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Theorem adderpqlem 10222
Description: Lemma for adderpq 10224. (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 7577 . . . . . 6 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
213ad2ant1 1126 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐴) ∈ N)
3 xp2nd 7578 . . . . . 6 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
433ad2ant3 1128 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐶) ∈ N)
5 mulclpi 10161 . . . . 5 (((1st𝐴) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐴) ·N (2nd𝐶)) ∈ N)
62, 4, 5syl2anc 584 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐴) ·N (2nd𝐶)) ∈ N)
7 xp1st 7577 . . . . . 6 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
873ad2ant3 1128 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐶) ∈ N)
9 xp2nd 7578 . . . . . 6 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
1093ad2ant1 1126 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐴) ∈ N)
11 mulclpi 10161 . . . . 5 (((1st𝐶) ∈ N ∧ (2nd𝐴) ∈ N) → ((1st𝐶) ·N (2nd𝐴)) ∈ N)
128, 10, 11syl2anc 584 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐶) ·N (2nd𝐴)) ∈ N)
13 addclpi 10160 . . . 4 ((((1st𝐴) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐴)) ∈ N) → (((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))) ∈ N)
146, 12, 13syl2anc 584 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))) ∈ N)
15 mulclpi 10161 . . . 4 (((2nd𝐴) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
1610, 4, 15syl2anc 584 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
17 xp1st 7577 . . . . . 6 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
18173ad2ant2 1127 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐵) ∈ N)
19 mulclpi 10161 . . . . 5 (((1st𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
2018, 4, 19syl2anc 584 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
21 xp2nd 7578 . . . . . 6 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
22213ad2ant2 1127 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐵) ∈ N)
23 mulclpi 10161 . . . . 5 (((1st𝐶) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
248, 22, 23syl2anc 584 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
25 addclpi 10160 . . . 4 ((((1st𝐵) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐵)) ∈ N) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
2620, 24, 25syl2anc 584 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
27 mulclpi 10161 . . . 4 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
2822, 4, 27syl2anc 584 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
29 enqbreq 10187 . . 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 835 . 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 10204 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 +pQ 𝐶) = ⟨(((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐶))⟩)
32313adant2 1124 . . 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 10204 . . . 4 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
34333adant1 1123 . . 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 4975 . 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 10188 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
37363adant3 1125 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
38 mulclpi 10161 . . . . 5 (((2nd𝐶) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐶) ·N (2nd𝐶)) ∈ N)
394, 4, 38syl2anc 584 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd𝐶) ·N (2nd𝐶)) ∈ N)
40 mulclpi 10161 . . . . 5 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
412, 22, 40syl2anc 584 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
42 mulcanpi 10168 . . . 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 584 . . 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 10164 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((2nd𝐶) ·N (2nd𝐶)))
45 fvex 6551 . . . . . . . . 9 (1st𝐴) ∈ V
46 fvex 6551 . . . . . . . . 9 (2nd𝐵) ∈ V
47 fvex 6551 . . . . . . . . 9 (2nd𝐶) ∈ V
48 mulcompi 10164 . . . . . . . . 9 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
49 mulasspi 10165 . . . . . . . . 9 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
5045, 46, 47, 48, 49, 47caov4 7235 . . . . . . . 8 (((1st𝐴) ·N (2nd𝐵)) ·N ((2nd𝐶) ·N (2nd𝐶))) = (((1st𝐴) ·N (2nd𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
5144, 50eqtri 2819 . . . . . . 7 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
52 fvex 6551 . . . . . . . . 9 (2nd𝐴) ∈ V
53 fvex 6551 . . . . . . . . 9 (1st𝐶) ∈ V
5452, 47, 53, 48, 49, 46caov4 7235 . . . . . . . 8 (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) = (((2nd𝐴) ·N (1st𝐶)) ·N ((2nd𝐶) ·N (2nd𝐵)))
55 mulcompi 10164 . . . . . . . . 9 ((2nd𝐴) ·N (1st𝐶)) = ((1st𝐶) ·N (2nd𝐴))
56 mulcompi 10164 . . . . . . . . 9 ((2nd𝐶) ·N (2nd𝐵)) = ((2nd𝐵) ·N (2nd𝐶))
5755, 56oveq12i 7028 . . . . . . . 8 (((2nd𝐴) ·N (1st𝐶)) ·N ((2nd𝐶) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐵) ·N (2nd𝐶)))
5854, 57eqtri 2819 . . . . . . 7 (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐵) ·N (2nd𝐶)))
5951, 58oveq12i 7028 . . . . . 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 10162 . . . . . 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 7048 . . . . . . 7 ((1st𝐴) ·N (2nd𝐶)) ∈ V
62 ovex 7048 . . . . . . 7 ((1st𝐶) ·N (2nd𝐴)) ∈ V
63 ovex 7048 . . . . . . 7 ((2nd𝐵) ·N (2nd𝐶)) ∈ V
64 distrpi 10166 . . . . . . 7 (𝑥 ·N (𝑦 +N 𝑧)) = ((𝑥 ·N 𝑦) +N (𝑥 ·N 𝑧))
6561, 62, 63, 48, 64caovdir 7238 . . . . . 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 2829 . . . . 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 10162 . . . . . 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 10165 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))))
69 mulcompi 10164 . . . . . . . . . 10 ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))) = (((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶))
70 mulasspi 10165 . . . . . . . . . . . 12 (((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) = ((2nd𝐴) ·N ((2nd𝐶) ·N (1st𝐵)))
71 mulcompi 10164 . . . . . . . . . . . 12 ((2nd𝐴) ·N ((2nd𝐶) ·N (1st𝐵))) = (((2nd𝐶) ·N (1st𝐵)) ·N (2nd𝐴))
72 mulasspi 10165 . . . . . . . . . . . 12 (((2nd𝐶) ·N (1st𝐵)) ·N (2nd𝐴)) = ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))
7370, 71, 723eqtrri 2824 . . . . . . . . . . 11 ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))) = (((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵))
7473oveq1i 7026 . . . . . . . . . 10 (((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) = ((((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) ·N (2nd𝐶))
7569, 74eqtri 2819 . . . . . . . . 9 ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))) = ((((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) ·N (2nd𝐶))
76 mulasspi 10165 . . . . . . . . 9 ((((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) ·N (2nd𝐶)) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶)))
7775, 76eqtri 2819 . . . . . . . 8 ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶)))
7868, 77eqtri 2819 . . . . . . 7 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶)))
7978oveq2i 7027 . . . . . 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 10166 . . . . . 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 2829 . . . . 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 2809 . . . 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 10161 . . . . . 6 ((((2nd𝐴) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐵)) ∈ N) → (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
8416, 24, 83syl2anc 584 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
85 mulclpi 10161 . . . . . 6 ((((2nd𝐶) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐴) ·N (2nd𝐵)) ∈ N) → (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) ∈ N)
8639, 41, 85syl2anc 584 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) ∈ N)
87 addcanpi 10167 . . . . 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 584 . . . 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 287 . . 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 308 . 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 313 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 207  w3a 1080   = wceq 1522  wcel 2081  cop 4478   class class class wbr 4962   × cxp 5441  cfv 6225  (class class class)co 7016  1st c1st 7543  2nd c2nd 7544  Ncnpi 10112   +N cpli 10113   ·N cmi 10114   +pQ cplpq 10116   ~Q ceq 10119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-oadd 7957  df-omul 7958  df-ni 10140  df-pli 10141  df-mi 10142  df-plpq 10176  df-enq 10179
This theorem is referenced by:  adderpq  10224
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