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Theorem mulerpq 9658
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 9632 . . . 4 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
2 nqercl 9632 . . . 4 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
3 mulpqnq 9642 . . . 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 9622 . . . . . 6 ~Q Er (N × N)
65a1i 11 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ~Q Er (N × N))
7 nqerrel 9633 . . . . . . 7 (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
87adantr 480 . . . . . 6 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐴 ~Q ([Q]‘𝐴))
9 elpqn 9626 . . . . . . . . 9 (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
101, 9syl 17 . . . . . . . 8 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
11 mulerpqlem 9656 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ ([Q]‘𝐴) ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵)))
12113exp 1256 . . . . . . . 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 221 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵))
16 nqerrel 9633 . . . . . . . 8 (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
1716adantl 481 . . . . . . 7 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 ~Q ([Q]‘𝐵))
18 elpqn 9626 . . . . . . . . . 10 (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
192, 18syl 17 . . . . . . . . 9 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
20 mulerpqlem 9656 . . . . . . . . . 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 1256 . . . . . . . . 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 221 . . . . . 6 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 ·pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) ·pQ ([Q]‘𝐴)))
25 mulcompq 9653 . . . . . 6 (𝐵 ·pQ ([Q]‘𝐴)) = (([Q]‘𝐴) ·pQ 𝐵)
26 mulcompq 9653 . . . . . 6 (([Q]‘𝐵) ·pQ ([Q]‘𝐴)) = (([Q]‘𝐴) ·pQ ([Q]‘𝐵))
2724, 25, 263brtr3g 4616 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)))
286, 15, 27ertrd 7645 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)))
29 mulpqf 9647 . . . . . 6 ·pQ :((N × N) × (N × N))⟶(N × N)
3029fovcl 6663 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ∈ (N × N))
3129fovcl 6663 . . . . . 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 9636 . . . . 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 691 . . . 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 221 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵))))
364, 35eqtr4d 2647 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵)))
37 0nnq 9625 . . . . . . . 8 ¬ ∅ ∈ Q
38 nqerf 9631 . . . . . . . . . . . 12 [Q]:(N × N)⟶Q
3938fdmi 5965 . . . . . . . . . . 11 dom [Q] = (N × N)
4039eleq2i 2680 . . . . . . . . . 10 (𝐴 ∈ dom [Q] ↔ 𝐴 ∈ (N × N))
41 ndmfv 6128 . . . . . . . . . 10 𝐴 ∈ dom [Q] → ([Q]‘𝐴) = ∅)
4240, 41sylnbir 320 . . . . . . . . 9 𝐴 ∈ (N × N) → ([Q]‘𝐴) = ∅)
4342eleq1d 2672 . . . . . . . 8 𝐴 ∈ (N × N) → (([Q]‘𝐴) ∈ Q ↔ ∅ ∈ Q))
4437, 43mtbiri 316 . . . . . . 7 𝐴 ∈ (N × N) → ¬ ([Q]‘𝐴) ∈ Q)
4544con4i 112 . . . . . 6 (([Q]‘𝐴) ∈ Q𝐴 ∈ (N × N))
4639eleq2i 2680 . . . . . . . . . 10 (𝐵 ∈ dom [Q] ↔ 𝐵 ∈ (N × N))
47 ndmfv 6128 . . . . . . . . . 10 𝐵 ∈ dom [Q] → ([Q]‘𝐵) = ∅)
4846, 47sylnbir 320 . . . . . . . . 9 𝐵 ∈ (N × N) → ([Q]‘𝐵) = ∅)
4948eleq1d 2672 . . . . . . . 8 𝐵 ∈ (N × N) → (([Q]‘𝐵) ∈ Q ↔ ∅ ∈ Q))
5037, 49mtbiri 316 . . . . . . 7 𝐵 ∈ (N × N) → ¬ ([Q]‘𝐵) ∈ Q)
5150con4i 112 . . . . . 6 (([Q]‘𝐵) ∈ Q𝐵 ∈ (N × N))
5245, 51anim12i 588 . . . . 5 ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
5352con3i 149 . . . 4 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ¬ (([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q))
54 mulnqf 9650 . . . . . 6 ·Q :(Q × Q)⟶Q
5554fdmi 5965 . . . . 5 dom ·Q = (Q × Q)
5655ndmov 6716 . . . 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 5067 . . . . . 6 ¬ ∅ ∈ (N × N)
5939eleq2i 2680 . . . . . 6 (∅ ∈ dom [Q] ↔ ∅ ∈ (N × N))
6058, 59mtbir 312 . . . . 5 ¬ ∅ ∈ dom [Q]
6129fdmi 5965 . . . . . . 7 dom ·pQ = ((N × N) × (N × N))
6261ndmov 6716 . . . . . 6 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ∅)
6362eleq1d 2672 . . . . 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 6128 . . . 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 2647 . 2 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵)))
6836, 67pm2.61i 175 1 (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  c0 3874   class class class wbr 4583   × cxp 5036  dom cdm 5038  cfv 5804  (class class class)co 6549   Er wer 7626  Ncnpi 9545   ·pQ cmpq 9550   ~Q ceq 9552  Qcnq 9553  [Q]cerq 9555   ·Q cmq 9557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ni 9573  df-mi 9575  df-lti 9576  df-mpq 9610  df-enq 9612  df-nq 9613  df-erq 9614  df-mq 9616  df-1nq 9617
This theorem is referenced by:  mulassnq  9660  distrnq  9662  recmulnq  9665
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