Theorem adderpq 10869

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

Step	Hyp	Ref	Expression
1		nqercl 10844	. . . 4 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
2		nqercl 10844	. . . 4 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
3		addpqnq 10851	. . . 4 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) +pQ ([Q]‘𝐵))))
4	1, 2, 3	syl2an 596	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) +pQ ([Q]‘𝐵))))
5		enqer 10834	. . . . . 6 ⊢ ~Q Er (N × N)
6	5	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ~Q Er (N × N))
7		nqerrel 10845	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
8	7	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐴 ~Q ([Q]‘𝐴))
9		elpqn 10838	. . . . . . . . 9 ⊢ (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
10	1, 9	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
11		adderpqlem 10867	. . . . . . . . 9 ⊢ ((𝐴 ∈ (N × N) ∧ ([Q]‘𝐴) ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ 𝐵)))
12	11	3exp 1119	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → (([Q]‘𝐴) ∈ (N × N) → (𝐵 ∈ (N × N) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ 𝐵)))))
13	10, 12	mpd 15	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (𝐵 ∈ (N × N) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ 𝐵))))
14	13	imp 406	. . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ 𝐵)))
15	8, 14	mpbid 232	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ 𝐵))
16		nqerrel 10845	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
17	16	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 ~Q ([Q]‘𝐵))
18		elpqn 10838	. . . . . . . . . 10 ⊢ (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
19	2, 18	syl 17	. . . . . . . . 9 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
20		adderpqlem 10867	. . . . . . . . . 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))))
21	20	3exp 1119	. . . . . . . . 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))))))
22	19, 21	mpd 15	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → (([Q]‘𝐴) ∈ (N × N) → (𝐵 ~Q ([Q]‘𝐵) ↔ (𝐵 +pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) +pQ ([Q]‘𝐴)))))
23	10, 22	mpan9 506	. . . . . . 7 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 ~Q ([Q]‘𝐵) ↔ (𝐵 +pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) +pQ ([Q]‘𝐴))))
24	17, 23	mpbid 232	. . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 +pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) +pQ ([Q]‘𝐴)))
25		addcompq 10863	. . . . . 6 ⊢ (𝐵 +pQ ([Q]‘𝐴)) = (([Q]‘𝐴) +pQ 𝐵)
26		addcompq 10863	. . . . . 6 ⊢ (([Q]‘𝐵) +pQ ([Q]‘𝐴)) = (([Q]‘𝐴) +pQ ([Q]‘𝐵))
27	24, 25, 26	3brtr3g 5128	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ ([Q]‘𝐵)))
28	6, 15, 27	ertrd 8648	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ ([Q]‘𝐵)))
29		addpqf 10857	. . . . . 6 ⊢ +pQ :((N × N) × (N × N))⟶(N × N)
30	29	fovcl 7481	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) ∈ (N × N))
31	29	fovcl 7481	. . . . . 6 ⊢ ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (([Q]‘𝐴) +pQ ([Q]‘𝐵)) ∈ (N × N))
32	10, 19, 31	syl2an 596	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +pQ ([Q]‘𝐵)) ∈ (N × N))
33		nqereq 10848	. . . . 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)))))
34	30, 32, 33	syl2anc 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)))))
35	28, 34	mpbid 232	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ([Q]‘(𝐴 +pQ 𝐵)) = ([Q]‘(([Q]‘𝐴) +pQ ([Q]‘𝐵))))
36	4, 35	eqtr4d 2767	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(𝐴 +pQ 𝐵)))
37		0nnq 10837	. . . . . . 7 ⊢ ¬ ∅ ∈ Q
38		nqerf 10843	. . . . . . . . . . 11 ⊢ [Q]:(N × N)⟶Q
39	38	fdmi 6667	. . . . . . . . . 10 ⊢ dom [Q] = (N × N)
40	39	eleq2i 2820	. . . . . . . . 9 ⊢ (𝐴 ∈ dom [Q] ↔ 𝐴 ∈ (N × N))
41		ndmfv 6859	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ dom [Q] → ([Q]‘𝐴) = ∅)
42	40, 41	sylnbir 331	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ (N × N) → ([Q]‘𝐴) = ∅)
43	42	eleq1d 2813	. . . . . . 7 ⊢ (¬ 𝐴 ∈ (N × N) → (([Q]‘𝐴) ∈ Q ↔ ∅ ∈ Q))
44	37, 43	mtbiri 327	. . . . . 6 ⊢ (¬ 𝐴 ∈ (N × N) → ¬ ([Q]‘𝐴) ∈ Q)
45	44	con4i 114	. . . . 5 ⊢ (([Q]‘𝐴) ∈ Q → 𝐴 ∈ (N × N))
46	39	eleq2i 2820	. . . . . . . . 9 ⊢ (𝐵 ∈ dom [Q] ↔ 𝐵 ∈ (N × N))
47		ndmfv 6859	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ dom [Q] → ([Q]‘𝐵) = ∅)
48	46, 47	sylnbir 331	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ (N × N) → ([Q]‘𝐵) = ∅)
49	48	eleq1d 2813	. . . . . . 7 ⊢ (¬ 𝐵 ∈ (N × N) → (([Q]‘𝐵) ∈ Q ↔ ∅ ∈ Q))
50	37, 49	mtbiri 327	. . . . . 6 ⊢ (¬ 𝐵 ∈ (N × N) → ¬ ([Q]‘𝐵) ∈ Q)
51	50	con4i 114	. . . . 5 ⊢ (([Q]‘𝐵) ∈ Q → 𝐵 ∈ (N × N))
52	45, 51	anim12i 613	. . . 4 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
53		addnqf 10861	. . . . . 6 ⊢ +Q :(Q × Q)⟶Q
54	53	fdmi 6667	. . . . 5 ⊢ dom +Q = (Q × Q)
55	54	ndmov 7537	. . . 4 ⊢ (¬ (([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ∅)
56	52, 55	nsyl5 159	. . 3 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ∅)
57		0nelxp 5657	. . . . . 6 ⊢ ¬ ∅ ∈ (N × N)
58	39	eleq2i 2820	. . . . . 6 ⊢ (∅ ∈ dom [Q] ↔ ∅ ∈ (N × N))
59	57, 58	mtbir 323	. . . . 5 ⊢ ¬ ∅ ∈ dom [Q]
60	29	fdmi 6667	. . . . . . 7 ⊢ dom +pQ = ((N × N) × (N × N))
61	60	ndmov 7537	. . . . . 6 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ∅)
62	61	eleq1d 2813	. . . . 5 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((𝐴 +pQ 𝐵) ∈ dom [Q] ↔ ∅ ∈ dom [Q]))
63	59, 62	mtbiri 327	. . . 4 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ¬ (𝐴 +pQ 𝐵) ∈ dom [Q])
64		ndmfv 6859	. . . 4 ⊢ (¬ (𝐴 +pQ 𝐵) ∈ dom [Q] → ([Q]‘(𝐴 +pQ 𝐵)) = ∅)
65	63, 64	syl 17	. . 3 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ([Q]‘(𝐴 +pQ 𝐵)) = ∅)
66	56, 65	eqtr4d 2767	. 2 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(𝐴 +pQ 𝐵)))
67	36, 66	pm2.61i 182	1 ⊢ (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(𝐴 +pQ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∅c0 4286   class class class wbr 5095   × cxp 5621  dom cdm 5623  ‘cfv 6486  (class class class)co 7353   Er wer 8629  Ncnpi 10757   +_pQ cplpq 10761   ~_Q ceq 10764  Qcnq 10765  [Q]cerq 10767   +_Q cplq 10768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-omul 8400  df-er 8632  df-ni 10785  df-pli 10786  df-mi 10787  df-lti 10788  df-plpq 10821  df-enq 10824  df-nq 10825  df-erq 10826  df-plq 10827  df-1nq 10829

This theorem is referenced by:  addassnq  10871  distrnq  10874  ltexnq  10888