Theorem mulerpqlem 10878

Description: Lemma for mulerpq 10880. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
mulerpqlem	⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (𝐴 ·pQ 𝐶) ~Q (𝐵 ·pQ 𝐶)))

Proof of Theorem mulerpqlem

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xp1st 7974	. . . . 5 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
2	1	3ad2ant1 1134	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐴) ∈ N)
3		xp1st 7974	. . . . 5 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
4	3	3ad2ant3 1136	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐶) ∈ N)
5		mulclpi 10816	. . . 4 ⊢ (((1st ‘𝐴) ∈ N ∧ (1st ‘𝐶) ∈ N) → ((1st ‘𝐴) ·N (1st ‘𝐶)) ∈ N)
6	2, 4, 5	syl2anc 585	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐴) ·N (1st ‘𝐶)) ∈ N)
7		xp2nd 7975	. . . . 5 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
8	7	3ad2ant1 1134	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐴) ∈ N)
9		xp2nd 7975	. . . . 5 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
10	9	3ad2ant3 1136	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐶) ∈ N)
11		mulclpi 10816	. . . 4 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐶)) ∈ N)
12	8, 10, 11	syl2anc 585	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd ‘𝐴) ·N (2nd ‘𝐶)) ∈ N)
13		xp1st 7974	. . . . 5 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
14	13	3ad2ant2 1135	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐵) ∈ N)
15		mulclpi 10816	. . . 4 ⊢ (((1st ‘𝐵) ∈ N ∧ (1st ‘𝐶) ∈ N) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
16	14, 4, 15	syl2anc 585	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
17		xp2nd 7975	. . . . 5 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
18	17	3ad2ant2 1135	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐵) ∈ N)
19		mulclpi 10816	. . . 4 ⊢ (((2nd ‘𝐵) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
20	18, 10, 19	syl2anc 585	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
21		enqbreq 10842	. . 3 ⊢ (((((1st ‘𝐴) ·N (1st ‘𝐶)) ∈ N ∧ ((2nd ‘𝐴) ·N (2nd ‘𝐶)) ∈ N) ∧ (((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N ∧ ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)) → (⟨((1st ‘𝐴) ·N (1st ‘𝐶)), ((2nd ‘𝐴) ·N (2nd ‘𝐶))⟩ ~Q ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩ ↔ (((1st ‘𝐴) ·N (1st ‘𝐶)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (1st ‘𝐶)))))
22	6, 12, 16, 20, 21	syl22anc 839	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (⟨((1st ‘𝐴) ·N (1st ‘𝐶)), ((2nd ‘𝐴) ·N (2nd ‘𝐶))⟩ ~Q ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩ ↔ (((1st ‘𝐴) ·N (1st ‘𝐶)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (1st ‘𝐶)))))
23		mulpipq2 10862	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st ‘𝐴) ·N (1st ‘𝐶)), ((2nd ‘𝐴) ·N (2nd ‘𝐶))⟩)
24	23	3adant2 1132	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st ‘𝐴) ·N (1st ‘𝐶)), ((2nd ‘𝐴) ·N (2nd ‘𝐶))⟩)
25		mulpipq2 10862	. . . 4 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
26	25	3adant1 1131	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
27	24, 26	breq12d 5098	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((𝐴 ·pQ 𝐶) ~Q (𝐵 ·pQ 𝐶) ↔ ⟨((1st ‘𝐴) ·N (1st ‘𝐶)), ((2nd ‘𝐴) ·N (2nd ‘𝐶))⟩ ~Q ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩))
28		enqbreq2 10843	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
29	28	3adant3 1133	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
30		mulclpi 10816	. . . . 5 ⊢ (((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
31	4, 10, 30	syl2anc 585	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
32		mulclpi 10816	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
33	2, 18, 32	syl2anc 585	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
34		mulcanpi 10823	. . . 4 ⊢ ((((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N ∧ ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N) → ((((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
35	31, 33, 34	syl2anc 585	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
36		mulcompi 10819	. . . . . 6 ⊢ (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐶)))
37		fvex 6853	. . . . . . 7 ⊢ (1st ‘𝐴) ∈ V
38		fvex 6853	. . . . . . 7 ⊢ (2nd ‘𝐵) ∈ V
39		fvex 6853	. . . . . . 7 ⊢ (1st ‘𝐶) ∈ V
40		mulcompi 10819	. . . . . . 7 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
41		mulasspi 10820	. . . . . . 7 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
42		fvex 6853	. . . . . . 7 ⊢ (2nd ‘𝐶) ∈ V
43	37, 38, 39, 40, 41, 42	caov4 7598	. . . . . 6 ⊢ (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐶))) = (((1st ‘𝐴) ·N (1st ‘𝐶)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))
44	36, 43	eqtri 2759	. . . . 5 ⊢ (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((1st ‘𝐴) ·N (1st ‘𝐶)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))
45		mulcompi 10819	. . . . . 6 ⊢ (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) = (((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐶)))
46		fvex 6853	. . . . . . 7 ⊢ (1st ‘𝐵) ∈ V
47		fvex 6853	. . . . . . 7 ⊢ (2nd ‘𝐴) ∈ V
48	46, 47, 39, 40, 41, 42	caov4 7598	. . . . . 6 ⊢ (((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐶))) = (((1st ‘𝐵) ·N (1st ‘𝐶)) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐶)))
49		mulcompi 10819	. . . . . 6 ⊢ (((1st ‘𝐵) ·N (1st ‘𝐶)) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐶))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (1st ‘𝐶)))
50	45, 48, 49	3eqtri 2763	. . . . 5 ⊢ (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (1st ‘𝐶)))
51	44, 50	eqeq12i 2754	. . . 4 ⊢ ((((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ↔ (((1st ‘𝐴) ·N (1st ‘𝐶)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (1st ‘𝐶))))
52	51	a1i 11	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ↔ (((1st ‘𝐴) ·N (1st ‘𝐶)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (1st ‘𝐶)))))
53	29, 35, 52	3bitr2d 307	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (((1st ‘𝐴) ·N (1st ‘𝐶)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (1st ‘𝐶)))))
54	22, 27, 53	3bitr4rd 312	1 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (𝐴 ·pQ 𝐶) ~Q (𝐵 ·pQ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ⟨cop 4573   class class class wbr 5085   × cxp 5629  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  Ncnpi 10767   ·_N cmi 10769   ·_pQ cmpq 10772   ~_Q ceq 10774

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 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-oadd 8409  df-omul 8410  df-ni 10795  df-mi 10797  df-mpq 10832  df-enq 10834

This theorem is referenced by:  mulerpq  10880