Theorem mulerpq 10368

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 10342	. . . 4 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
2		nqercl 10342	. . . 4 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
3		mulpqnq 10352	. . . 4 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵))))
4	1, 2, 3	syl2an 598	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵))))
5		enqer 10332	. . . . . 6 ⊢ ~Q Er (N × N)
6	5	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ~Q Er (N × N))
7		nqerrel 10343	. . . . . . 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 10336	. . . . . . . . 9 ⊢ (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
10	1, 9	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
11		mulerpqlem 10366	. . . . . . . . 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 1116	. . . . . . . 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 235	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ 𝐵))
16		nqerrel 10343	. . . . . . . 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 10336	. . . . . . . . . 10 ⊢ (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
19	2, 18	syl 17	. . . . . . . . 9 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
20		mulerpqlem 10366	. . . . . . . . . 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 1116	. . . . . . . . 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 510	. . . . . . 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 235	. . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 ·pQ ([Q]‘𝐴)) ~Q (([Q]‘𝐵) ·pQ ([Q]‘𝐴)))
25		mulcompq 10363	. . . . . 6 ⊢ (𝐵 ·pQ ([Q]‘𝐴)) = (([Q]‘𝐴) ·pQ 𝐵)
26		mulcompq 10363	. . . . . 6 ⊢ (([Q]‘𝐵) ·pQ ([Q]‘𝐴)) = (([Q]‘𝐴) ·pQ ([Q]‘𝐵))
27	24, 25, 26	3brtr3g 5063	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)))
28	6, 15, 27	ertrd 8288	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ~Q (([Q]‘𝐴) ·pQ ([Q]‘𝐵)))
29		mulpqf 10357	. . . . . 6 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N)
30	29	fovcl 7258	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ∈ (N × N))
31	29	fovcl 7258	. . . . . 6 ⊢ ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (([Q]‘𝐴) ·pQ ([Q]‘𝐵)) ∈ (N × N))
32	10, 19, 31	syl2an 598	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·pQ ([Q]‘𝐵)) ∈ (N × N))
33		nqereq 10346	. . . . 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 587	. . . 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 235	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘(([Q]‘𝐴) ·pQ ([Q]‘𝐵))))
36	4, 35	eqtr4d 2836	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵)))
37		0nnq 10335	. . . . . . 7 ⊢ ¬ ∅ ∈ Q
38		nqerf 10341	. . . . . . . . . . 11 ⊢ [Q]:(N × N)⟶Q
39	38	fdmi 6498	. . . . . . . . . 10 ⊢ dom [Q] = (N × N)
40	39	eleq2i 2881	. . . . . . . . 9 ⊢ (𝐴 ∈ dom [Q] ↔ 𝐴 ∈ (N × N))
41		ndmfv 6675	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ dom [Q] → ([Q]‘𝐴) = ∅)
42	40, 41	sylnbir 334	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ (N × N) → ([Q]‘𝐴) = ∅)
43	42	eleq1d 2874	. . . . . . 7 ⊢ (¬ 𝐴 ∈ (N × N) → (([Q]‘𝐴) ∈ Q ↔ ∅ ∈ Q))
44	37, 43	mtbiri 330	. . . . . 6 ⊢ (¬ 𝐴 ∈ (N × N) → ¬ ([Q]‘𝐴) ∈ Q)
45	44	con4i 114	. . . . 5 ⊢ (([Q]‘𝐴) ∈ Q → 𝐴 ∈ (N × N))
46	39	eleq2i 2881	. . . . . . . . 9 ⊢ (𝐵 ∈ dom [Q] ↔ 𝐵 ∈ (N × N))
47		ndmfv 6675	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ dom [Q] → ([Q]‘𝐵) = ∅)
48	46, 47	sylnbir 334	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ (N × N) → ([Q]‘𝐵) = ∅)
49	48	eleq1d 2874	. . . . . . 7 ⊢ (¬ 𝐵 ∈ (N × N) → (([Q]‘𝐵) ∈ Q ↔ ∅ ∈ Q))
50	37, 49	mtbiri 330	. . . . . 6 ⊢ (¬ 𝐵 ∈ (N × N) → ¬ ([Q]‘𝐵) ∈ Q)
51	50	con4i 114	. . . . 5 ⊢ (([Q]‘𝐵) ∈ Q → 𝐵 ∈ (N × N))
52	45, 51	anim12i 615	. . . 4 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
53		mulnqf 10360	. . . . . 6 ⊢ ·Q :(Q × Q)⟶Q
54	53	fdmi 6498	. . . . 5 ⊢ dom ·Q = (Q × Q)
55	54	ndmov 7312	. . . 4 ⊢ (¬ (([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ∅)
56	52, 55	nsyl5 162	. . 3 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ∅)
57		0nelxp 5553	. . . . . 6 ⊢ ¬ ∅ ∈ (N × N)
58	39	eleq2i 2881	. . . . . 6 ⊢ (∅ ∈ dom [Q] ↔ ∅ ∈ (N × N))
59	57, 58	mtbir 326	. . . . 5 ⊢ ¬ ∅ ∈ dom [Q]
60	29	fdmi 6498	. . . . . . 7 ⊢ dom ·pQ = ((N × N) × (N × N))
61	60	ndmov 7312	. . . . . 6 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ∅)
62	61	eleq1d 2874	. . . . 5 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((𝐴 ·pQ 𝐵) ∈ dom [Q] ↔ ∅ ∈ dom [Q]))
63	59, 62	mtbiri 330	. . . 4 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ¬ (𝐴 ·pQ 𝐵) ∈ dom [Q])
64		ndmfv 6675	. . . 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 2836	. 2 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵)))
67	36, 66	pm2.61i 185	1 ⊢ (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∅c0 4243   class class class wbr 5030   × cxp 5517  dom cdm 5519  ‘cfv 6324  (class class class)co 7135   Er wer 8269  Ncnpi 10255   ·_pQ cmpq 10260   ~_Q ceq 10262  Qcnq 10263  [Q]cerq 10265   ·_Q cmq 10267

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-omul 8090  df-er 8272  df-ni 10283  df-mi 10285  df-lti 10286  df-mpq 10320  df-enq 10322  df-nq 10323  df-erq 10324  df-mq 10326  df-1nq 10327

This theorem is referenced by:  mulassnq  10370  distrnq  10372  recmulnq  10375