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Theorem mulerpqlem 9729
Description: Lemma for mulerpq 9731. (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 7150 . . . . 5 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
213ad2ant1 1080 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐴) ∈ N)
3 xp1st 7150 . . . . 5 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
433ad2ant3 1082 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐶) ∈ N)
5 mulclpi 9667 . . . 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 7151 . . . . 5 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
873ad2ant1 1080 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐴) ∈ N)
9 xp2nd 7151 . . . . 5 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
1093ad2ant3 1082 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐶) ∈ N)
11 mulclpi 9667 . . . 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 7150 . . . . 5 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
14133ad2ant2 1081 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐵) ∈ N)
15 mulclpi 9667 . . . 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 7151 . . . . 5 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
18173ad2ant2 1081 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐵) ∈ N)
19 mulclpi 9667 . . . 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 9693 . . 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 9713 . . . 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 9713 . . . 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 4631 . 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 9694 . . . 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 9667 . . . . 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 9667 . . . . 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 9674 . . . 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 9670 . . . . . 6 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐶)))
37 fvex 6163 . . . . . . 7 (1st𝐴) ∈ V
38 fvex 6163 . . . . . . 7 (2nd𝐵) ∈ V
39 fvex 6163 . . . . . . 7 (1st𝐶) ∈ V
40 mulcompi 9670 . . . . . . 7 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
41 mulasspi 9671 . . . . . . 7 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
42 fvex 6163 . . . . . . 7 (2nd𝐶) ∈ V
4337, 38, 39, 40, 41, 42caov4 6825 . . . . . 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 9670 . . . . . 6 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐵) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐶)))
46 fvex 6163 . . . . . . 7 (1st𝐵) ∈ V
47 fvex 6163 . . . . . . 7 (2nd𝐴) ∈ V
4846, 47, 39, 40, 41, 42caov4 6825 . . . . . 6 (((1st𝐵) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐶))) = (((1st𝐵) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐶)))
49 mulcompi 9670 . . . . . 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 4159   class class class wbr 4618   × cxp 5077  cfv 5852  (class class class)co 6610  1st c1st 7118  2nd c2nd 7119  Ncnpi 9618   ·N cmi 9620   ·pQ cmpq 9623   ~Q ceq 9625
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 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-oadd 7516  df-omul 7517  df-ni 9646  df-mi 9648  df-mpq 9683  df-enq 9685
This theorem is referenced by:  mulerpq  9731
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