Theorem adderpqlem 10957

Description: Lemma for adderpq 10959. (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.

Step	Hyp	Ref	Expression
1		xp1st 8027	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
2	1	3ad2ant1 1151	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐴) ∈ N)
3		xp2nd 8028	. . . . . 6 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
4	3	3ad2ant3 1153	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐶) ∈ N)
5		mulclpi 10896	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐶)) ∈ N)
6	2, 4, 5	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐴) ·N (2nd ‘𝐶)) ∈ N)
7		xp1st 8027	. . . . . 6 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
8	7	3ad2ant3 1153	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐶) ∈ N)
9		xp2nd 8028	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
10	9	3ad2ant1 1151	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐴) ∈ N)
11		mulclpi 10896	. . . . 5 ⊢ (((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((1st ‘𝐶) ·N (2nd ‘𝐴)) ∈ N)
12	8, 10, 11	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐶) ·N (2nd ‘𝐴)) ∈ N)
13		addclpi 10895	. . . 4 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐶)) ∈ N ∧ ((1st ‘𝐶) ·N (2nd ‘𝐴)) ∈ N) → (((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ∈ N)
14	6, 12, 13	syl2anc 596	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ∈ N)
15		mulclpi 10896	. . . 4 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐶)) ∈ N)
16	10, 4, 15	syl2anc 596	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd ‘𝐴) ·N (2nd ‘𝐶)) ∈ N)
17		xp1st 8027	. . . . . 6 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
18	17	3ad2ant2 1152	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st ‘𝐵) ∈ N)
19		mulclpi 10896	. . . . 5 ⊢ (((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((1st ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
20	18, 4, 19	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
21		xp2nd 8028	. . . . . 6 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
22	21	3ad2ant2 1152	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd ‘𝐵) ∈ N)
23		mulclpi 10896	. . . . 5 ⊢ (((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐶) ·N (2nd ‘𝐵)) ∈ N)
24	8, 22, 23	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐶) ·N (2nd ‘𝐵)) ∈ N)
25		addclpi 10895	. . . 4 ⊢ ((((1st ‘𝐵) ·N (2nd ‘𝐶)) ∈ N ∧ ((1st ‘𝐶) ·N (2nd ‘𝐵)) ∈ N) → (((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ∈ N)
26	20, 24, 25	syl2anc 596	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ∈ N)
27		mulclpi 10896	. . . 4 ⊢ (((2nd ‘𝐵) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
28	22, 4, 27	syl2anc 596	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
29		enqbreq 10922	. . 3 ⊢ ((((((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ∈ N ∧ ((2nd ‘𝐴) ·N (2nd ‘𝐶)) ∈ N) ∧ ((((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ∈ N ∧ ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)) → (⟨(((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐶))⟩ ~Q ⟨(((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩ ↔ ((((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N (((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))))))
30	14, 16, 26, 28, 29	syl22anc 852	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (⟨(((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐶))⟩ ~Q ⟨(((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩ ↔ ((((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N (((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))))))
31		addpipq2 10939	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 +pQ 𝐶) = ⟨(((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐶))⟩)
32	31	3adant2 1149	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 +pQ 𝐶) = ⟨(((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐶))⟩)
33		addpipq2 10939	. . . 4 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
34	33	3adant1 1148	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
35	32, 34	breq12d 5127	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((𝐴 +pQ 𝐶) ~Q (𝐵 +pQ 𝐶) ↔ ⟨(((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐶))⟩ ~Q ⟨(((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩))
36		enqbreq2 10923	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
37	36	3adant3 1150	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
38		mulclpi 10896	. . . . 5 ⊢ (((2nd ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
39	4, 4, 38	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
40		mulclpi 10896	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
41	2, 22, 40	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
42		mulcanpi 10903	. . . 4 ⊢ ((((2nd ‘𝐶) ·N (2nd ‘𝐶)) ∈ N ∧ ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N) → ((((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
43	39, 41, 42	syl2anc 596	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
44		mulclpi 10896	. . . . . 6 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ∈ N ∧ ((1st ‘𝐶) ·N (2nd ‘𝐵)) ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ∈ N)
45	16, 24, 44	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ∈ N)
46		mulclpi 10896	. . . . . 6 ⊢ ((((2nd ‘𝐶) ·N (2nd ‘𝐶)) ∈ N ∧ ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N) → (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) ∈ N)
47	39, 41, 46	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) ∈ N)
48		addcanpi 10902	. . . . 5 ⊢ (((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ∈ N ∧ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) ∈ N) → (((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵)))) = ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) ↔ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))))
49	45, 47, 48	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵)))) = ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) ↔ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))))
50		mulcompi 10899	. . . . . . . 8 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐶)))
51		fvex 6901	. . . . . . . . 9 ⊢ (1st ‘𝐴) ∈ V
52		fvex 6901	. . . . . . . . 9 ⊢ (2nd ‘𝐵) ∈ V
53		fvex 6901	. . . . . . . . 9 ⊢ (2nd ‘𝐶) ∈ V
54		mulcompi 10899	. . . . . . . . 9 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
55		mulasspi 10900	. . . . . . . . 9 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
56	51, 52, 53, 54, 55, 53	caov4 7654	. . . . . . . 8 ⊢ (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐶))) = (((1st ‘𝐴) ·N (2nd ‘𝐶)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))
57	50, 56	eqtri 2789	. . . . . . 7 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((1st ‘𝐴) ·N (2nd ‘𝐶)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))
58		fvex 6901	. . . . . . . . 9 ⊢ (2nd ‘𝐴) ∈ V
59		fvex 6901	. . . . . . . . 9 ⊢ (1st ‘𝐶) ∈ V
60	58, 53, 59, 54, 55, 52	caov4 7654	. . . . . . . 8 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) = (((2nd ‘𝐴) ·N (1st ‘𝐶)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐵)))
61		mulcompi 10899	. . . . . . . . 9 ⊢ ((2nd ‘𝐴) ·N (1st ‘𝐶)) = ((1st ‘𝐶) ·N (2nd ‘𝐴))
62		mulcompi 10899	. . . . . . . . 9 ⊢ ((2nd ‘𝐶) ·N (2nd ‘𝐵)) = ((2nd ‘𝐵) ·N (2nd ‘𝐶))
63	61, 62	oveq12i 7435	. . . . . . . 8 ⊢ (((2nd ‘𝐴) ·N (1st ‘𝐶)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐵))) = (((1st ‘𝐶) ·N (2nd ‘𝐴)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))
64	60, 63	eqtri 2789	. . . . . . 7 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) = (((1st ‘𝐶) ·N (2nd ‘𝐴)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))
65	57, 64	oveq12i 7435	. . . . . 6 ⊢ ((((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵)))) = ((((1st ‘𝐴) ·N (2nd ‘𝐶)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N (((1st ‘𝐶) ·N (2nd ‘𝐴)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))))
66		addcompi 10897	. . . . . 6 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵)))) = ((((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))))
67		ovex 7456	. . . . . . 7 ⊢ ((1st ‘𝐴) ·N (2nd ‘𝐶)) ∈ V
68		ovex 7456	. . . . . . 7 ⊢ ((1st ‘𝐶) ·N (2nd ‘𝐴)) ∈ V
69		ovex 7456	. . . . . . 7 ⊢ ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ V
70		distrpi 10901	. . . . . . 7 ⊢ (𝑥 ·N (𝑦 +N 𝑧)) = ((𝑥 ·N 𝑦) +N (𝑥 ·N 𝑧))
71	67, 68, 69, 54, 70	caovdir 7657	. . . . . 6 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) = ((((1st ‘𝐴) ·N (2nd ‘𝐶)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N (((1st ‘𝐶) ·N (2nd ‘𝐴)) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))))
72	65, 66, 71	3eqtr4i 2799	. . . . 5 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵)))) = ((((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))
73		addcompi 10897	. . . . . 6 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐶)))) = ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐶))) +N (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))))
74		mulasspi 10900	. . . . . . . 8 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) = ((2nd ‘𝐶) ·N ((2nd ‘𝐶) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
75		mulcompi 10899	. . . . . . . . . 10 ⊢ ((2nd ‘𝐶) ·N ((2nd ‘𝐶) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) = (((2nd ‘𝐶) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ·N (2nd ‘𝐶))
76		mulasspi 10900	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N (1st ‘𝐵)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐶) ·N (1st ‘𝐵)))
77		mulcompi 10899	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝐴) ·N ((2nd ‘𝐶) ·N (1st ‘𝐵))) = (((2nd ‘𝐶) ·N (1st ‘𝐵)) ·N (2nd ‘𝐴))
78		mulasspi 10900	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝐶) ·N (1st ‘𝐵)) ·N (2nd ‘𝐴)) = ((2nd ‘𝐶) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))
79	76, 77, 78	3eqtrri 2794	. . . . . . . . . . 11 ⊢ ((2nd ‘𝐶) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N (1st ‘𝐵))
80	79	oveq1i 7433	. . . . . . . . . 10 ⊢ (((2nd ‘𝐶) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ·N (2nd ‘𝐶)) = ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N (1st ‘𝐵)) ·N (2nd ‘𝐶))
81	75, 80	eqtri 2789	. . . . . . . . 9 ⊢ ((2nd ‘𝐶) ·N ((2nd ‘𝐶) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) = ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N (1st ‘𝐵)) ·N (2nd ‘𝐶))
82		mulasspi 10900	. . . . . . . . 9 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N (1st ‘𝐵)) ·N (2nd ‘𝐶)) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐶)))
83	81, 82	eqtri 2789	. . . . . . . 8 ⊢ ((2nd ‘𝐶) ·N ((2nd ‘𝐶) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐶)))
84	74, 83	eqtri 2789	. . . . . . 7 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐶)))
85	84	oveq2i 7434	. . . . . 6 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) = ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐶))))
86		distrpi 10901	. . . . . 6 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N (((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵)))) = ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐶))) +N (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))))
87	73, 85, 86	3eqtr4i 2799	. . . . 5 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N (((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))))
88	72, 87	eqeq12i 2784	. . . 4 ⊢ (((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵)))) = ((((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) ↔ ((((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N (((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵)))))
89	49, 88	bitr3di 289	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ↔ ((((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N (((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))))))
90	37, 43, 89	3bitr2d 310	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((((1st ‘𝐴) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) = (((2nd ‘𝐴) ·N (2nd ‘𝐶)) ·N (((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))))))
91	30, 35, 90	3bitr4rd 315	1 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (𝐴 +pQ 𝐶) ~Q (𝐵 +pQ 𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ⟨cop 4600   class class class wbr 5114   × cxp 5664  ‘cfv 6543  (class class class)co 7423  1^st c1st 7993  2^nd c2nd 7994  Ncnpi 10847   +_N cpli 10848   ·_N cmi 10849   +_pQ cplpq 10851   ~_Q ceq 10854

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-oadd 8466  df-omul 8467  df-ni 10875  df-pli 10876  df-mi 10877  df-plpq 10911  df-enq 10914

This theorem is used by:  adderpq  10959