Metamath Proof Explorer

Theorem nn0opth2

Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of Quine p. 124. See nn0opthi . (Contributed by NM, 22-Jul-2004)

Ref Expression
Assertion nn0opth2 ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ∧ ( 𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ) ) → ( ( ( ( 𝐴 + 𝐵 ) ↑ 2 ) + 𝐵 ) = ( ( ( 𝐶 + 𝐷 ) ↑ 2 ) + 𝐷 ) ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) )

Proof

Step Hyp Ref Expression
1 oveq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) → ( 𝐴 + 𝐵 ) = ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + 𝐵 ) )
2 1 oveq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) → ( ( 𝐴 + 𝐵 ) ↑ 2 ) = ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + 𝐵 ) ↑ 2 ) )
3 2 oveq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) → ( ( ( 𝐴 + 𝐵 ) ↑ 2 ) + 𝐵 ) = ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + 𝐵 ) ↑ 2 ) + 𝐵 ) )
4 3 eqeq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) → ( ( ( ( 𝐴 + 𝐵 ) ↑ 2 ) + 𝐵 ) = ( ( ( 𝐶 + 𝐷 ) ↑ 2 ) + 𝐷 ) ↔ ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + 𝐵 ) ↑ 2 ) + 𝐵 ) = ( ( ( 𝐶 + 𝐷 ) ↑ 2 ) + 𝐷 ) ) )
5 eqeq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) → ( 𝐴 = 𝐶 ↔ if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = 𝐶 ) )
6 5 anbi1d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) → ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ↔ ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = 𝐶 ∧ 𝐵 = 𝐷 ) ) )
7 4 6 bibi12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) → ( ( ( ( ( 𝐴 + 𝐵 ) ↑ 2 ) + 𝐵 ) = ( ( ( 𝐶 + 𝐷 ) ↑ 2 ) + 𝐷 ) ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ↔ ( ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + 𝐵 ) ↑ 2 ) + 𝐵 ) = ( ( ( 𝐶 + 𝐷 ) ↑ 2 ) + 𝐷 ) ↔ ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) )
8 oveq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) → ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + 𝐵 ) = ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) )
9 8 oveq1d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) → ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + 𝐵 ) ↑ 2 ) = ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) ↑ 2 ) )
10 id ⊢ ( 𝐵 = if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) → 𝐵 = if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) )
11 9 10 oveq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) → ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + 𝐵 ) ↑ 2 ) + 𝐵 ) = ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) ↑ 2 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) )
12 11 eqeq1d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) → ( ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + 𝐵 ) ↑ 2 ) + 𝐵 ) = ( ( ( 𝐶 + 𝐷 ) ↑ 2 ) + 𝐷 ) ↔ ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) ↑ 2 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) = ( ( ( 𝐶 + 𝐷 ) ↑ 2 ) + 𝐷 ) ) )
13 eqeq1 ⊢ ( 𝐵 = if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) → ( 𝐵 = 𝐷 ↔ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = 𝐷 ) )
14 13 anbi2d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) → ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = 𝐶 ∧ 𝐵 = 𝐷 ) ↔ ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = 𝐶 ∧ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = 𝐷 ) ) )
15 12 14 bibi12d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) → ( ( ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + 𝐵 ) ↑ 2 ) + 𝐵 ) = ( ( ( 𝐶 + 𝐷 ) ↑ 2 ) + 𝐷 ) ↔ ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = 𝐶 ∧ 𝐵 = 𝐷 ) ) ↔ ( ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) ↑ 2 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) = ( ( ( 𝐶 + 𝐷 ) ↑ 2 ) + 𝐷 ) ↔ ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = 𝐶 ∧ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = 𝐷 ) ) ) )
16 oveq1 ⊢ ( 𝐶 = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) → ( 𝐶 + 𝐷 ) = ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + 𝐷 ) )
17 16 oveq1d ⊢ ( 𝐶 = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) → ( ( 𝐶 + 𝐷 ) ↑ 2 ) = ( ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + 𝐷 ) ↑ 2 ) )
18 17 oveq1d ⊢ ( 𝐶 = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) → ( ( ( 𝐶 + 𝐷 ) ↑ 2 ) + 𝐷 ) = ( ( ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + 𝐷 ) ↑ 2 ) + 𝐷 ) )
19 18 eqeq2d ⊢ ( 𝐶 = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) → ( ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) ↑ 2 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) = ( ( ( 𝐶 + 𝐷 ) ↑ 2 ) + 𝐷 ) ↔ ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) ↑ 2 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) = ( ( ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + 𝐷 ) ↑ 2 ) + 𝐷 ) ) )
20 eqeq2 ⊢ ( 𝐶 = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) → ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = 𝐶 ↔ if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) ) )
21 20 anbi1d ⊢ ( 𝐶 = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) → ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = 𝐶 ∧ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = 𝐷 ) ↔ ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) ∧ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = 𝐷 ) ) )
22 19 21 bibi12d ⊢ ( 𝐶 = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) → ( ( ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) ↑ 2 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) = ( ( ( 𝐶 + 𝐷 ) ↑ 2 ) + 𝐷 ) ↔ ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = 𝐶 ∧ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = 𝐷 ) ) ↔ ( ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) ↑ 2 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) = ( ( ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + 𝐷 ) ↑ 2 ) + 𝐷 ) ↔ ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) ∧ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = 𝐷 ) ) ) )
23 oveq2 ⊢ ( 𝐷 = if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) → ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + 𝐷 ) = ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) )
24 23 oveq1d ⊢ ( 𝐷 = if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) → ( ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + 𝐷 ) ↑ 2 ) = ( ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) ↑ 2 ) )
25 id ⊢ ( 𝐷 = if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) → 𝐷 = if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) )
26 24 25 oveq12d ⊢ ( 𝐷 = if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) → ( ( ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + 𝐷 ) ↑ 2 ) + 𝐷 ) = ( ( ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) ↑ 2 ) + if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) )
27 26 eqeq2d ⊢ ( 𝐷 = if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) → ( ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) ↑ 2 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) = ( ( ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + 𝐷 ) ↑ 2 ) + 𝐷 ) ↔ ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) ↑ 2 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) = ( ( ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) ↑ 2 ) + if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) ) )
28 eqeq2 ⊢ ( 𝐷 = if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) → ( if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = 𝐷 ↔ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) )
29 28 anbi2d ⊢ ( 𝐷 = if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) → ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) ∧ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = 𝐷 ) ↔ ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) ∧ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) ) )
30 27 29 bibi12d ⊢ ( 𝐷 = if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) → ( ( ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) ↑ 2 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) = ( ( ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + 𝐷 ) ↑ 2 ) + 𝐷 ) ↔ ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) ∧ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = 𝐷 ) ) ↔ ( ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) ↑ 2 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) = ( ( ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) ↑ 2 ) + if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) ↔ ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) ∧ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) ) ) )
31 0nn0 ⊢ 0 ∈ ℕ0
32 31 elimel ⊢ if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) ∈ ℕ0
33 31 elimel ⊢ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ∈ ℕ0
34 31 elimel ⊢ if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) ∈ ℕ0
35 31 elimel ⊢ if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ∈ ℕ0
36 32 33 34 35 nn0opth2i ⊢ ( ( ( ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) ↑ 2 ) + if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) ) = ( ( ( if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) + if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) ↑ 2 ) + if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) ↔ ( if ( 𝐴 ∈ ℕ0 , 𝐴 , 0 ) = if ( 𝐶 ∈ ℕ0 , 𝐶 , 0 ) ∧ if ( 𝐵 ∈ ℕ0 , 𝐵 , 0 ) = if ( 𝐷 ∈ ℕ0 , 𝐷 , 0 ) ) )
37 7 15 22 30 36 dedth4h ⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ∧ ( 𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ) ) → ( ( ( ( 𝐴 + 𝐵 ) ↑ 2 ) + 𝐵 ) = ( ( ( 𝐶 + 𝐷 ) ↑ 2 ) + 𝐷 ) ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) )