Theorem mulerpqlem 10928

Description: Lemma for mulerpq 10930. (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 8006	. . . . 5 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
2	1	3ad2ant1 1149	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐴) ∈ N)
3		xp1st 8006	. . . . 5 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
4	3	3ad2ant3 1151	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐶) ∈ N)
5		mulclpi 10866	. . . 4 ⊢ (((1st ‘𝐴) ∈ N ∧ (1st ‘𝐶) ∈ N) → ((1st ‘𝐴) ·N (1st ‘𝐶)) ∈ N)
6	2, 4, 5	syl2anc 595	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐴) ·N (1st ‘𝐶)) ∈ N)
7		xp2nd 8007	. . . . 5 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
8	7	3ad2ant1 1149	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐴) ∈ N)
9		xp2nd 8007	. . . . 5 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
10	9	3ad2ant3 1151	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐶) ∈ N)
11		mulclpi 10866	. . . 4 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐶)) ∈ N)
12	8, 10, 11	syl2anc 595	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd ‘𝐴) ·N (2nd ‘𝐶)) ∈ N)
13		xp1st 8006	. . . . 5 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
14	13	3ad2ant2 1150	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐵) ∈ N)
15		mulclpi 10866	. . . 4 ⊢ (((1st ‘𝐵) ∈ N ∧ (1st ‘𝐶) ∈ N) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
16	14, 4, 15	syl2anc 595	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
17		xp2nd 8007	. . . . 5 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
18	17	3ad2ant2 1150	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐵) ∈ N)
19		mulclpi 10866	. . . 4 ⊢ (((2nd ‘𝐵) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
20	18, 10, 19	syl2anc 595	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
21		enqbreq 10892	. . 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 851	. 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 10912	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st ‘𝐴) ·N (1st ‘𝐶)), ((2nd ‘𝐴) ·N (2nd ‘𝐶))⟩)
24	23	3adant2 1147	. . 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 10912	. . . 4 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
26	25	3adant1 1146	. . 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 5117	. 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 10893	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
29	28	3adant3 1148	. . 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 10866	. . . . 5 ⊢ (((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
31	4, 10, 30	syl2anc 595	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
32		mulclpi 10866	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
33	2, 18, 32	syl2anc 595	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
34		mulcanpi 10873	. . . 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 595	. . 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 10869	. . . . . 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 6884	. . . . . . 7 ⊢ (1st ‘𝐴) ∈ V
38		fvex 6884	. . . . . . 7 ⊢ (2nd ‘𝐵) ∈ V
39		fvex 6884	. . . . . . 7 ⊢ (1st ‘𝐶) ∈ V
40		mulcompi 10869	. . . . . . 7 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
41		mulasspi 10870	. . . . . . 7 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
42		fvex 6884	. . . . . . 7 ⊢ (2nd ‘𝐶) ∈ V
43	37, 38, 39, 40, 41, 42	caov4 7631	. . . . . 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 2788	. . . . 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 10869	. . . . . 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 6884	. . . . . . 7 ⊢ (1st ‘𝐵) ∈ V
47		fvex 6884	. . . . . . 7 ⊢ (2nd ‘𝐴) ∈ V
48	46, 47, 39, 40, 41, 42	caov4 7631	. . . . . 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 10869	. . . . . 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 2792	. . . . 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 2783	. . . 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 310	. 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 315	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 209   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ⟨cop 4591   class class class wbr 5104   × cxp 5649  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  Ncnpi 10817   ·_N cmi 10819   ·_pQ cmpq 10822   ~_Q ceq 10824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-oadd 8445  df-omul 8446  df-ni 10845  df-mi 10847  df-mpq 10882  df-enq 10884

This theorem is referenced by:  mulerpq  10930