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Theorem mulerpqlem 9737
Description: Lemma for mulerpq 9739. (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.
StepHypRef Expression
1 xp1st 7158 . . . . 5 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
213ad2ant1 1080 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐴) ∈ N)
3 xp1st 7158 . . . . 5 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
433ad2ant3 1082 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐶) ∈ N)
5 mulclpi 9675 . . . 4 (((1st𝐴) ∈ N ∧ (1st𝐶) ∈ N) → ((1st𝐴) ·N (1st𝐶)) ∈ N)
62, 4, 5syl2anc 692 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐴) ·N (1st𝐶)) ∈ N)
7 xp2nd 7159 . . . . 5 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
873ad2ant1 1080 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐴) ∈ N)
9 xp2nd 7159 . . . . 5 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
1093ad2ant3 1082 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐶) ∈ N)
11 mulclpi 9675 . . . 4 (((2nd𝐴) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
128, 10, 11syl2anc 692 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
13 xp1st 7158 . . . . 5 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
14133ad2ant2 1081 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐵) ∈ N)
15 mulclpi 9675 . . . 4 (((1st𝐵) ∈ N ∧ (1st𝐶) ∈ N) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
1614, 4, 15syl2anc 692 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
17 xp2nd 7159 . . . . 5 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
18173ad2ant2 1081 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐵) ∈ N)
19 mulclpi 9675 . . . 4 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
2018, 10, 19syl2anc 692 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
21 enqbreq 9701 . . 3 (((((1st𝐴) ·N (1st𝐶)) ∈ N ∧ ((2nd𝐴) ·N (2nd𝐶)) ∈ N) ∧ (((1st𝐵) ·N (1st𝐶)) ∈ N ∧ ((2nd𝐵) ·N (2nd𝐶)) ∈ N)) → (⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩ ~Q ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩ ↔ (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))))
226, 12, 16, 20, 21syl22anc 1324 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩ ~Q ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩ ↔ (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))))
23 mulpipq2 9721 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩)
24233adant2 1078 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩)
25 mulpipq2 9721 . . . 4 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
26253adant1 1077 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
2724, 26breq12d 4636 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((𝐴 ·pQ 𝐶) ~Q (𝐵 ·pQ 𝐶) ↔ ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩ ~Q ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩))
28 enqbreq2 9702 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
29283adant3 1079 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
30 mulclpi 9675 . . . . 5 (((1st𝐶) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
314, 10, 30syl2anc 692 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
32 mulclpi 9675 . . . . 5 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
332, 18, 32syl2anc 692 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
34 mulcanpi 9682 . . . 4 ((((1st𝐶) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐴) ·N (2nd𝐵)) ∈ N) → ((((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
3531, 33, 34syl2anc 692 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
36 mulcompi 9678 . . . . . 6 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐶)))
37 fvex 6168 . . . . . . 7 (1st𝐴) ∈ V
38 fvex 6168 . . . . . . 7 (2nd𝐵) ∈ V
39 fvex 6168 . . . . . . 7 (1st𝐶) ∈ V
40 mulcompi 9678 . . . . . . 7 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
41 mulasspi 9679 . . . . . . 7 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
42 fvex 6168 . . . . . . 7 (2nd𝐶) ∈ V
4337, 38, 39, 40, 41, 42caov4 6830 . . . . . 6 (((1st𝐴) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐶))) = (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
4436, 43eqtri 2643 . . . . 5 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
45 mulcompi 9678 . . . . . 6 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐵) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐶)))
46 fvex 6168 . . . . . . 7 (1st𝐵) ∈ V
47 fvex 6168 . . . . . . 7 (2nd𝐴) ∈ V
4846, 47, 39, 40, 41, 42caov4 6830 . . . . . 6 (((1st𝐵) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐶))) = (((1st𝐵) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐶)))
49 mulcompi 9678 . . . . . 6 (((1st𝐵) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))
5045, 48, 493eqtri 2647 . . . . 5 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))
5144, 50eqeq12i 2635 . . . 4 ((((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶))))
5251a1i 11 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))))
5329, 35, 523bitr2d 296 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))))
5422, 27, 533bitr4rd 301 1 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (𝐴 ·pQ 𝐶) ~Q (𝐵 ·pQ 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1480  wcel 1987  cop 4161   class class class wbr 4623   × cxp 5082  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  Ncnpi 9626   ·N cmi 9628   ·pQ cmpq 9631   ~Q ceq 9633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-oadd 7524  df-omul 7525  df-ni 9654  df-mi 9656  df-mpq 9691  df-enq 9693
This theorem is referenced by:  mulerpq  9739
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