Theorem adderpq 10850

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 10825	. . . 4 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
2		nqercl 10825	. . . 4 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
3		addpqnq 10832	. . . 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 10815	. . . . . 6 ⊢ ~Q Er (N × N)
6	5	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ~Q Er (N × N))
7		nqerrel 10826	. . . . . . 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 10819	. . . . . . . . 9 ⊢ (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
10	1, 9	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
11		adderpqlem 10848	. . . . . . . . 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 10826	. . . . . . . 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 10819	. . . . . . . . . 10 ⊢ (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
19	2, 18	syl 17	. . . . . . . . 9 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
20		adderpqlem 10848	. . . . . . . . . 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 10844	. . . . . 6 ⊢ (𝐵 +pQ ([Q]‘𝐴)) = (([Q]‘𝐴) +pQ 𝐵)
26		addcompq 10844	. . . . . 6 ⊢ (([Q]‘𝐵) +pQ ([Q]‘𝐴)) = (([Q]‘𝐴) +pQ ([Q]‘𝐵))
27	24, 25, 26	3brtr3g 5125	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ ([Q]‘𝐵)))
28	6, 15, 27	ertrd 8641	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ ([Q]‘𝐵)))
29		addpqf 10838	. . . . . 6 ⊢ +pQ :((N × N) × (N × N))⟶(N × N)
30	29	fovcl 7477	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) ∈ (N × N))
31	29	fovcl 7477	. . . . . 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 10829	. . . . 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 10818	. . . . . . 7 ⊢ ¬ ∅ ∈ Q
38		nqerf 10824	. . . . . . . . . . 11 ⊢ [Q]:(N × N)⟶Q
39	38	fdmi 6663	. . . . . . . . . 10 ⊢ dom [Q] = (N × N)
40	39	eleq2i 2820	. . . . . . . . 9 ⊢ (𝐴 ∈ dom [Q] ↔ 𝐴 ∈ (N × N))
41		ndmfv 6855	. . . . . . . . 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 6855	. . . . . . . . 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 10842	. . . . . 6 ⊢ +Q :(Q × Q)⟶Q
54	53	fdmi 6663	. . . . 5 ⊢ dom +Q = (Q × Q)
55	54	ndmov 7533	. . . 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 5653	. . . . . 6 ⊢ ¬ ∅ ∈ (N × N)
58	39	eleq2i 2820	. . . . . 6 ⊢ (∅ ∈ dom [Q] ↔ ∅ ∈ (N × N))
59	57, 58	mtbir 323	. . . . 5 ⊢ ¬ ∅ ∈ dom [Q]
60	29	fdmi 6663	. . . . . . 7 ⊢ dom +pQ = ((N × N) × (N × N))
61	60	ndmov 7533	. . . . . 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 6855	. . . 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 4284   class class class wbr 5092   × cxp 5617  dom cdm 5619  ‘cfv 6482  (class class class)co 7349   Er wer 8622  Ncnpi 10738   +_pQ cplpq 10742   ~_Q ceq 10745  Qcnq 10746  [Q]cerq 10748   +_Q cplq 10749

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 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-oadd 8392  df-omul 8393  df-er 8625  df-ni 10766  df-pli 10767  df-mi 10768  df-lti 10769  df-plpq 10802  df-enq 10805  df-nq 10806  df-erq 10807  df-plq 10808  df-1nq 10810

This theorem is referenced by:  addassnq  10852  distrnq  10855  ltexnq  10869