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Theorem adderpq 10899
Description: Addition is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
adderpq (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(𝐴 +pQ 𝐵))

Proof of Theorem adderpq
StepHypRef Expression
1 nqercl 10874 . . . 4 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
2 nqercl 10874 . . . 4 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
3 addpqnq 10881 . . . 4 ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) +pQ ([Q]‘𝐵))))
41, 2, 3syl2an 597 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) +pQ ([Q]‘𝐵))))
5 enqer 10864 . . . . . 6 ~Q Er (N × N)
65a1i 11 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ~Q Er (N × N))
7 nqerrel 10875 . . . . . . 7 (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
87adantr 482 . . . . . 6 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐴 ~Q ([Q]‘𝐴))
9 elpqn 10868 . . . . . . . . 9 (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
101, 9syl 17 . . . . . . . 8 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
11 adderpqlem 10897 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ ([Q]‘𝐴) ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ 𝐵)))
12113exp 1120 . . . . . . . 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 408 . . . . . 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 10875 . . . . . . . 8 (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
1716adantl 483 . . . . . . 7 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 ~Q ([Q]‘𝐵))
18 elpqn 10868 . . . . . . . . . 10 (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
192, 18syl 17 . . . . . . . . 9 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
20 adderpqlem 10897 . . . . . . . . . 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 1120 . . . . . . . . 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 508 . . . . . . 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 addcompq 10893 . . . . . 6 (𝐵 +pQ ([Q]‘𝐴)) = (([Q]‘𝐴) +pQ 𝐵)
26 addcompq 10893 . . . . . 6 (([Q]‘𝐵) +pQ ([Q]‘𝐴)) = (([Q]‘𝐴) +pQ ([Q]‘𝐵))
2724, 25, 263brtr3g 5143 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ ([Q]‘𝐵)))
286, 15, 27ertrd 8671 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ ([Q]‘𝐵)))
29 addpqf 10887 . . . . . 6 +pQ :((N × N) × (N × N))⟶(N × N)
3029fovcl 7489 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) ∈ (N × N))
3129fovcl 7489 . . . . . 6 ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (([Q]‘𝐴) +pQ ([Q]‘𝐵)) ∈ (N × N))
3210, 19, 31syl2an 597 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +pQ ([Q]‘𝐵)) ∈ (N × N))
33 nqereq 10878 . . . . 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 585 . . . 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 2780 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(𝐴 +pQ 𝐵)))
37 0nnq 10867 . . . . . . 7 ¬ ∅ ∈ Q
38 nqerf 10873 . . . . . . . . . . 11 [Q]:(N × N)⟶Q
3938fdmi 6685 . . . . . . . . . 10 dom [Q] = (N × N)
4039eleq2i 2830 . . . . . . . . 9 (𝐴 ∈ dom [Q] ↔ 𝐴 ∈ (N × N))
41 ndmfv 6882 . . . . . . . . 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 6882 . . . . . . . . 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 614 . . . 4 ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
53 addnqf 10891 . . . . . 6 +Q :(Q × Q)⟶Q
5453fdmi 6685 . . . . 5 dom +Q = (Q × Q)
5554ndmov 7543 . . . 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 5672 . . . . . 6 ¬ ∅ ∈ (N × N)
5839eleq2i 2830 . . . . . 6 (∅ ∈ dom [Q] ↔ ∅ ∈ (N × N))
5957, 58mtbir 323 . . . . 5 ¬ ∅ ∈ dom [Q]
6029fdmi 6685 . . . . . . 7 dom +pQ = ((N × N) × (N × N))
6160ndmov 7543 . . . . . 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 6882 . . . 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 2780 . 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 397   = wceq 1542  wcel 2107  c0 4287   class class class wbr 5110   × cxp 5636  dom cdm 5638  cfv 6501  (class class class)co 7362   Er wer 8652  Ncnpi 10787   +pQ cplpq 10791   ~Q ceq 10794  Qcnq 10795  [Q]cerq 10797   +Q cplq 10798
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-omul 8422  df-er 8655  df-ni 10815  df-pli 10816  df-mi 10817  df-lti 10818  df-plpq 10851  df-enq 10854  df-nq 10855  df-erq 10856  df-plq 10857  df-1nq 10859
This theorem is referenced by:  addassnq  10901  distrnq  10904  ltexnq  10918
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