Theorem mulerpqlem 10380

Description: Lemma for mulerpq 10382. (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 7724	. . . . 5 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
2	1	3ad2ant1 1129	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐴) ∈ N)
3		xp1st 7724	. . . . 5 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
4	3	3ad2ant3 1131	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐶) ∈ N)
5		mulclpi 10318	. . . 4 ⊢ (((1st ‘𝐴) ∈ N ∧ (1st ‘𝐶) ∈ N) → ((1st ‘𝐴) ·N (1st ‘𝐶)) ∈ N)
6	2, 4, 5	syl2anc 586	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐴) ·N (1st ‘𝐶)) ∈ N)
7		xp2nd 7725	. . . . 5 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
8	7	3ad2ant1 1129	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐴) ∈ N)
9		xp2nd 7725	. . . . 5 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
10	9	3ad2ant3 1131	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐶) ∈ N)
11		mulclpi 10318	. . . 4 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐶)) ∈ N)
12	8, 10, 11	syl2anc 586	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd ‘𝐴) ·N (2nd ‘𝐶)) ∈ N)
13		xp1st 7724	. . . . 5 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
14	13	3ad2ant2 1130	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐵) ∈ N)
15		mulclpi 10318	. . . 4 ⊢ (((1st ‘𝐵) ∈ N ∧ (1st ‘𝐶) ∈ N) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
16	14, 4, 15	syl2anc 586	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
17		xp2nd 7725	. . . . 5 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
18	17	3ad2ant2 1130	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐵) ∈ N)
19		mulclpi 10318	. . . 4 ⊢ (((2nd ‘𝐵) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
20	18, 10, 19	syl2anc 586	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
21		enqbreq 10344	. . 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 836	. 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 10364	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st ‘𝐴) ·N (1st ‘𝐶)), ((2nd ‘𝐴) ·N (2nd ‘𝐶))⟩)
24	23	3adant2 1127	. . 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 10364	. . . 4 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
26	25	3adant1 1126	. . 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 5082	. 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 10345	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
29	28	3adant3 1128	. . 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 10318	. . . . 5 ⊢ (((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
31	4, 10, 30	syl2anc 586	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
32		mulclpi 10318	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
33	2, 18, 32	syl2anc 586	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
34		mulcanpi 10325	. . . 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 586	. . 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 10321	. . . . . 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 6686	. . . . . . 7 ⊢ (1st ‘𝐴) ∈ V
38		fvex 6686	. . . . . . 7 ⊢ (2nd ‘𝐵) ∈ V
39		fvex 6686	. . . . . . 7 ⊢ (1st ‘𝐶) ∈ V
40		mulcompi 10321	. . . . . . 7 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
41		mulasspi 10322	. . . . . . 7 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
42		fvex 6686	. . . . . . 7 ⊢ (2nd ‘𝐶) ∈ V
43	37, 38, 39, 40, 41, 42	caov4 7382	. . . . . 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 2847	. . . . 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 10321	. . . . . 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 6686	. . . . . . 7 ⊢ (1st ‘𝐵) ∈ V
47		fvex 6686	. . . . . . 7 ⊢ (2nd ‘𝐴) ∈ V
48	46, 47, 39, 40, 41, 42	caov4 7382	. . . . . 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 10321	. . . . . 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 2851	. . . . 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 2839	. . . 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 309	. 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 314	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 208   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ⟨cop 4576   class class class wbr 5069   × cxp 5556  ‘cfv 6358  (class class class)co 7159  1^st c1st 7690  2^nd c2nd 7691  Ncnpi 10269   ·_N cmi 10271   ·_pQ cmpq 10274   ~_Q ceq 10276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-oadd 8109  df-omul 8110  df-ni 10297  df-mi 10299  df-mpq 10334  df-enq 10336

This theorem is referenced by:  mulerpq  10382