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Theorem adderpqlem 10364
Description: Lemma for adderpq 10366. (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 7710 . . . . . 6 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
213ad2ant1 1125 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐴) ∈ N)
3 xp2nd 7711 . . . . . 6 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
433ad2ant3 1127 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐶) ∈ N)
5 mulclpi 10303 . . . . 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 7710 . . . . . 6 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
873ad2ant3 1127 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐶) ∈ N)
9 xp2nd 7711 . . . . . 6 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
1093ad2ant1 1125 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐴) ∈ N)
11 mulclpi 10303 . . . . 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 10302 . . . 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 10303 . . . 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 7710 . . . . . 6 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
18173ad2ant2 1126 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐵) ∈ N)
19 mulclpi 10303 . . . . 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 7711 . . . . . 6 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
22213ad2ant2 1126 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐵) ∈ N)
23 mulclpi 10303 . . . . 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 10302 . . . 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 10303 . . . 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 10329 . . 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 834 . 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 10346 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 +pQ 𝐶) = ⟨(((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐶))⟩)
32313adant2 1123 . . 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 10346 . . . 4 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
34333adant1 1122 . . 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 5070 . 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 10330 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
37363adant3 1124 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
38 mulclpi 10303 . . . . 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 10303 . . . . 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 10310 . . . 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 10306 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((2nd𝐶) ·N (2nd𝐶)))
45 fvex 6676 . . . . . . . . 9 (1st𝐴) ∈ V
46 fvex 6676 . . . . . . . . 9 (2nd𝐵) ∈ V
47 fvex 6676 . . . . . . . . 9 (2nd𝐶) ∈ V
48 mulcompi 10306 . . . . . . . . 9 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
49 mulasspi 10307 . . . . . . . . 9 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
5045, 46, 47, 48, 49, 47caov4 7368 . . . . . . . 8 (((1st𝐴) ·N (2nd𝐵)) ·N ((2nd𝐶) ·N (2nd𝐶))) = (((1st𝐴) ·N (2nd𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
5144, 50eqtri 2841 . . . . . . 7 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
52 fvex 6676 . . . . . . . . 9 (2nd𝐴) ∈ V
53 fvex 6676 . . . . . . . . 9 (1st𝐶) ∈ V
5452, 47, 53, 48, 49, 46caov4 7368 . . . . . . . 8 (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) = (((2nd𝐴) ·N (1st𝐶)) ·N ((2nd𝐶) ·N (2nd𝐵)))
55 mulcompi 10306 . . . . . . . . 9 ((2nd𝐴) ·N (1st𝐶)) = ((1st𝐶) ·N (2nd𝐴))
56 mulcompi 10306 . . . . . . . . 9 ((2nd𝐶) ·N (2nd𝐵)) = ((2nd𝐵) ·N (2nd𝐶))
5755, 56oveq12i 7157 . . . . . . . 8 (((2nd𝐴) ·N (1st𝐶)) ·N ((2nd𝐶) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐵) ·N (2nd𝐶)))
5854, 57eqtri 2841 . . . . . . 7 (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐵) ·N (2nd𝐶)))
5951, 58oveq12i 7157 . . . . . 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 10304 . . . . . 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 7178 . . . . . . 7 ((1st𝐴) ·N (2nd𝐶)) ∈ V
62 ovex 7178 . . . . . . 7 ((1st𝐶) ·N (2nd𝐴)) ∈ V
63 ovex 7178 . . . . . . 7 ((2nd𝐵) ·N (2nd𝐶)) ∈ V
64 distrpi 10308 . . . . . . 7 (𝑥 ·N (𝑦 +N 𝑧)) = ((𝑥 ·N 𝑦) +N (𝑥 ·N 𝑧))
6561, 62, 63, 48, 64caovdir 7371 . . . . . 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 2851 . . . . 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 10304 . . . . . 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 10307 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))))
69 mulcompi 10306 . . . . . . . . . 10 ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))) = (((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶))
70 mulasspi 10307 . . . . . . . . . . . 12 (((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) = ((2nd𝐴) ·N ((2nd𝐶) ·N (1st𝐵)))
71 mulcompi 10306 . . . . . . . . . . . 12 ((2nd𝐴) ·N ((2nd𝐶) ·N (1st𝐵))) = (((2nd𝐶) ·N (1st𝐵)) ·N (2nd𝐴))
72 mulasspi 10307 . . . . . . . . . . . 12 (((2nd𝐶) ·N (1st𝐵)) ·N (2nd𝐴)) = ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))
7370, 71, 723eqtrri 2846 . . . . . . . . . . 11 ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))) = (((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵))
7473oveq1i 7155 . . . . . . . . . 10 (((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) = ((((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) ·N (2nd𝐶))
7569, 74eqtri 2841 . . . . . . . . 9 ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))) = ((((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) ·N (2nd𝐶))
76 mulasspi 10307 . . . . . . . . 9 ((((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) ·N (2nd𝐶)) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶)))
7775, 76eqtri 2841 . . . . . . . 8 ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶)))
7868, 77eqtri 2841 . . . . . . 7 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶)))
7978oveq2i 7156 . . . . . 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 10308 . . . . . 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 2851 . . . . 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 2833 . . . 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 10303 . . . . . 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 10303 . . . . . 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 10309 . . . . 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 1079   = wceq 1528  wcel 2105  cop 4563   class class class wbr 5057   × cxp 5546  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  Ncnpi 10254   +N cpli 10255   ·N cmi 10256   +pQ cplpq 10258   ~Q ceq 10261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-oadd 8095  df-omul 8096  df-ni 10282  df-pli 10283  df-mi 10284  df-plpq 10318  df-enq 10321
This theorem is referenced by:  adderpq  10366
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