Theorem mulerpqlem 10998

Description: Lemma for mulerpq 11000. (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 8035	. . . . 5 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
2	1	3ad2ant1 1130	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐴) ∈ N)
3		xp1st 8035	. . . . 5 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
4	3	3ad2ant3 1132	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐶) ∈ N)
5		mulclpi 10936	. . . 4 ⊢ (((1st ‘𝐴) ∈ N ∧ (1st ‘𝐶) ∈ N) → ((1st ‘𝐴) ·N (1st ‘𝐶)) ∈ N)
6	2, 4, 5	syl2anc 582	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐴) ·N (1st ‘𝐶)) ∈ N)
7		xp2nd 8036	. . . . 5 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
8	7	3ad2ant1 1130	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐴) ∈ N)
9		xp2nd 8036	. . . . 5 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
10	9	3ad2ant3 1132	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐶) ∈ N)
11		mulclpi 10936	. . . 4 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐶)) ∈ N)
12	8, 10, 11	syl2anc 582	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd ‘𝐴) ·N (2nd ‘𝐶)) ∈ N)
13		xp1st 8035	. . . . 5 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
14	13	3ad2ant2 1131	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐵) ∈ N)
15		mulclpi 10936	. . . 4 ⊢ (((1st ‘𝐵) ∈ N ∧ (1st ‘𝐶) ∈ N) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
16	14, 4, 15	syl2anc 582	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
17		xp2nd 8036	. . . . 5 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
18	17	3ad2ant2 1131	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐵) ∈ N)
19		mulclpi 10936	. . . 4 ⊢ (((2nd ‘𝐵) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
20	18, 10, 19	syl2anc 582	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
21		enqbreq 10962	. . 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 837	. 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 10982	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st ‘𝐴) ·N (1st ‘𝐶)), ((2nd ‘𝐴) ·N (2nd ‘𝐶))⟩)
24	23	3adant2 1128	. . 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 10982	. . . 4 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
26	25	3adant1 1127	. . 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 5166	. 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 10963	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
29	28	3adant3 1129	. . 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 10936	. . . . 5 ⊢ (((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
31	4, 10, 30	syl2anc 582	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
32		mulclpi 10936	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
33	2, 18, 32	syl2anc 582	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
34		mulcanpi 10943	. . . 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 582	. . 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 10939	. . . . . 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 6914	. . . . . . 7 ⊢ (1st ‘𝐴) ∈ V
38		fvex 6914	. . . . . . 7 ⊢ (2nd ‘𝐵) ∈ V
39		fvex 6914	. . . . . . 7 ⊢ (1st ‘𝐶) ∈ V
40		mulcompi 10939	. . . . . . 7 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
41		mulasspi 10940	. . . . . . 7 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
42		fvex 6914	. . . . . . 7 ⊢ (2nd ‘𝐶) ∈ V
43	37, 38, 39, 40, 41, 42	caov4 7657	. . . . . 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 2754	. . . . 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 10939	. . . . . 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 6914	. . . . . . 7 ⊢ (1st ‘𝐵) ∈ V
47		fvex 6914	. . . . . . 7 ⊢ (2nd ‘𝐴) ∈ V
48	46, 47, 39, 40, 41, 42	caov4 7657	. . . . . 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 10939	. . . . . 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 2758	. . . . 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 2744	. . . 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 306	. 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 311	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 205   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  ⟨cop 4639   class class class wbr 5153   × cxp 5680  ‘cfv 6554  (class class class)co 7424  1^st c1st 8001  2^nd c2nd 8002  Ncnpi 10887   ·_N cmi 10889   ·_pQ cmpq 10892   ~_Q ceq 10894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-oadd 8500  df-omul 8501  df-ni 10915  df-mi 10917  df-mpq 10952  df-enq 10954

This theorem is referenced by:  mulerpq  11000