . . . . 6 ·pQ :((N × N) × (N × N))⟶(N × N)
3029fovcl 6807 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ∈ (N × N))
3129fovcl 6807 . . . . . 6 ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (([Q]‘𝐴) ·pQ ([Q]‘𝐵)) ∈ (N × N))
3210, 19, 31syl2an 493 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·pQ ([Q]‘𝐵)) ∈ (N × N))
33 nqereq 9795 . . . . 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 694 . . . 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 2688 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵)))
37 0nnq 9784 . . . . . . . 8 ¬ ∅ ∈ Q
38 nqerf 9790 . . . . . . . . . . . 12 [Q]:(N × N)⟶Q
3938fdmi 6090 . . . . . . . . . . 11 dom [Q] = (N × N)
4039eleq2i 2722 . . . . . . . . . 10 (𝐴 ∈ dom [Q] ↔ 𝐴 ∈ (N × N))
41 ndmfv 6256 . . . . . . . . . 10 𝐴 ∈ dom [Q] → ([Q]‘𝐴) = ∅)
4240, 41sylnbir 320 . . . . . . . . 9 𝐴 ∈ (N × N) → ([Q]‘𝐴) = ∅)
4342eleq1d 2715 . . . . . . . 8 𝐴 ∈ (N × N) → (([Q]‘𝐴) ∈ Q ↔ ∅ ∈ Q))
4437, 43mtbiri 316 . . . . . . 7 𝐴 ∈ (N × N) → ¬ ([Q]‘𝐴) ∈ Q)
4544con4i 113 . . . . . 6 (([Q]‘𝐴) ∈ Q𝐴 ∈ (N × N))
4639eleq2i 2722 . . . . . . . . . 10 (𝐵 ∈ dom [Q] ↔ 𝐵 ∈ (N × N))
47 ndmfv 6256 . . . . . . . . . 10 𝐵 ∈ dom [Q] → ([Q]‘𝐵) = ∅)
4846, 47sylnbir 320 . . . . . . . . 9 𝐵 ∈ (N × N) → ([Q]‘𝐵) = ∅)
4948eleq1d 2715 . . . . . . . 8 𝐵 ∈ (N × N) → (([Q]‘𝐵) ∈ Q ↔ ∅ ∈ Q))
5037, 49mtbiri 316 . . . . . . 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 9809 . . . . . 6 ·Q :(Q × Q)⟶Q
5554fdmi 6090 . . . . 5 dom ·Q = (Q × Q)
5655ndmov 6860 . . . 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 5177 . . . . . 6 ¬ ∅ ∈ (N × N)
5939eleq2i 2722 . . . . . 6 (∅ ∈ dom [Q] ↔ ∅ ∈ (N × N))
6058, 59mtbir 312 . . . . 5 ¬ ∅ ∈ dom [Q]
6129fdmi 6090 . . . . . . 7 dom ·pQ = ((N × N) × (N × N))
6261ndmov 6860 . . . . . 6 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ∅)
6362eleq1d 2715 . . . . 5 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((𝐴 ·pQ 𝐵) ∈ dom [Q] ↔ ∅ ∈ dom [Q]))
6460, 63mtbiri 316 . . . 4 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ¬ (𝐴 ·pQ 𝐵) ∈ dom [Q])
65 ndmfv 6256 . . . 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 2688 . 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 383   = wceq 1523  wcel 2030  c0 3948   class class class wbr 4685   × cxp 5141  dom cdm 5143  cfv 5926  (class class class)co 6690   Er wer 7784  Ncnpi 9704   ·pQ cmpq 9709   ~Q ceq 9711  Qcnq 9712  [Q]cerq 9714   ·Q cmq 9716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ni 9732  df-mi 9734  df-lti 9735  df-mpq 9769  df-enq 9771  df-nq 9772  df-erq 9773  df-mq 9775  df-1nq 9776
This theorem is referenced by:  mulassnq  9819  distrnq  9821  recmulnq  9824
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