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Theorem zseo 28424
Description: A surreal integer is either even or odd. (Contributed by Scott Fenton, 19-Aug-2025.)
Assertion
Ref Expression
zseo (𝑁 ∈ ℤs → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s )))
Distinct variable group:   𝑥,𝑁

Proof of Theorem zseo
Dummy variables 𝑦 𝑧 𝑤 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elzs 28388 . 2 (𝑁 ∈ ℤs ↔ ∃𝑦 ∈ ℕs𝑧 ∈ ℕs 𝑁 = (𝑦 -s 𝑧))
2 nnn0s 28350 . . . . . 6 (𝑦 ∈ ℕs𝑦 ∈ ℕ0s)
3 n0seo 28423 . . . . . 6 (𝑦 ∈ ℕ0s → (∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∨ ∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s )))
42, 3syl 17 . . . . 5 (𝑦 ∈ ℕs → (∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∨ ∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s )))
5 nnn0s 28350 . . . . . 6 (𝑧 ∈ ℕs𝑧 ∈ ℕ0s)
6 n0seo 28423 . . . . . 6 (𝑧 ∈ ℕ0s → (∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡) ∨ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )))
75, 6syl 17 . . . . 5 (𝑧 ∈ ℕs → (∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡) ∨ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )))
8 reeanv 3235 . . . . . . . 8 (∃𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s (𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) ↔ (∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)))
9 n0zs 28393 . . . . . . . . . . . . 13 (𝑤 ∈ ℕ0s𝑤 ∈ ℤs)
109adantr 480 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → 𝑤 ∈ ℤs)
11 n0zs 28393 . . . . . . . . . . . . 13 (𝑡 ∈ ℕ0s𝑡 ∈ ℤs)
1211adantl 481 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → 𝑡 ∈ ℤs)
1310, 12zsubscld 28400 . . . . . . . . . . 11 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (𝑤 -s 𝑡) ∈ ℤs)
14 2sno 28421 . . . . . . . . . . . . . 14 2s No
1514a1i 11 . . . . . . . . . . . . 13 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → 2s No )
16 n0sno 28346 . . . . . . . . . . . . . 14 (𝑤 ∈ ℕ0s𝑤 No )
1716adantr 480 . . . . . . . . . . . . 13 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → 𝑤 No )
18 n0sno 28346 . . . . . . . . . . . . . 14 (𝑡 ∈ ℕ0s𝑡 No )
1918adantl 481 . . . . . . . . . . . . 13 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → 𝑡 No )
2015, 17, 19subsdid 28202 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (2s ·s (𝑤 -s 𝑡)) = ((2s ·s 𝑤) -s (2s ·s 𝑡)))
2120eqcomd 2746 . . . . . . . . . . 11 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s (𝑤 -s 𝑡)))
22 oveq2 7456 . . . . . . . . . . . 12 (𝑥 = (𝑤 -s 𝑡) → (2s ·s 𝑥) = (2s ·s (𝑤 -s 𝑡)))
2322rspceeqv 3658 . . . . . . . . . . 11 (((𝑤 -s 𝑡) ∈ ℤs ∧ ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s (𝑤 -s 𝑡))) → ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s 𝑥))
2413, 21, 23syl2anc 583 . . . . . . . . . 10 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s 𝑥))
25 oveq12 7457 . . . . . . . . . . . 12 ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → (𝑦 -s 𝑧) = ((2s ·s 𝑤) -s (2s ·s 𝑡)))
2625eqeq1d 2742 . . . . . . . . . . 11 ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → ((𝑦 -s 𝑧) = (2s ·s 𝑥) ↔ ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s 𝑥)))
2726rexbidv 3185 . . . . . . . . . 10 ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s 𝑥)))
2824, 27syl5ibrcom 247 . . . . . . . . 9 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥)))
2928rexlimivv 3207 . . . . . . . 8 (∃𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s (𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥))
308, 29sylbir 235 . . . . . . 7 ((∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥))
3130orcd 872 . . . . . 6 ((∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
32 reeanv 3235 . . . . . . . 8 (∃𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s (𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = (2s ·s 𝑡)) ↔ (∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)))
3315, 17mulscld 28179 . . . . . . . . . . . . 13 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (2s ·s 𝑤) ∈ No )
34 1sno 27890 . . . . . . . . . . . . . 14 1s No
3534a1i 11 . . . . . . . . . . . . 13 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → 1s No )
3615, 19mulscld 28179 . . . . . . . . . . . . 13 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (2s ·s 𝑡) ∈ No )
3733, 35, 36addsubsd 28130 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s 1s ))
3821oveq1d 7463 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s 1s ))
3937, 38eqtrd 2780 . . . . . . . . . . 11 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s (𝑤 -s 𝑡)) +s 1s ))
4022oveq1d 7463 . . . . . . . . . . . 12 (𝑥 = (𝑤 -s 𝑡) → ((2s ·s 𝑥) +s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s 1s ))
4140rspceeqv 3658 . . . . . . . . . . 11 (((𝑤 -s 𝑡) ∈ ℤs ∧ (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s (𝑤 -s 𝑡)) +s 1s )) → ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s 𝑥) +s 1s ))
4213, 39, 41syl2anc 583 . . . . . . . . . 10 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s 𝑥) +s 1s ))
43 oveq12 7457 . . . . . . . . . . . 12 ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = (2s ·s 𝑡)) → (𝑦 -s 𝑧) = (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)))
4443eqeq1d 2742 . . . . . . . . . . 11 ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = (2s ·s 𝑡)) → ((𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ) ↔ (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s 𝑥) +s 1s )))
4544rexbidv 3185 . . . . . . . . . 10 ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = (2s ·s 𝑡)) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s 𝑥) +s 1s )))
4642, 45syl5ibrcom 247 . . . . . . . . 9 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
4746rexlimivv 3207 . . . . . . . 8 (∃𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s (𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ))
4832, 47sylbir 235 . . . . . . 7 ((∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ))
4948olcd 873 . . . . . 6 ((∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
50 reeanv 3235 . . . . . . . 8 (∃𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s (𝑦 = (2s ·s 𝑤) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) ↔ (∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )))
51 1zs 28395 . . . . . . . . . . . . 13 1s ∈ ℤs
5251a1i 11 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → 1s ∈ ℤs)
5313, 52zsubscld 28400 . . . . . . . . . . 11 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((𝑤 -s 𝑡) -s 1s ) ∈ ℤs)
5413znod 28387 . . . . . . . . . . . . . . 15 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (𝑤 -s 𝑡) ∈ No )
5515, 54, 35subsdid 28202 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (2s ·s ((𝑤 -s 𝑡) -s 1s )) = ((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )))
5655oveq1d 7463 . . . . . . . . . . . . 13 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((2s ·s ((𝑤 -s 𝑡) -s 1s )) +s 1s ) = (((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )) +s 1s ))
57 mulsrid 28157 . . . . . . . . . . . . . . . . 17 (2s No → (2s ·s 1s ) = 2s)
5814, 57ax-mp 5 . . . . . . . . . . . . . . . 16 (2s ·s 1s ) = 2s
5958oveq2i 7459 . . . . . . . . . . . . . . 15 ((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )) = ((2s ·s (𝑤 -s 𝑡)) -s 2s)
6059oveq1i 7458 . . . . . . . . . . . . . 14 (((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )) +s 1s ) = (((2s ·s (𝑤 -s 𝑡)) -s 2s) +s 1s )
6115, 54mulscld 28179 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (2s ·s (𝑤 -s 𝑡)) ∈ No )
6261, 35, 15addsubsd 28130 . . . . . . . . . . . . . . 15 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (((2s ·s (𝑤 -s 𝑡)) +s 1s ) -s 2s) = (((2s ·s (𝑤 -s 𝑡)) -s 2s) +s 1s ))
6361, 35, 15addsubsassd 28129 . . . . . . . . . . . . . . 15 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (((2s ·s (𝑤 -s 𝑡)) +s 1s ) -s 2s) = ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)))
6462, 63eqtr3d 2782 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (((2s ·s (𝑤 -s 𝑡)) -s 2s) +s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)))
6560, 64eqtrid 2792 . . . . . . . . . . . . 13 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )) +s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)))
6656, 65eqtrd 2780 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((2s ·s ((𝑤 -s 𝑡) -s 1s )) +s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)))
67 subscl 28110 . . . . . . . . . . . . . . . . . 18 (( 1s No ∧ 2s No ) → ( 1s -s 2s) ∈ No )
6834, 14, 67mp2an 691 . . . . . . . . . . . . . . . . 17 ( 1s -s 2s) ∈ No
69 negnegs 28094 . . . . . . . . . . . . . . . . 17 (( 1s -s 2s) ∈ No → ( -us ‘( -us ‘( 1s -s 2s))) = ( 1s -s 2s))
7068, 69ax-mp 5 . . . . . . . . . . . . . . . 16 ( -us ‘( -us ‘( 1s -s 2s))) = ( 1s -s 2s)
7134a1i 11 . . . . . . . . . . . . . . . . . . . 20 (⊤ → 1s No )
7214a1i 11 . . . . . . . . . . . . . . . . . . . 20 (⊤ → 2s No )
7371, 72negsubsdi2d 28128 . . . . . . . . . . . . . . . . . . 19 (⊤ → ( -us ‘( 1s -s 2s)) = (2s -s 1s ))
7473mptru 1544 . . . . . . . . . . . . . . . . . 18 ( -us ‘( 1s -s 2s)) = (2s -s 1s )
75 1p1e2s 28418 . . . . . . . . . . . . . . . . . . 19 ( 1s +s 1s ) = 2s
76 subadds 28118 . . . . . . . . . . . . . . . . . . . 20 ((2s No ∧ 1s No ∧ 1s No ) → ((2s -s 1s ) = 1s ↔ ( 1s +s 1s ) = 2s))
7714, 34, 34, 76mp3an 1461 . . . . . . . . . . . . . . . . . . 19 ((2s -s 1s ) = 1s ↔ ( 1s +s 1s ) = 2s)
7875, 77mpbir 231 . . . . . . . . . . . . . . . . . 18 (2s -s 1s ) = 1s
7974, 78eqtri 2768 . . . . . . . . . . . . . . . . 17 ( -us ‘( 1s -s 2s)) = 1s
8079fveq2i 6923 . . . . . . . . . . . . . . . 16 ( -us ‘( -us ‘( 1s -s 2s))) = ( -us ‘ 1s )
8170, 80eqtr3i 2770 . . . . . . . . . . . . . . 15 ( 1s -s 2s) = ( -us ‘ 1s )
8281oveq2i 7459 . . . . . . . . . . . . . 14 ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)) = ((2s ·s (𝑤 -s 𝑡)) +s ( -us ‘ 1s ))
8361, 35subsvald 28109 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 -s 𝑡)) -s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s ( -us ‘ 1s )))
8482, 83eqtr4id 2799 . . . . . . . . . . . . 13 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)) = ((2s ·s (𝑤 -s 𝑡)) -s 1s ))
8520oveq1d 7463 . . . . . . . . . . . . 13 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 -s 𝑡)) -s 1s ) = (((2s ·s 𝑤) -s (2s ·s 𝑡)) -s 1s ))
8684, 85eqtrd 2780 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)) = (((2s ·s 𝑤) -s (2s ·s 𝑡)) -s 1s ))
8733, 36, 35subsubs4d 28142 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) -s (2s ·s 𝑡)) -s 1s ) = ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )))
8866, 86, 873eqtrrd 2785 . . . . . . . . . . 11 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )) = ((2s ·s ((𝑤 -s 𝑡) -s 1s )) +s 1s ))
89 oveq2 7456 . . . . . . . . . . . . 13 (𝑥 = ((𝑤 -s 𝑡) -s 1s ) → (2s ·s 𝑥) = (2s ·s ((𝑤 -s 𝑡) -s 1s )))
9089oveq1d 7463 . . . . . . . . . . . 12 (𝑥 = ((𝑤 -s 𝑡) -s 1s ) → ((2s ·s 𝑥) +s 1s ) = ((2s ·s ((𝑤 -s 𝑡) -s 1s )) +s 1s ))
9190rspceeqv 3658 . . . . . . . . . . 11 ((((𝑤 -s 𝑡) -s 1s ) ∈ ℤs ∧ ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )) = ((2s ·s ((𝑤 -s 𝑡) -s 1s )) +s 1s )) → ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )) = ((2s ·s 𝑥) +s 1s ))
9253, 88, 91syl2anc 583 . . . . . . . . . 10 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )) = ((2s ·s 𝑥) +s 1s ))
93 oveq12 7457 . . . . . . . . . . . 12 ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → (𝑦 -s 𝑧) = ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )))
9493eqeq1d 2742 . . . . . . . . . . 11 ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → ((𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ) ↔ ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )) = ((2s ·s 𝑥) +s 1s )))
9594rexbidv 3185 . . . . . . . . . 10 ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )) = ((2s ·s 𝑥) +s 1s )))
9692, 95syl5ibrcom 247 . . . . . . . . 9 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
9796rexlimivv 3207 . . . . . . . 8 (∃𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s (𝑦 = (2s ·s 𝑤) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ))
9850, 97sylbir 235 . . . . . . 7 ((∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ))
9998olcd 873 . . . . . 6 ((∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
100 reeanv 3235 . . . . . . . 8 (∃𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s (𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) ↔ (∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )))
10133, 35, 36, 35addsubs4d 28148 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s ( 1s -s 1s )))
102 subsid 28117 . . . . . . . . . . . . . . 15 ( 1s No → ( 1s -s 1s ) = 0s )
10334, 102ax-mp 5 . . . . . . . . . . . . . 14 ( 1s -s 1s ) = 0s
104103oveq2i 7459 . . . . . . . . . . . . 13 (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s ( 1s -s 1s )) = (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s 0s )
10533, 36subscld 28111 . . . . . . . . . . . . . . 15 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((2s ·s 𝑤) -s (2s ·s 𝑡)) ∈ No )
106105addsridd 28016 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s 0s ) = ((2s ·s 𝑤) -s (2s ·s 𝑡)))
107106, 21eqtrd 2780 . . . . . . . . . . . . 13 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s 0s ) = (2s ·s (𝑤 -s 𝑡)))
108104, 107eqtrid 2792 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s ( 1s -s 1s )) = (2s ·s (𝑤 -s 𝑡)))
109101, 108eqtrd 2780 . . . . . . . . . . 11 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s (𝑤 -s 𝑡)))
11022rspceeqv 3658 . . . . . . . . . . 11 (((𝑤 -s 𝑡) ∈ ℤs ∧ (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s (𝑤 -s 𝑡))) → ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s 𝑥))
11113, 109, 110syl2anc 583 . . . . . . . . . 10 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s 𝑥))
112 oveq12 7457 . . . . . . . . . . . 12 ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → (𝑦 -s 𝑧) = (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )))
113112eqeq1d 2742 . . . . . . . . . . 11 ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → ((𝑦 -s 𝑧) = (2s ·s 𝑥) ↔ (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s 𝑥)))
114113rexbidv 3185 . . . . . . . . . 10 ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s 𝑥)))
115111, 114syl5ibrcom 247 . . . . . . . . 9 ((𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s) → ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥)))
116115rexlimivv 3207 . . . . . . . 8 (∃𝑤 ∈ ℕ0s𝑡 ∈ ℕ0s (𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥))
117100, 116sylbir 235 . . . . . . 7 ((∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥))
118117orcd 872 . . . . . 6 ((∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
11931, 49, 99, 118ccase 1038 . . . . 5 (((∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∨ ∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s )) ∧ (∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡) ∨ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s ))) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
1204, 7, 119syl2an 595 . . . 4 ((𝑦 ∈ ℕs𝑧 ∈ ℕs) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
121 eqeq1 2744 . . . . . 6 (𝑁 = (𝑦 -s 𝑧) → (𝑁 = (2s ·s 𝑥) ↔ (𝑦 -s 𝑧) = (2s ·s 𝑥)))
122121rexbidv 3185 . . . . 5 (𝑁 = (𝑦 -s 𝑧) → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥)))
123 eqeq1 2744 . . . . . 6 (𝑁 = (𝑦 -s 𝑧) → (𝑁 = ((2s ·s 𝑥) +s 1s ) ↔ (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
124123rexbidv 3185 . . . . 5 (𝑁 = (𝑦 -s 𝑧) → (∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
125122, 124orbi12d 917 . . . 4 (𝑁 = (𝑦 -s 𝑧) → ((∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s )) ↔ (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ))))
126120, 125syl5ibrcom 247 . . 3 ((𝑦 ∈ ℕs𝑧 ∈ ℕs) → (𝑁 = (𝑦 -s 𝑧) → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s ))))
127126rexlimivv 3207 . 2 (∃𝑦 ∈ ℕs𝑧 ∈ ℕs 𝑁 = (𝑦 -s 𝑧) → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s )))
1281, 127sylbi 217 1 (𝑁 ∈ ℤs → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 846   = wceq 1537  wtru 1538  wcel 2108  wrex 3076  cfv 6573  (class class class)co 7448   No csur 27702   0s c0s 27885   1s c1s 27886   +s cadds 28010   -us cnegs 28069   -s csubs 28070   ·s cmuls 28150  0scnn0s 28336  scnns 28337  sczs 28382  2sc2s 28412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-muls 28151  df-n0s 28338  df-nns 28339  df-zs 28383  df-2s 28413
This theorem is referenced by:  zs12bday  28442
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