Theorem adderpq 10911

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 10886	. . . 4 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
2		nqercl 10886	. . . 4 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
3		addpqnq 10893	. . . 4 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) +pQ ([Q]‘𝐵))))
4	1, 2, 3	syl2an 605	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) +pQ ([Q]‘𝐵))))
5		enqer 10876	. . . . . 6 ⊢ ~Q Er (N × N)
6	5	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ~Q Er (N × N))
7		nqerrel 10887	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
8	7	adantr 484	. . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐴 ~Q ([Q]‘𝐴))
9		elpqn 10880	. . . . . . . . 9 ⊢ (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
10	1, 9	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
11		adderpqlem 10909	. . . . . . . . 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 1131	. . . . . . . 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 410	. . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ 𝐵)))
15	8, 14	mpbid 234	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ 𝐵))
16		nqerrel 10887	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
17	16	adantl 485	. . . . . . 7 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 ~Q ([Q]‘𝐵))
18		elpqn 10880	. . . . . . . . . 10 ⊢ (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
19	2, 18	syl 17	. . . . . . . . 9 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
20		adderpqlem 10909	. . . . . . . . . 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 1131	. . . . . . . . 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 514	. . . . . . 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 234	. . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 +pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) +pQ ([Q]‘𝐴)))
25		addcompq 10905	. . . . . 6 ⊢ (𝐵 +pQ ([Q]‘𝐴)) = (([Q]‘𝐴) +pQ 𝐵)
26		addcompq 10905	. . . . . 6 ⊢ (([Q]‘𝐵) +pQ ([Q]‘𝐴)) = (([Q]‘𝐴) +pQ ([Q]‘𝐵))
27	24, 25, 26	3brtr3g 5132	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ ([Q]‘𝐵)))
28	6, 15, 27	ertrd 8690	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) ~Q (([Q]‘𝐴) +pQ ([Q]‘𝐵)))
29		addpqf 10899	. . . . . 6 ⊢ +pQ :((N × N) × (N × N))⟶(N × N)
30	29	fovcl 7520	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) ∈ (N × N))
31	29	fovcl 7520	. . . . . 6 ⊢ ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (([Q]‘𝐴) +pQ ([Q]‘𝐵)) ∈ (N × N))
32	10, 19, 31	syl2an 605	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +pQ ([Q]‘𝐵)) ∈ (N × N))
33		nqereq 10890	. . . . 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 593	. . . 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 234	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ([Q]‘(𝐴 +pQ 𝐵)) = ([Q]‘(([Q]‘𝐴) +pQ ([Q]‘𝐵))))
36	4, 35	eqtr4d 2799	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(𝐴 +pQ 𝐵)))
37		0nnq 10879	. . . . . . 7 ⊢ ¬ ∅ ∈ Q
38		nqerf 10885	. . . . . . . . . . 11 ⊢ [Q]:(N × N)⟶Q
39	38	fdmi 6699	. . . . . . . . . 10 ⊢ dom [Q] = (N × N)
40	39	eleq2i 2853	. . . . . . . . 9 ⊢ (𝐴 ∈ dom [Q] ↔ 𝐴 ∈ (N × N))
41		ndmfv 6895	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ dom [Q] → ([Q]‘𝐴) = ∅)
42	40, 41	sylnbir 333	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ (N × N) → ([Q]‘𝐴) = ∅)
43	42	eleq1d 2846	. . . . . . 7 ⊢ (¬ 𝐴 ∈ (N × N) → (([Q]‘𝐴) ∈ Q ↔ ∅ ∈ Q))
44	37, 43	mtbiri 329	. . . . . 6 ⊢ (¬ 𝐴 ∈ (N × N) → ¬ ([Q]‘𝐴) ∈ Q)
45	44	con4i 114	. . . . 5 ⊢ (([Q]‘𝐴) ∈ Q → 𝐴 ∈ (N × N))
46	39	eleq2i 2853	. . . . . . . . 9 ⊢ (𝐵 ∈ dom [Q] ↔ 𝐵 ∈ (N × N))
47		ndmfv 6895	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ dom [Q] → ([Q]‘𝐵) = ∅)
48	46, 47	sylnbir 333	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ (N × N) → ([Q]‘𝐵) = ∅)
49	48	eleq1d 2846	. . . . . . 7 ⊢ (¬ 𝐵 ∈ (N × N) → (([Q]‘𝐵) ∈ Q ↔ ∅ ∈ Q))
50	37, 49	mtbiri 329	. . . . . 6 ⊢ (¬ 𝐵 ∈ (N × N) → ¬ ([Q]‘𝐵) ∈ Q)
51	50	con4i 114	. . . . 5 ⊢ (([Q]‘𝐵) ∈ Q → 𝐵 ∈ (N × N))
52	45, 51	anim12i 622	. . . 4 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
53		addnqf 10903	. . . . . 6 ⊢ +Q :(Q × Q)⟶Q
54	53	fdmi 6699	. . . . 5 ⊢ dom +Q = (Q × Q)
55	54	ndmov 7576	. . . 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 5679	. . . . . 6 ⊢ ¬ ∅ ∈ (N × N)
58	39	eleq2i 2853	. . . . . 6 ⊢ (∅ ∈ dom [Q] ↔ ∅ ∈ (N × N))
59	57, 58	mtbir 325	. . . . 5 ⊢ ¬ ∅ ∈ dom [Q]
60	29	fdmi 6699	. . . . . . 7 ⊢ dom +pQ = ((N × N) × (N × N))
61	60	ndmov 7576	. . . . . 6 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ∅)
62	61	eleq1d 2846	. . . . 5 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((𝐴 +pQ 𝐵) ∈ dom [Q] ↔ ∅ ∈ dom [Q]))
63	59, 62	mtbiri 329	. . . 4 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ¬ (𝐴 +pQ 𝐵) ∈ dom [Q])
64		ndmfv 6895	. . . 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 2799	. 2 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(𝐴 +pQ 𝐵)))
67	36, 66	pm2.61i 183	1 ⊢ (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(𝐴 +pQ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∅c0 4285   class class class wbr 5099   × cxp 5643  dom cdm 5645  ‘cfv 6517  (class class class)co 7392   Er wer 8670  Ncnpi 10799   +_pQ cplpq 10803   ~_Q ceq 10806  Qcnq 10807  [Q]cerq 10809   +_Q cplq 10810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-oadd 8436  df-omul 8437  df-er 8673  df-ni 10827  df-pli 10828  df-mi 10829  df-lti 10830  df-plpq 10863  df-enq 10866  df-nq 10867  df-erq 10868  df-plq 10869  df-1nq 10871

This theorem is referenced by:  addassnq  10913  distrnq  10916  ltexnq  10930