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Theorem recut 28428
Description: The cut involved in defining surreal reals is a genuine cut. (Contributed by Scott Fenton, 15-Apr-2025.)
Assertion
Ref Expression
recut (𝐴 No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
Distinct variable group:   𝑥,𝐴,𝑛

Proof of Theorem recut
Dummy variables 𝑦 𝑧 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnsex 28323 . . . 4 s ∈ V
21abrexex 7987 . . 3 {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ∈ V
32a1i 11 . 2 (𝐴 No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ∈ V)
41abrexex 7987 . . 3 {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ∈ V
54a1i 11 . 2 (𝐴 No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ∈ V)
6 1sno 27872 . . . . . . . 8 1s No
76a1i 11 . . . . . . 7 (𝑛 ∈ ℕs → 1s No )
8 nnsno 28329 . . . . . . 7 (𝑛 ∈ ℕs𝑛 No )
9 nnne0s 28340 . . . . . . 7 (𝑛 ∈ ℕs𝑛 ≠ 0s )
107, 8, 9divscld 28248 . . . . . 6 (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
11 subscl 28092 . . . . . 6 ((𝐴 No ∧ ( 1s /su 𝑛) ∈ No ) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
1210, 11sylan2 593 . . . . 5 ((𝐴 No 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
13 eleq1 2829 . . . . 5 (𝑥 = (𝐴 -s ( 1s /su 𝑛)) → (𝑥 No ↔ (𝐴 -s ( 1s /su 𝑛)) ∈ No ))
1412, 13syl5ibrcom 247 . . . 4 ((𝐴 No 𝑛 ∈ ℕs) → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) → 𝑥 No ))
1514rexlimdva 3155 . . 3 (𝐴 No → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) → 𝑥 No ))
1615abssdv 4068 . 2 (𝐴 No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No )
17 addscl 28014 . . . . . 6 ((𝐴 No ∧ ( 1s /su 𝑛) ∈ No ) → (𝐴 +s ( 1s /su 𝑛)) ∈ No )
1810, 17sylan2 593 . . . . 5 ((𝐴 No 𝑛 ∈ ℕs) → (𝐴 +s ( 1s /su 𝑛)) ∈ No )
19 eleq1 2829 . . . . 5 (𝑥 = (𝐴 +s ( 1s /su 𝑛)) → (𝑥 No ↔ (𝐴 +s ( 1s /su 𝑛)) ∈ No ))
2018, 19syl5ibrcom 247 . . . 4 ((𝐴 No 𝑛 ∈ ℕs) → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) → 𝑥 No ))
2120rexlimdva 3155 . . 3 (𝐴 No → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) → 𝑥 No ))
2221abssdv 4068 . 2 (𝐴 No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No )
23 vex 3484 . . . . . . 7 𝑦 ∈ V
24 eqeq1 2741 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
2524rexbidv 3179 . . . . . . 7 (𝑥 = 𝑦 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
2623, 25elab 3679 . . . . . 6 (𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)))
27 vex 3484 . . . . . . 7 𝑧 ∈ V
28 eqeq1 2741 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
2928rexbidv 3179 . . . . . . . 8 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
30 oveq2 7439 . . . . . . . . . . 11 (𝑛 = 𝑚 → ( 1s /su 𝑛) = ( 1s /su 𝑚))
3130oveq2d 7447 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐴 +s ( 1s /su 𝑛)) = (𝐴 +s ( 1s /su 𝑚)))
3231eqeq2d 2748 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
3332cbvrexvw 3238 . . . . . . . 8 (∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚)))
3429, 33bitrdi 287 . . . . . . 7 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
3527, 34elab 3679 . . . . . 6 (𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ↔ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚)))
3626, 35anbi12i 628 . . . . 5 ((𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ↔ (∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
37 reeanv 3229 . . . . 5 (∃𝑛 ∈ ℕs𝑚 ∈ ℕs (𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝐴 +s ( 1s /su 𝑚))) ↔ (∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
3836, 37bitr4i 278 . . . 4 ((𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ↔ ∃𝑛 ∈ ℕs𝑚 ∈ ℕs (𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
39 simpl 482 . . . . . . . . 9 ((𝐴 No 𝑛 ∈ ℕs) → 𝐴 No )
4010adantl 481 . . . . . . . . 9 ((𝐴 No 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
4139, 40subsvald 28091 . . . . . . . 8 ((𝐴 No 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
4241adantrr 717 . . . . . . 7 ((𝐴 No ∧ (𝑛 ∈ ℕs𝑚 ∈ ℕs)) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
4310negscld 28069 . . . . . . . . . . 11 (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) ∈ No )
4443adantr 480 . . . . . . . . . 10 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) ∈ No )
45 0sno 27871 . . . . . . . . . . 11 0s No
4645a1i 11 . . . . . . . . . 10 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → 0s No )
476a1i 11 . . . . . . . . . . . 12 (𝑚 ∈ ℕs → 1s No )
48 nnsno 28329 . . . . . . . . . . . 12 (𝑚 ∈ ℕs𝑚 No )
49 nnne0s 28340 . . . . . . . . . . . 12 (𝑚 ∈ ℕs𝑚 ≠ 0s )
5047, 48, 49divscld 28248 . . . . . . . . . . 11 (𝑚 ∈ ℕs → ( 1s /su 𝑚) ∈ No )
5150adantl 481 . . . . . . . . . 10 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → ( 1s /su 𝑚) ∈ No )
52 id 22 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕs𝑛 ∈ ℕs)
5352nnsrecgt0d 28356 . . . . . . . . . . . . 13 (𝑛 ∈ ℕs → 0s <s ( 1s /su 𝑛))
5445a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕs → 0s No )
5554, 10sltnegd 28079 . . . . . . . . . . . . 13 (𝑛 ∈ ℕs → ( 0s <s ( 1s /su 𝑛) ↔ ( -us ‘( 1s /su 𝑛)) <s ( -us ‘ 0s )))
5653, 55mpbid 232 . . . . . . . . . . . 12 (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) <s ( -us ‘ 0s ))
57 negs0s 28058 . . . . . . . . . . . 12 ( -us ‘ 0s ) = 0s
5856, 57breqtrdi 5184 . . . . . . . . . . 11 (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) <s 0s )
5958adantr 480 . . . . . . . . . 10 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) <s 0s )
60 id 22 . . . . . . . . . . . 12 (𝑚 ∈ ℕs𝑚 ∈ ℕs)
6160nnsrecgt0d 28356 . . . . . . . . . . 11 (𝑚 ∈ ℕs → 0s <s ( 1s /su 𝑚))
6261adantl 481 . . . . . . . . . 10 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → 0s <s ( 1s /su 𝑚))
6344, 46, 51, 59, 62slttrd 27804 . . . . . . . . 9 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) <s ( 1s /su 𝑚))
6463adantl 481 . . . . . . . 8 ((𝐴 No ∧ (𝑛 ∈ ℕs𝑚 ∈ ℕs)) → ( -us ‘( 1s /su 𝑛)) <s ( 1s /su 𝑚))
6544adantl 481 . . . . . . . . 9 ((𝐴 No ∧ (𝑛 ∈ ℕs𝑚 ∈ ℕs)) → ( -us ‘( 1s /su 𝑛)) ∈ No )
6650ad2antll 729 . . . . . . . . 9 ((𝐴 No ∧ (𝑛 ∈ ℕs𝑚 ∈ ℕs)) → ( 1s /su 𝑚) ∈ No )
67 simpl 482 . . . . . . . . 9 ((𝐴 No ∧ (𝑛 ∈ ℕs𝑚 ∈ ℕs)) → 𝐴 No )
6865, 66, 67sltadd2d 28030 . . . . . . . 8 ((𝐴 No ∧ (𝑛 ∈ ℕs𝑚 ∈ ℕs)) → (( -us ‘( 1s /su 𝑛)) <s ( 1s /su 𝑚) ↔ (𝐴 +s ( -us ‘( 1s /su 𝑛))) <s (𝐴 +s ( 1s /su 𝑚))))
6964, 68mpbid 232 . . . . . . 7 ((𝐴 No ∧ (𝑛 ∈ ℕs𝑚 ∈ ℕs)) → (𝐴 +s ( -us ‘( 1s /su 𝑛))) <s (𝐴 +s ( 1s /su 𝑚)))
7042, 69eqbrtrd 5165 . . . . . 6 ((𝐴 No ∧ (𝑛 ∈ ℕs𝑚 ∈ ℕs)) → (𝐴 -s ( 1s /su 𝑛)) <s (𝐴 +s ( 1s /su 𝑚)))
71 breq12 5148 . . . . . 6 ((𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝐴 +s ( 1s /su 𝑚))) → (𝑦 <s 𝑧 ↔ (𝐴 -s ( 1s /su 𝑛)) <s (𝐴 +s ( 1s /su 𝑚))))
7270, 71syl5ibrcom 247 . . . . 5 ((𝐴 No ∧ (𝑛 ∈ ℕs𝑚 ∈ ℕs)) → ((𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝐴 +s ( 1s /su 𝑚))) → 𝑦 <s 𝑧))
7372rexlimdvva 3213 . . . 4 (𝐴 No → (∃𝑛 ∈ ℕs𝑚 ∈ ℕs (𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝐴 +s ( 1s /su 𝑚))) → 𝑦 <s 𝑧))
7438, 73biimtrid 242 . . 3 (𝐴 No → ((𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) → 𝑦 <s 𝑧))
75743impib 1117 . 2 ((𝐴 No 𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) → 𝑦 <s 𝑧)
763, 5, 16, 22, 75ssltd 27836 1 (𝐴 No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480   class class class wbr 5143  cfv 6561  (class class class)co 7431   No csur 27684   <s cslt 27685   <<s csslt 27825   0s c0s 27867   1s c1s 27868   +s cadds 27992   -us cnegs 28051   -s csubs 28052   /su cdivs 28213  scnns 28319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-dc 10486
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-muls 28133  df-divs 28214  df-n0s 28320  df-nns 28321
This theorem is referenced by:  renegscl  28430  readdscl  28431  remulscl  28434
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