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Theorem adderpqlem 9655
Description: Lemma for adderpq 9657. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
adderpqlem ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (𝐴 +pQ 𝐶) ~Q (𝐵 +pQ 𝐶)))

Proof of Theorem adderpqlem
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 7089 . . . . . 6 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
213ad2ant1 1075 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐴) ∈ N)
3 xp2nd 7090 . . . . . 6 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
433ad2ant3 1077 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐶) ∈ N)
5 mulclpi 9594 . . . . 5 (((1st𝐴) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐴) ·N (2nd𝐶)) ∈ N)
62, 4, 5syl2anc 691 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐴) ·N (2nd𝐶)) ∈ N)
7 xp1st 7089 . . . . . 6 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
873ad2ant3 1077 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐶) ∈ N)
9 xp2nd 7090 . . . . . 6 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
1093ad2ant1 1075 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐴) ∈ N)
11 mulclpi 9594 . . . . 5 (((1st𝐶) ∈ N ∧ (2nd𝐴) ∈ N) → ((1st𝐶) ·N (2nd𝐴)) ∈ N)
128, 10, 11syl2anc 691 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐶) ·N (2nd𝐴)) ∈ N)
13 addclpi 9593 . . . 4 ((((1st𝐴) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐴)) ∈ N) → (((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))) ∈ N)
146, 12, 13syl2anc 691 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))) ∈ N)
15 mulclpi 9594 . . . 4 (((2nd𝐴) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
1610, 4, 15syl2anc 691 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
17 xp1st 7089 . . . . . 6 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
18173ad2ant2 1076 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐵) ∈ N)
19 mulclpi 9594 . . . . 5 (((1st𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
2018, 4, 19syl2anc 691 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
21 xp2nd 7090 . . . . . 6 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
22213ad2ant2 1076 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐵) ∈ N)
23 mulclpi 9594 . . . . 5 (((1st𝐶) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
248, 22, 23syl2anc 691 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
25 addclpi 9593 . . . 4 ((((1st𝐵) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐵)) ∈ N) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
2620, 24, 25syl2anc 691 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
27 mulclpi 9594 . . . 4 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
2822, 4, 27syl2anc 691 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
29 enqbreq 9620 . . 3 ((((((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))) ∈ N ∧ ((2nd𝐴) ·N (2nd𝐶)) ∈ N) ∧ ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N ∧ ((2nd𝐵) ·N (2nd𝐶)) ∈ N)) → (⟨(((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐶))⟩ ~Q ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩ ↔ ((((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))))))
3014, 16, 26, 28, 29syl22anc 1319 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (⟨(((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐶))⟩ ~Q ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩ ↔ ((((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))))))
31 addpipq2 9637 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 +pQ 𝐶) = ⟨(((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐶))⟩)
32313adant2 1073 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 +pQ 𝐶) = ⟨(((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐶))⟩)
33 addpipq2 9637 . . . 4 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
34333adant1 1072 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
3532, 34breq12d 4596 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((𝐴 +pQ 𝐶) ~Q (𝐵 +pQ 𝐶) ↔ ⟨(((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐶))⟩ ~Q ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩))
36 enqbreq2 9621 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
37363adant3 1074 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
38 mulclpi 9594 . . . . 5 (((2nd𝐶) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐶) ·N (2nd𝐶)) ∈ N)
394, 4, 38syl2anc 691 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd𝐶) ·N (2nd𝐶)) ∈ N)
40 mulclpi 9594 . . . . 5 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
412, 22, 40syl2anc 691 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
42 mulcanpi 9601 . . . 4 ((((2nd𝐶) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐴) ·N (2nd𝐵)) ∈ N) → ((((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
4339, 41, 42syl2anc 691 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
44 mulcompi 9597 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((2nd𝐶) ·N (2nd𝐶)))
45 fvex 6113 . . . . . . . . 9 (1st𝐴) ∈ V
46 fvex 6113 . . . . . . . . 9 (2nd𝐵) ∈ V
47 fvex 6113 . . . . . . . . 9 (2nd𝐶) ∈ V
48 mulcompi 9597 . . . . . . . . 9 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
49 mulasspi 9598 . . . . . . . . 9 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
5045, 46, 47, 48, 49, 47caov4 6763 . . . . . . . 8 (((1st𝐴) ·N (2nd𝐵)) ·N ((2nd𝐶) ·N (2nd𝐶))) = (((1st𝐴) ·N (2nd𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
5144, 50eqtri 2632 . . . . . . 7 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
52 fvex 6113 . . . . . . . . 9 (2nd𝐴) ∈ V
53 fvex 6113 . . . . . . . . 9 (1st𝐶) ∈ V
5452, 47, 53, 48, 49, 46caov4 6763 . . . . . . . 8 (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) = (((2nd𝐴) ·N (1st𝐶)) ·N ((2nd𝐶) ·N (2nd𝐵)))
55 mulcompi 9597 . . . . . . . . 9 ((2nd𝐴) ·N (1st𝐶)) = ((1st𝐶) ·N (2nd𝐴))
56 mulcompi 9597 . . . . . . . . 9 ((2nd𝐶) ·N (2nd𝐵)) = ((2nd𝐵) ·N (2nd𝐶))
5755, 56oveq12i 6561 . . . . . . . 8 (((2nd𝐴) ·N (1st𝐶)) ·N ((2nd𝐶) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐵) ·N (2nd𝐶)))
5854, 57eqtri 2632 . . . . . . 7 (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐵) ·N (2nd𝐶)))
5951, 58oveq12i 6561 . . . . . 6 ((((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) +N (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵)))) = ((((1st𝐴) ·N (2nd𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶))) +N (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐵) ·N (2nd𝐶))))
60 addcompi 9595 . . . . . 6 ((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) = ((((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) +N (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))))
61 ovex 6577 . . . . . . 7 ((1st𝐴) ·N (2nd𝐶)) ∈ V
62 ovex 6577 . . . . . . 7 ((1st𝐶) ·N (2nd𝐴)) ∈ V
63 ovex 6577 . . . . . . 7 ((2nd𝐵) ·N (2nd𝐶)) ∈ V
64 distrpi 9599 . . . . . . 7 (𝑥 ·N (𝑦 +N 𝑧)) = ((𝑥 ·N 𝑦) +N (𝑥 ·N 𝑧))
6561, 62, 63, 48, 64caovdir 6766 . . . . . 6 ((((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))) ·N ((2nd𝐵) ·N (2nd𝐶))) = ((((1st𝐴) ·N (2nd𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶))) +N (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐵) ·N (2nd𝐶))))
6659, 60, 653eqtr4i 2642 . . . . 5 ((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) = ((((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))) ·N ((2nd𝐵) ·N (2nd𝐶)))
67 addcompi 9595 . . . . . 6 ((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶)))) = ((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶))) +N (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))))
68 mulasspi 9598 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))))
69 mulcompi 9597 . . . . . . . . . 10 ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))) = (((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶))
70 mulasspi 9598 . . . . . . . . . . . 12 (((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) = ((2nd𝐴) ·N ((2nd𝐶) ·N (1st𝐵)))
71 mulcompi 9597 . . . . . . . . . . . 12 ((2nd𝐴) ·N ((2nd𝐶) ·N (1st𝐵))) = (((2nd𝐶) ·N (1st𝐵)) ·N (2nd𝐴))
72 mulasspi 9598 . . . . . . . . . . . 12 (((2nd𝐶) ·N (1st𝐵)) ·N (2nd𝐴)) = ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))
7370, 71, 723eqtrri 2637 . . . . . . . . . . 11 ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))) = (((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵))
7473oveq1i 6559 . . . . . . . . . 10 (((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) = ((((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) ·N (2nd𝐶))
7569, 74eqtri 2632 . . . . . . . . 9 ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))) = ((((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) ·N (2nd𝐶))
76 mulasspi 9598 . . . . . . . . 9 ((((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) ·N (2nd𝐶)) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶)))
7775, 76eqtri 2632 . . . . . . . 8 ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶)))
7868, 77eqtri 2632 . . . . . . 7 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶)))
7978oveq2i 6560 . . . . . 6 ((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))) = ((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶))))
80 distrpi 9599 . . . . . 6 (((2nd𝐴) ·N (2nd𝐶)) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))) = ((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶))) +N (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))))
8167, 79, 803eqtr4i 2642 . . . . 5 ((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))) = (((2nd𝐴) ·N (2nd𝐶)) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))))
8266, 81eqeq12i 2624 . . . 4 (((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) = ((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))) ↔ ((((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))))
83 mulclpi 9594 . . . . . 6 ((((2nd𝐴) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐵)) ∈ N) → (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
8416, 24, 83syl2anc 691 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
85 mulclpi 9594 . . . . . 6 ((((2nd𝐶) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐴) ·N (2nd𝐵)) ∈ N) → (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) ∈ N)
8639, 41, 85syl2anc 691 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) ∈ N)
87 addcanpi 9600 . . . . 5 (((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) ∈ N ∧ (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) ∈ N) → (((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) = ((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))) ↔ (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))))
8884, 86, 87syl2anc 691 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) = ((((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))) ↔ (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))))
8982, 88syl5rbbr 274 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ ((((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))))))
9037, 43, 893bitr2d 295 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))))))
9130, 35, 903bitr4rd 300 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 195  w3a 1031   = wceq 1475  wcel 1977  cop 4131   class class class wbr 4583   × cxp 5036  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Ncnpi 9545   +N cpli 9546   ·N cmi 9547   +pQ cplpq 9549   ~Q ceq 9552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452  df-ni 9573  df-pli 9574  df-mi 9575  df-plpq 9609  df-enq 9612
This theorem is referenced by:  adderpq  9657
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