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Theorem 4sqlem2

Description: Lemma for 4sq . Change bound variables in S . (Contributed by Mario Carneiro, 14-Jul-2014)

Ref Expression
Hypothesis 4sq.1 ⊢ 𝑆 = { 𝑛 ∣ ∃ 𝑥 ∈ ℤ ∃ 𝑦 ∈ ℤ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) }
Assertion 4sqlem2 ( 𝐴 ∈ 𝑆 ↔ ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) )

Proof

Step Hyp Ref Expression
1 4sq.1 ⊢ 𝑆 = { 𝑛 ∣ ∃ 𝑥 ∈ ℤ ∃ 𝑦 ∈ ℤ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) }
2 1 eleq2i ⊢ ( 𝐴 ∈ 𝑆 ↔ 𝐴 ∈ { 𝑛 ∣ ∃ 𝑥 ∈ ℤ ∃ 𝑦 ∈ ℤ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) } )
3 id ⊢ ( 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) → 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) )
4 ovex ⊢ ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) ∈ V
5 3 4 eqeltrdi ⊢ ( 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) → 𝐴 ∈ V )
6 5 a1i ⊢ ( ( ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ∧ ( 𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ ) ) → ( 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) → 𝐴 ∈ V ) )
7 6 rexlimdvva ⊢ ( ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) → ( ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) → 𝐴 ∈ V ) )
8 7 rexlimivv ⊢ ( ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) → 𝐴 ∈ V )
9 oveq1 ⊢ ( 𝑥 = 𝑎 → ( 𝑥 ↑ 2 ) = ( 𝑎 ↑ 2 ) )
10 9 oveq1d ⊢ ( 𝑥 = 𝑎 → ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) = ( ( 𝑎 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) )
11 10 oveq1d ⊢ ( 𝑥 = 𝑎 → ( ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) = ( ( ( 𝑎 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) )
12 11 eqeq2d ⊢ ( 𝑥 = 𝑎 → ( 𝑛 = ( ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ↔ 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ) )
13 12 2rexbidv ⊢ ( 𝑥 = 𝑎 → ( ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ↔ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ) )
14 oveq1 ⊢ ( 𝑦 = 𝑏 → ( 𝑦 ↑ 2 ) = ( 𝑏 ↑ 2 ) )
15 14 oveq2d ⊢ ( 𝑦 = 𝑏 → ( ( 𝑎 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) = ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) )
16 15 oveq1d ⊢ ( 𝑦 = 𝑏 → ( ( ( 𝑎 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) )
17 16 eqeq2d ⊢ ( 𝑦 = 𝑏 → ( 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ↔ 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ) )
18 17 2rexbidv ⊢ ( 𝑦 = 𝑏 → ( ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ↔ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ) )
19 13 18 cbvrex2vw ⊢ ( ∃ 𝑥 ∈ ℤ ∃ 𝑦 ∈ ℤ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ↔ ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) )
20 oveq1 ⊢ ( 𝑧 = 𝑐 → ( 𝑧 ↑ 2 ) = ( 𝑐 ↑ 2 ) )
21 20 oveq1d ⊢ ( 𝑧 = 𝑐 → ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) = ( ( 𝑐 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) )
22 21 oveq2d ⊢ ( 𝑧 = 𝑐 → ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) )
23 22 eqeq2d ⊢ ( 𝑧 = 𝑐 → ( 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ↔ 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ) )
24 oveq1 ⊢ ( 𝑤 = 𝑑 → ( 𝑤 ↑ 2 ) = ( 𝑑 ↑ 2 ) )
25 24 oveq2d ⊢ ( 𝑤 = 𝑑 → ( ( 𝑐 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) = ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) )
26 25 oveq2d ⊢ ( 𝑤 = 𝑑 → ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) )
27 26 eqeq2d ⊢ ( 𝑤 = 𝑑 → ( 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ↔ 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) ) )
28 23 27 cbvrex2vw ⊢ ( ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ↔ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) )
29 eqeq1 ⊢ ( 𝑛 = 𝐴 → ( 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) ↔ 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) ) )
30 29 2rexbidv ⊢ ( 𝑛 = 𝐴 → ( ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) ↔ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) ) )
31 28 30 bitrid ⊢ ( 𝑛 = 𝐴 → ( ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ↔ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) ) )
32 31 2rexbidv ⊢ ( 𝑛 = 𝐴 → ( ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ↔ ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) ) )
33 19 32 bitrid ⊢ ( 𝑛 = 𝐴 → ( ∃ 𝑥 ∈ ℤ ∃ 𝑦 ∈ ℤ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) ↔ ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) ) )
34 8 33 elab3 ⊢ ( 𝐴 ∈ { 𝑛 ∣ ∃ 𝑥 ∈ ℤ ∃ 𝑦 ∈ ℤ ∃ 𝑧 ∈ ℤ ∃ 𝑤 ∈ ℤ 𝑛 = ( ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) + ( ( 𝑧 ↑ 2 ) + ( 𝑤 ↑ 2 ) ) ) } ↔ ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) )
35 2 34 bitri ⊢ ( 𝐴 ∈ 𝑆 ↔ ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ ∃ 𝑐 ∈ ℤ ∃ 𝑑 ∈ ℤ 𝐴 = ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) + ( ( 𝑐 ↑ 2 ) + ( 𝑑 ↑ 2 ) ) ) )