Theorem adderpq 10879

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 10854	. . . 4 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
2		nqercl 10854	. . . 4 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
3		addpqnq 10861	. . . 4 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) +pQ ([Q]‘𝐵))))
4	1, 2, 3	syl2an 597	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) +pQ ([Q]‘𝐵))))
5		enqer 10844	. . . . . 6 ⊢ ~Q Er (N × N)
6	5	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ~Q Er (N × N))
7		nqerrel 10855	. . . . . . 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 10848	. . . . . . . . 9 ⊢ (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
10	1, 9	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
11		adderpqlem 10877	. . . . . . . . 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 1120	. . . . . . . 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 10855	. . . . . . . 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 10848	. . . . . . . . . 10 ⊢ (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
19	2, 18	syl 17	. . . . . . . . 9 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
20		adderpqlem 10877	. . . . . . . . . 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 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))))))
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 10873	. . . . . 6 ⊢ (𝐵 +pQ ([Q]‘𝐴)) = (([Q]‘𝐴) +pQ 𝐵)
26		addcompq 10873	. . . . . 6 ⊢ (([Q]‘𝐵) +pQ ([Q]‘𝐴)) = (([Q]‘𝐴) +pQ ([Q]‘𝐵))
27	24, 25, 26	3brtr3g 5133	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ ([Q]‘𝐵)))
28	6, 15, 27	ertrd 8662	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ ([Q]‘𝐵)))
29		addpqf 10867	. . . . . 6 ⊢ +pQ :((N × N) × (N × N))⟶(N × N)
30	29	fovcl 7496	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) ∈ (N × N))
31	29	fovcl 7496	. . . . . 6 ⊢ ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (([Q]‘𝐴) +pQ ([Q]‘𝐵)) ∈ (N × N))
32	10, 19, 31	syl2an 597	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +pQ ([Q]‘𝐵)) ∈ (N × N))
33		nqereq 10858	. . . . 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 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)))))
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 2775	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(𝐴 +pQ 𝐵)))
37		0nnq 10847	. . . . . . 7 ⊢ ¬ ∅ ∈ Q
38		nqerf 10853	. . . . . . . . . . 11 ⊢ [Q]:(N × N)⟶Q
39	38	fdmi 6681	. . . . . . . . . 10 ⊢ dom [Q] = (N × N)
40	39	eleq2i 2829	. . . . . . . . 9 ⊢ (𝐴 ∈ dom [Q] ↔ 𝐴 ∈ (N × N))
41		ndmfv 6874	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ dom [Q] → ([Q]‘𝐴) = ∅)
42	40, 41	sylnbir 331	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ (N × N) → ([Q]‘𝐴) = ∅)
43	42	eleq1d 2822	. . . . . . 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 2829	. . . . . . . . 9 ⊢ (𝐵 ∈ dom [Q] ↔ 𝐵 ∈ (N × N))
47		ndmfv 6874	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ dom [Q] → ([Q]‘𝐵) = ∅)
48	46, 47	sylnbir 331	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ (N × N) → ([Q]‘𝐵) = ∅)
49	48	eleq1d 2822	. . . . . . 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 614	. . . 4 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
53		addnqf 10871	. . . . . 6 ⊢ +Q :(Q × Q)⟶Q
54	53	fdmi 6681	. . . . 5 ⊢ dom +Q = (Q × Q)
55	54	ndmov 7552	. . . 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 5666	. . . . . 6 ⊢ ¬ ∅ ∈ (N × N)
58	39	eleq2i 2829	. . . . . 6 ⊢ (∅ ∈ dom [Q] ↔ ∅ ∈ (N × N))
59	57, 58	mtbir 323	. . . . 5 ⊢ ¬ ∅ ∈ dom [Q]
60	29	fdmi 6681	. . . . . . 7 ⊢ dom +pQ = ((N × N) × (N × N))
61	60	ndmov 7552	. . . . . 6 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ∅)
62	61	eleq1d 2822	. . . . 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 6874	. . . 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 2775	. 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 1542   ∈ wcel 2114  ∅c0 4287   class class class wbr 5100   × cxp 5630  dom cdm 5632  ‘cfv 6500  (class class class)co 7368   Er wer 8642  Ncnpi 10767   +_pQ cplpq 10771   ~_Q ceq 10774  Qcnq 10775  [Q]cerq 10777   +_Q cplq 10778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-omul 8412  df-er 8645  df-ni 10795  df-pli 10796  df-mi 10797  df-lti 10798  df-plpq 10831  df-enq 10834  df-nq 10835  df-erq 10836  df-plq 10837  df-1nq 10839

This theorem is referenced by:  addassnq  10881  distrnq  10884  ltexnq  10898