Theorem mulerpq 10711

Description: Multiplication is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
mulerpq	⊢ (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵))

Proof of Theorem mulerpq

Step	Hyp	Ref	Expression
1		nqercl 10685	. . . 4 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
2		nqercl 10685	. . . 4 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
3		mulpqnq 10695	. . . 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 10675	. . . . . 6 ⊢ ~Q Er (N × N)
6	5	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ~Q Er (N × N))
7		nqerrel 10686	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
8	7	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐴 ~Q ([Q]‘𝐴))
9		elpqn 10679	. . . . . . . . 9 ⊢ (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
10	1, 9	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
11		mulerpqlem 10709	. . . . . . . . 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 1118	. . . . . . . 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 407	. . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵)))
15	8, 14	mpbid 231	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵))
16		nqerrel 10686	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
17	16	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 ~Q ([Q]‘𝐵))
18		elpqn 10679	. . . . . . . . . 10 ⊢ (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
19	2, 18	syl 17	. . . . . . . . 9 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
20		mulerpqlem 10709	. . . . . . . . . 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 1118	. . . . . . . . 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 507	. . . . . . 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 231	. . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 ·pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) ·pQ ([Q]‘𝐴)))
25		mulcompq 10706	. . . . . 6 ⊢ (𝐵 ·pQ ([Q]‘𝐴)) = (([Q]‘𝐴) ·pQ 𝐵)
26		mulcompq 10706	. . . . . 6 ⊢ (([Q]‘𝐵) ·pQ ([Q]‘𝐴)) = (([Q]‘𝐴) ·pQ ([Q]‘𝐵))
27	24, 25, 26	3brtr3g 5109	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)))
28	6, 15, 27	ertrd 8512	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)))
29		mulpqf 10700	. . . . . 6 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N)
30	29	fovcl 7402	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ∈ (N × N))
31	29	fovcl 7402	. . . . . 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 10689	. . . . 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 231	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵))))
36	4, 35	eqtr4d 2781	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵)))
37		0nnq 10678	. . . . . . 7 ⊢ ¬ ∅ ∈ Q
38		nqerf 10684	. . . . . . . . . . 11 ⊢ [Q]:(N × N)⟶Q
39	38	fdmi 6614	. . . . . . . . . 10 ⊢ dom [Q] = (N × N)
40	39	eleq2i 2830	. . . . . . . . 9 ⊢ (𝐴 ∈ dom [Q] ↔ 𝐴 ∈ (N × N))
41		ndmfv 6806	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ dom [Q] → ([Q]‘𝐴) = ∅)
42	40, 41	sylnbir 331	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ (N × N) → ([Q]‘𝐴) = ∅)
43	42	eleq1d 2823	. . . . . . 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 2830	. . . . . . . . 9 ⊢ (𝐵 ∈ dom [Q] ↔ 𝐵 ∈ (N × N))
47		ndmfv 6806	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ dom [Q] → ([Q]‘𝐵) = ∅)
48	46, 47	sylnbir 331	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ (N × N) → ([Q]‘𝐵) = ∅)
49	48	eleq1d 2823	. . . . . . 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		mulnqf 10703	. . . . . 6 ⊢ ·Q :(Q × Q)⟶Q
54	53	fdmi 6614	. . . . 5 ⊢ dom ·Q = (Q × Q)
55	54	ndmov 7456	. . . 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 5625	. . . . . 6 ⊢ ¬ ∅ ∈ (N × N)
58	39	eleq2i 2830	. . . . . 6 ⊢ (∅ ∈ dom [Q] ↔ ∅ ∈ (N × N))
59	57, 58	mtbir 323	. . . . 5 ⊢ ¬ ∅ ∈ dom [Q]
60	29	fdmi 6614	. . . . . . 7 ⊢ dom ·pQ = ((N × N) × (N × N))
61	60	ndmov 7456	. . . . . 6 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ∅)
62	61	eleq1d 2823	. . . . 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 6806	. . . 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 2781	. 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 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∅c0 4258   class class class wbr 5076   × cxp 5589  dom cdm 5591  ‘cfv 6435  (class class class)co 7277   Er wer 8493  Ncnpi 10598   ·_pQ cmpq 10603   ~_Q ceq 10605  Qcnq 10606  [Q]cerq 10608   ·_Q cmq 10610

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pr 5354  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5077  df-opab 5139  df-mpt 5160  df-tr 5194  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-pred 6204  df-ord 6271  df-on 6272  df-lim 6273  df-suc 6274  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-f1 6440  df-fo 6441  df-f1o 6442  df-fv 6443  df-ov 7280  df-oprab 7281  df-mpo 7282  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8095  df-wrecs 8126  df-recs 8200  df-rdg 8239  df-1o 8295  df-oadd 8299  df-omul 8300  df-er 8496  df-ni 10626  df-mi 10628  df-lti 10629  df-mpq 10663  df-enq 10665  df-nq 10666  df-erq 10667  df-mq 10669  df-1nq 10670

This theorem is referenced by:  mulassnq  10713  distrnq  10715  recmulnq  10718