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Theorem adderpqlem 10890
Description: Lemma for adderpq 10892. (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 7953 . . . . . 6 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
213ad2ant1 1133 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐴) ∈ N)
3 xp2nd 7954 . . . . . 6 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
433ad2ant3 1135 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐶) ∈ N)
5 mulclpi 10829 . . . . 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 7953 . . . . . 6 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
873ad2ant3 1135 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐶) ∈ N)
9 xp2nd 7954 . . . . . 6 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
1093ad2ant1 1133 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐴) ∈ N)
11 mulclpi 10829 . . . . 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 10828 . . . 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 10829 . . . 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 7953 . . . . . 6 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
18173ad2ant2 1134 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐵) ∈ N)
19 mulclpi 10829 . . . . 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 7954 . . . . . 6 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
22213ad2ant2 1134 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐵) ∈ N)
23 mulclpi 10829 . . . . 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 10828 . . . 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 10829 . . . 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 10855 . . 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 837 . 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 10872 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 +pQ 𝐶) = ⟨(((1st𝐴) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐶))⟩)
32313adant2 1131 . . 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 10872 . . . 4 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
34333adant1 1130 . . 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 5118 . 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 10856 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
37363adant3 1132 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
38 mulclpi 10829 . . . . 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 10829 . . . . 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 10836 . . . 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 mulclpi 10829 . . . . . 6 ((((2nd𝐴) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐵)) ∈ N) → (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
4516, 24, 44syl2anc 584 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
46 mulclpi 10829 . . . . . 6 ((((2nd𝐶) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐴) ·N (2nd𝐵)) ∈ N) → (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) ∈ N)
4739, 41, 46syl2anc 584 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) ∈ N)
48 addcanpi 10835 . . . . 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)))))
4945, 47, 48syl2anc 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)))))
50 mulcompi 10832 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((2nd𝐶) ·N (2nd𝐶)))
51 fvex 6855 . . . . . . . . 9 (1st𝐴) ∈ V
52 fvex 6855 . . . . . . . . 9 (2nd𝐵) ∈ V
53 fvex 6855 . . . . . . . . 9 (2nd𝐶) ∈ V
54 mulcompi 10832 . . . . . . . . 9 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
55 mulasspi 10833 . . . . . . . . 9 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
5651, 52, 53, 54, 55, 53caov4 7585 . . . . . . . 8 (((1st𝐴) ·N (2nd𝐵)) ·N ((2nd𝐶) ·N (2nd𝐶))) = (((1st𝐴) ·N (2nd𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
5750, 56eqtri 2764 . . . . . . 7 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
58 fvex 6855 . . . . . . . . 9 (2nd𝐴) ∈ V
59 fvex 6855 . . . . . . . . 9 (1st𝐶) ∈ V
6058, 53, 59, 54, 55, 52caov4 7585 . . . . . . . 8 (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) = (((2nd𝐴) ·N (1st𝐶)) ·N ((2nd𝐶) ·N (2nd𝐵)))
61 mulcompi 10832 . . . . . . . . 9 ((2nd𝐴) ·N (1st𝐶)) = ((1st𝐶) ·N (2nd𝐴))
62 mulcompi 10832 . . . . . . . . 9 ((2nd𝐶) ·N (2nd𝐵)) = ((2nd𝐵) ·N (2nd𝐶))
6361, 62oveq12i 7369 . . . . . . . 8 (((2nd𝐴) ·N (1st𝐶)) ·N ((2nd𝐶) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐵) ·N (2nd𝐶)))
6460, 63eqtri 2764 . . . . . . 7 (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐶) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐵) ·N (2nd𝐶)))
6557, 64oveq12i 7369 . . . . . 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))))
66 addcompi 10830 . . . . . 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))))
67 ovex 7390 . . . . . . 7 ((1st𝐴) ·N (2nd𝐶)) ∈ V
68 ovex 7390 . . . . . . 7 ((1st𝐶) ·N (2nd𝐴)) ∈ V
69 ovex 7390 . . . . . . 7 ((2nd𝐵) ·N (2nd𝐶)) ∈ V
70 distrpi 10834 . . . . . . 7 (𝑥 ·N (𝑦 +N 𝑧)) = ((𝑥 ·N 𝑦) +N (𝑥 ·N 𝑧))
7167, 68, 69, 54, 70caovdir 7588 . . . . . 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))))
7265, 66, 713eqtr4i 2774 . . . . 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)))
73 addcompi 10830 . . . . . 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))))
74 mulasspi 10833 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))))
75 mulcompi 10832 . . . . . . . . . 10 ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))) = (((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶))
76 mulasspi 10833 . . . . . . . . . . . 12 (((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) = ((2nd𝐴) ·N ((2nd𝐶) ·N (1st𝐵)))
77 mulcompi 10832 . . . . . . . . . . . 12 ((2nd𝐴) ·N ((2nd𝐶) ·N (1st𝐵))) = (((2nd𝐶) ·N (1st𝐵)) ·N (2nd𝐴))
78 mulasspi 10833 . . . . . . . . . . . 12 (((2nd𝐶) ·N (1st𝐵)) ·N (2nd𝐴)) = ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))
7976, 77, 783eqtrri 2769 . . . . . . . . . . 11 ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))) = (((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵))
8079oveq1i 7367 . . . . . . . . . 10 (((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) = ((((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) ·N (2nd𝐶))
8175, 80eqtri 2764 . . . . . . . . 9 ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))) = ((((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) ·N (2nd𝐶))
82 mulasspi 10833 . . . . . . . . 9 ((((2nd𝐴) ·N (2nd𝐶)) ·N (1st𝐵)) ·N (2nd𝐶)) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶)))
8381, 82eqtri 2764 . . . . . . . 8 ((2nd𝐶) ·N ((2nd𝐶) ·N ((1st𝐵) ·N (2nd𝐴)))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶)))
8474, 83eqtri 2764 . . . . . . 7 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐶)))
8584oveq2i 7368 . . . . . 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))))
86 distrpi 10834 . . . . . 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))))
8773, 85, 863eqtr4i 2774 . . . . 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))))
8872, 87eqeq12i 2754 . . . 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)))))
8949, 88bitr3di 285 . . 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 306 . 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 311 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 205  w3a 1087   = wceq 1541  wcel 2106  cop 4592   class class class wbr 5105   × cxp 5631  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Ncnpi 10780   +N cpli 10781   ·N cmi 10782   +pQ cplpq 10784   ~Q ceq 10787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-oadd 8416  df-omul 8417  df-ni 10808  df-pli 10809  df-mi 10810  df-plpq 10844  df-enq 10847
This theorem is referenced by:  adderpq  10892
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