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Theorem zscut 28402
Description: A cut expression for surreal integers. (Contributed by Scott Fenton, 20-Aug-2025.)
Assertion
Ref Expression
zscut (𝐴 ∈ ℤs𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))

Proof of Theorem zscut
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elzn0s 28393 . 2 (𝐴 ∈ ℤs ↔ (𝐴 No ∧ (𝐴 ∈ ℕ0s ∨ ( -us𝐴) ∈ ℕ0s)))
2 n0scut 28347 . . . . 5 (𝐴 ∈ ℕ0s𝐴 = ({(𝐴 -s 1s )} |s ∅))
3 n0sno 28337 . . . . . . . 8 (𝐴 ∈ ℕ0s𝐴 No )
4 1sno 27881 . . . . . . . . 9 1s No
54a1i 11 . . . . . . . 8 (𝐴 ∈ ℕ0s → 1s No )
63, 5subscld 28102 . . . . . . 7 (𝐴 ∈ ℕ0s → (𝐴 -s 1s ) ∈ No )
7 snelpwi 5466 . . . . . . 7 ((𝐴 -s 1s ) ∈ No → {(𝐴 -s 1s )} ∈ 𝒫 No )
8 nulssgt 27852 . . . . . . 7 ({(𝐴 -s 1s )} ∈ 𝒫 No → {(𝐴 -s 1s )} <<s ∅)
96, 7, 83syl 18 . . . . . 6 (𝐴 ∈ ℕ0s → {(𝐴 -s 1s )} <<s ∅)
10 slerflex 27817 . . . . . . . 8 ((𝐴 -s 1s ) ∈ No → (𝐴 -s 1s ) ≤s (𝐴 -s 1s ))
116, 10syl 17 . . . . . . 7 (𝐴 ∈ ℕ0s → (𝐴 -s 1s ) ≤s (𝐴 -s 1s ))
12 ovex 7478 . . . . . . . . 9 (𝐴 -s 1s ) ∈ V
13 breq1 5172 . . . . . . . . . 10 (𝑥 = (𝐴 -s 1s ) → (𝑥 ≤s 𝑦 ↔ (𝐴 -s 1s ) ≤s 𝑦))
1413rexbidv 3181 . . . . . . . . 9 (𝑥 = (𝐴 -s 1s ) → (∃𝑦 ∈ {(𝐴 -s 1s )}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(𝐴 -s 1s )} (𝐴 -s 1s ) ≤s 𝑦))
1512, 14ralsn 4705 . . . . . . . 8 (∀𝑥 ∈ {(𝐴 -s 1s )}∃𝑦 ∈ {(𝐴 -s 1s )}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(𝐴 -s 1s )} (𝐴 -s 1s ) ≤s 𝑦)
16 breq2 5173 . . . . . . . . 9 (𝑦 = (𝐴 -s 1s ) → ((𝐴 -s 1s ) ≤s 𝑦 ↔ (𝐴 -s 1s ) ≤s (𝐴 -s 1s )))
1712, 16rexsn 4706 . . . . . . . 8 (∃𝑦 ∈ {(𝐴 -s 1s )} (𝐴 -s 1s ) ≤s 𝑦 ↔ (𝐴 -s 1s ) ≤s (𝐴 -s 1s ))
1815, 17bitri 275 . . . . . . 7 (∀𝑥 ∈ {(𝐴 -s 1s )}∃𝑦 ∈ {(𝐴 -s 1s )}𝑥 ≤s 𝑦 ↔ (𝐴 -s 1s ) ≤s (𝐴 -s 1s ))
1911, 18sylibr 234 . . . . . 6 (𝐴 ∈ ℕ0s → ∀𝑥 ∈ {(𝐴 -s 1s )}∃𝑦 ∈ {(𝐴 -s 1s )}𝑥 ≤s 𝑦)
20 ral0 4532 . . . . . . 7 𝑥 ∈ ∅ ∃𝑦 ∈ {(𝐴 +s 1s )}𝑦 ≤s 𝑥
2120a1i 11 . . . . . 6 (𝐴 ∈ ℕ0s → ∀𝑥 ∈ ∅ ∃𝑦 ∈ {(𝐴 +s 1s )}𝑦 ≤s 𝑥)
223sltm1d 28140 . . . . . . . 8 (𝐴 ∈ ℕ0s → (𝐴 -s 1s ) <s 𝐴)
236, 3, 22ssltsn 27846 . . . . . . 7 (𝐴 ∈ ℕ0s → {(𝐴 -s 1s )} <<s {𝐴})
242sneqd 4660 . . . . . . 7 (𝐴 ∈ ℕ0s → {𝐴} = {({(𝐴 -s 1s )} |s ∅)})
2523, 24breqtrd 5195 . . . . . 6 (𝐴 ∈ ℕ0s → {(𝐴 -s 1s )} <<s {({(𝐴 -s 1s )} |s ∅)})
263, 5addscld 28022 . . . . . . . 8 (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ No )
273sltp1d 28057 . . . . . . . 8 (𝐴 ∈ ℕ0s𝐴 <s (𝐴 +s 1s ))
283, 26, 27ssltsn 27846 . . . . . . 7 (𝐴 ∈ ℕ0s → {𝐴} <<s {(𝐴 +s 1s )})
2924, 28eqbrtrrd 5193 . . . . . 6 (𝐴 ∈ ℕ0s → {({(𝐴 -s 1s )} |s ∅)} <<s {(𝐴 +s 1s )})
309, 19, 21, 25, 29cofcut1d 27964 . . . . 5 (𝐴 ∈ ℕ0s → ({(𝐴 -s 1s )} |s ∅) = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
312, 30eqtrd 2774 . . . 4 (𝐴 ∈ ℕ0s𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
3231adantl 481 . . 3 ((𝐴 No 𝐴 ∈ ℕ0s) → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
33 negsfn 28064 . . . . . . . 8 -us Fn No
34 simpl 482 . . . . . . . . 9 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → 𝐴 No )
354a1i 11 . . . . . . . . 9 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → 1s No )
3634, 35addscld 28022 . . . . . . . 8 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → (𝐴 +s 1s ) ∈ No )
37 fnsnfv 6999 . . . . . . . 8 (( -us Fn No ∧ (𝐴 +s 1s ) ∈ No ) → {( -us ‘(𝐴 +s 1s ))} = ( -us “ {(𝐴 +s 1s )}))
3833, 36, 37sylancr 586 . . . . . . 7 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 +s 1s ))} = ( -us “ {(𝐴 +s 1s )}))
39 negsdi 28091 . . . . . . . . . 10 ((𝐴 No ∧ 1s No ) → ( -us ‘(𝐴 +s 1s )) = (( -us𝐴) +s ( -us ‘ 1s )))
4034, 4, 39sylancl 585 . . . . . . . . 9 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 +s 1s )) = (( -us𝐴) +s ( -us ‘ 1s )))
41 n0sno 28337 . . . . . . . . . . 11 (( -us𝐴) ∈ ℕ0s → ( -us𝐴) ∈ No )
4241adantl 481 . . . . . . . . . 10 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ( -us𝐴) ∈ No )
4342, 35subsvald 28100 . . . . . . . . 9 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → (( -us𝐴) -s 1s ) = (( -us𝐴) +s ( -us ‘ 1s )))
4440, 43eqtr4d 2777 . . . . . . . 8 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 +s 1s )) = (( -us𝐴) -s 1s ))
4544sneqd 4660 . . . . . . 7 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 +s 1s ))} = {(( -us𝐴) -s 1s )})
4638, 45eqtr3d 2776 . . . . . 6 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ( -us “ {(𝐴 +s 1s )}) = {(( -us𝐴) -s 1s )})
4734, 35subscld 28102 . . . . . . . 8 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → (𝐴 -s 1s ) ∈ No )
48 fnsnfv 6999 . . . . . . . 8 (( -us Fn No ∧ (𝐴 -s 1s ) ∈ No ) → {( -us ‘(𝐴 -s 1s ))} = ( -us “ {(𝐴 -s 1s )}))
4933, 47, 48sylancr 586 . . . . . . 7 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 -s 1s ))} = ( -us “ {(𝐴 -s 1s )}))
5035, 34subsvald 28100 . . . . . . . . 9 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ( 1s -s 𝐴) = ( 1s +s ( -us𝐴)))
5134, 35negsubsdi2d 28119 . . . . . . . . 9 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 -s 1s )) = ( 1s -s 𝐴))
5242, 35addscomd 28009 . . . . . . . . 9 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → (( -us𝐴) +s 1s ) = ( 1s +s ( -us𝐴)))
5350, 51, 523eqtr4d 2784 . . . . . . . 8 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 -s 1s )) = (( -us𝐴) +s 1s ))
5453sneqd 4660 . . . . . . 7 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 -s 1s ))} = {(( -us𝐴) +s 1s )})
5549, 54eqtr3d 2776 . . . . . 6 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ( -us “ {(𝐴 -s 1s )}) = {(( -us𝐴) +s 1s )})
5646, 55oveq12d 7463 . . . . 5 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → (( -us “ {(𝐴 +s 1s )}) |s ( -us “ {(𝐴 -s 1s )})) = ({(( -us𝐴) -s 1s )} |s {(( -us𝐴) +s 1s )}))
5734sltm1d 28140 . . . . . . . 8 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → (𝐴 -s 1s ) <s 𝐴)
5834sltp1d 28057 . . . . . . . 8 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → 𝐴 <s (𝐴 +s 1s ))
5947, 34, 36, 57, 58slttrd 27813 . . . . . . 7 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → (𝐴 -s 1s ) <s (𝐴 +s 1s ))
6047, 36, 59ssltsn 27846 . . . . . 6 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → {(𝐴 -s 1s )} <<s {(𝐴 +s 1s )})
61 eqidd 2735 . . . . . 6 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}) = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
6260, 61negsunif 28096 . . . . 5 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ( -us ‘({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})) = (( -us “ {(𝐴 +s 1s )}) |s ( -us “ {(𝐴 -s 1s )})))
63 n0scut 28347 . . . . . . 7 (( -us𝐴) ∈ ℕ0s → ( -us𝐴) = ({(( -us𝐴) -s 1s )} |s ∅))
644a1i 11 . . . . . . . . . 10 (( -us𝐴) ∈ ℕ0s → 1s No )
6541, 64subscld 28102 . . . . . . . . 9 (( -us𝐴) ∈ ℕ0s → (( -us𝐴) -s 1s ) ∈ No )
66 snelpwi 5466 . . . . . . . . 9 ((( -us𝐴) -s 1s ) ∈ No → {(( -us𝐴) -s 1s )} ∈ 𝒫 No )
67 nulssgt 27852 . . . . . . . . 9 ({(( -us𝐴) -s 1s )} ∈ 𝒫 No → {(( -us𝐴) -s 1s )} <<s ∅)
6865, 66, 673syl 18 . . . . . . . 8 (( -us𝐴) ∈ ℕ0s → {(( -us𝐴) -s 1s )} <<s ∅)
69 slerflex 27817 . . . . . . . . . 10 ((( -us𝐴) -s 1s ) ∈ No → (( -us𝐴) -s 1s ) ≤s (( -us𝐴) -s 1s ))
7065, 69syl 17 . . . . . . . . 9 (( -us𝐴) ∈ ℕ0s → (( -us𝐴) -s 1s ) ≤s (( -us𝐴) -s 1s ))
71 ovex 7478 . . . . . . . . . . 11 (( -us𝐴) -s 1s ) ∈ V
72 breq1 5172 . . . . . . . . . . . 12 (𝑥 = (( -us𝐴) -s 1s ) → (𝑥 ≤s 𝑦 ↔ (( -us𝐴) -s 1s ) ≤s 𝑦))
7372rexbidv 3181 . . . . . . . . . . 11 (𝑥 = (( -us𝐴) -s 1s ) → (∃𝑦 ∈ {(( -us𝐴) -s 1s )}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(( -us𝐴) -s 1s )} (( -us𝐴) -s 1s ) ≤s 𝑦))
7471, 73ralsn 4705 . . . . . . . . . 10 (∀𝑥 ∈ {(( -us𝐴) -s 1s )}∃𝑦 ∈ {(( -us𝐴) -s 1s )}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(( -us𝐴) -s 1s )} (( -us𝐴) -s 1s ) ≤s 𝑦)
75 breq2 5173 . . . . . . . . . . 11 (𝑦 = (( -us𝐴) -s 1s ) → ((( -us𝐴) -s 1s ) ≤s 𝑦 ↔ (( -us𝐴) -s 1s ) ≤s (( -us𝐴) -s 1s )))
7671, 75rexsn 4706 . . . . . . . . . 10 (∃𝑦 ∈ {(( -us𝐴) -s 1s )} (( -us𝐴) -s 1s ) ≤s 𝑦 ↔ (( -us𝐴) -s 1s ) ≤s (( -us𝐴) -s 1s ))
7774, 76bitri 275 . . . . . . . . 9 (∀𝑥 ∈ {(( -us𝐴) -s 1s )}∃𝑦 ∈ {(( -us𝐴) -s 1s )}𝑥 ≤s 𝑦 ↔ (( -us𝐴) -s 1s ) ≤s (( -us𝐴) -s 1s ))
7870, 77sylibr 234 . . . . . . . 8 (( -us𝐴) ∈ ℕ0s → ∀𝑥 ∈ {(( -us𝐴) -s 1s )}∃𝑦 ∈ {(( -us𝐴) -s 1s )}𝑥 ≤s 𝑦)
79 ral0 4532 . . . . . . . . 9 𝑥 ∈ ∅ ∃𝑦 ∈ {(( -us𝐴) +s 1s )}𝑦 ≤s 𝑥
8079a1i 11 . . . . . . . 8 (( -us𝐴) ∈ ℕ0s → ∀𝑥 ∈ ∅ ∃𝑦 ∈ {(( -us𝐴) +s 1s )}𝑦 ≤s 𝑥)
8141sltm1d 28140 . . . . . . . . . 10 (( -us𝐴) ∈ ℕ0s → (( -us𝐴) -s 1s ) <s ( -us𝐴))
8265, 41, 81ssltsn 27846 . . . . . . . . 9 (( -us𝐴) ∈ ℕ0s → {(( -us𝐴) -s 1s )} <<s {( -us𝐴)})
8363sneqd 4660 . . . . . . . . 9 (( -us𝐴) ∈ ℕ0s → {( -us𝐴)} = {({(( -us𝐴) -s 1s )} |s ∅)})
8482, 83breqtrd 5195 . . . . . . . 8 (( -us𝐴) ∈ ℕ0s → {(( -us𝐴) -s 1s )} <<s {({(( -us𝐴) -s 1s )} |s ∅)})
8541, 64addscld 28022 . . . . . . . . . 10 (( -us𝐴) ∈ ℕ0s → (( -us𝐴) +s 1s ) ∈ No )
8641sltp1d 28057 . . . . . . . . . 10 (( -us𝐴) ∈ ℕ0s → ( -us𝐴) <s (( -us𝐴) +s 1s ))
8741, 85, 86ssltsn 27846 . . . . . . . . 9 (( -us𝐴) ∈ ℕ0s → {( -us𝐴)} <<s {(( -us𝐴) +s 1s )})
8883, 87eqbrtrrd 5193 . . . . . . . 8 (( -us𝐴) ∈ ℕ0s → {({(( -us𝐴) -s 1s )} |s ∅)} <<s {(( -us𝐴) +s 1s )})
8968, 78, 80, 84, 88cofcut1d 27964 . . . . . . 7 (( -us𝐴) ∈ ℕ0s → ({(( -us𝐴) -s 1s )} |s ∅) = ({(( -us𝐴) -s 1s )} |s {(( -us𝐴) +s 1s )}))
9063, 89eqtrd 2774 . . . . . 6 (( -us𝐴) ∈ ℕ0s → ( -us𝐴) = ({(( -us𝐴) -s 1s )} |s {(( -us𝐴) +s 1s )}))
9190adantl 481 . . . . 5 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ( -us𝐴) = ({(( -us𝐴) -s 1s )} |s {(( -us𝐴) +s 1s )}))
9256, 62, 913eqtr4rd 2785 . . . 4 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ( -us𝐴) = ( -us ‘({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})))
9360scutcld 27857 . . . . 5 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}) ∈ No )
94 negs11 28090 . . . . 5 ((𝐴 No ∧ ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}) ∈ No ) → (( -us𝐴) = ( -us ‘({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})) ↔ 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})))
9593, 94syldan 590 . . . 4 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → (( -us𝐴) = ( -us ‘({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})) ↔ 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})))
9692, 95mpbid 232 . . 3 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
9732, 96jaodan 958 . 2 ((𝐴 No ∧ (𝐴 ∈ ℕ0s ∨ ( -us𝐴) ∈ ℕ0s)) → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
981, 97sylbi 217 1 (𝐴 ∈ ℤs𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 846   = wceq 1537  wcel 2103  wral 3063  wrex 3072  c0 4347  𝒫 cpw 4622  {csn 4648   class class class wbr 5169  cima 5702   Fn wfn 6567  cfv 6572  (class class class)co 7445   No csur 27693   ≤s csle 27798   <<s csslt 27834   |s cscut 27836   1s c1s 27877   +s cadds 28001   -us cnegs 28060   -s csubs 28061  0scnn0s 28327  sczs 28373
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  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 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-se 5655  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-2o 8519  df-nadd 8718  df-no 27696  df-slt 27697  df-bday 27698  df-sle 27799  df-sslt 27835  df-scut 27837  df-0s 27878  df-1s 27879  df-made 27895  df-old 27896  df-left 27898  df-right 27899  df-norec 27980  df-norec2 27991  df-adds 28002  df-negs 28062  df-subs 28063  df-n0s 28329  df-nns 28330  df-zs 28374
This theorem is referenced by:  zs12bday  28433
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