MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  recut Structured version   Visualization version   GIF version

Theorem recut 28443
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 28338 . . . 4 s ∈ V
21abrexex 7986 . . 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 7986 . . 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 27887 . . . . . . . 8 1s No
76a1i 11 . . . . . . 7 (𝑛 ∈ ℕs → 1s No )
8 nnsno 28344 . . . . . . 7 (𝑛 ∈ ℕs𝑛 No )
9 nnne0s 28355 . . . . . . 7 (𝑛 ∈ ℕs𝑛 ≠ 0s )
107, 8, 9divscld 28263 . . . . . 6 (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
11 subscl 28107 . . . . . 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 2827 . . . . 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 3153 . . 3 (𝐴 No → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) → 𝑥 No ))
1615abssdv 4078 . 2 (𝐴 No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No )
17 addscl 28029 . . . . . 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 2827 . . . . 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 3153 . . 3 (𝐴 No → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) → 𝑥 No ))
2221abssdv 4078 . 2 (𝐴 No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No )
23 vex 3482 . . . . . . 7 𝑦 ∈ V
24 eqeq1 2739 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
2524rexbidv 3177 . . . . . . 7 (𝑥 = 𝑦 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
2623, 25elab 3681 . . . . . 6 (𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)))
27 vex 3482 . . . . . . 7 𝑧 ∈ V
28 eqeq1 2739 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
2928rexbidv 3177 . . . . . . . 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 2746 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
3332cbvrexvw 3236 . . . . . . . 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 3681 . . . . . 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 3227 . . . . 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 28106 . . . . . . . 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 28084 . . . . . . . . . . 11 (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) ∈ No )
4443adantr 480 . . . . . . . . . 10 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) ∈ No )
45 0sno 27886 . . . . . . . . . . 11 0s No
4645a1i 11 . . . . . . . . . 10 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → 0s No )
476a1i 11 . . . . . . . . . . . 12 (𝑚 ∈ ℕs → 1s No )
48 nnsno 28344 . . . . . . . . . . . 12 (𝑚 ∈ ℕs𝑚 No )
49 nnne0s 28355 . . . . . . . . . . . 12 (𝑚 ∈ ℕs𝑚 ≠ 0s )
5047, 48, 49divscld 28263 . . . . . . . . . . 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 28371 . . . . . . . . . . . . 13 (𝑛 ∈ ℕs → 0s <s ( 1s /su 𝑛))
5445a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕs → 0s No )
5554, 10sltnegd 28094 . . . . . . . . . . . . 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 28073 . . . . . . . . . . . 12 ( -us ‘ 0s ) = 0s
5856, 57breqtrdi 5189 . . . . . . . . . . 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 28371 . . . . . . . . . . 11 (𝑚 ∈ ℕs → 0s <s ( 1s /su 𝑚))
6261adantl 481 . . . . . . . . . 10 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → 0s <s ( 1s /su 𝑚))
6344, 46, 51, 59, 62slttrd 27819 . . . . . . . . 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 28045 . . . . . . . 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 5170 . . . . . 6 ((𝐴 No ∧ (𝑛 ∈ ℕs𝑚 ∈ ℕs)) → (𝐴 -s ( 1s /su 𝑛)) <s (𝐴 +s ( 1s /su 𝑚)))
71 breq12 5153 . . . . . 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 3211 . . . 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 1115 . 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 27851 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 1537  wcel 2106  {cab 2712  wrex 3068  Vcvv 3478   class class class wbr 5148  cfv 6563  (class class class)co 7431   No csur 27699   <s cslt 27700   <<s csslt 27840   0s c0s 27882   1s c1s 27883   +s cadds 28007   -us cnegs 28066   -s csubs 28067   /su cdivs 28228  scnns 28334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-dc 10484
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148  df-divs 28229  df-n0s 28335  df-nns 28336
This theorem is referenced by:  renegscl  28445  readdscl  28446  remulscl  28449
  Copyright terms: Public domain W3C validator