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

Theorem nosupinfsep 27777
Description: Given two sets of surreals, a surreal 𝑊 separates them iff its restriction to the maximum of dom 𝑆 and dom 𝑇 separates them. Corollary 4.4 of [Lipparini] p. 7. (Contributed by Scott Fenton, 9-Aug-2024.)
Hypotheses
Ref Expression
nosupinfsep.1 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
nosupinfsep.2 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
nosupinfsep (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → ((∀𝑎𝐴 𝑎 <s 𝑊 ∧ ∀𝑏𝐵 𝑊 <s 𝑏) ↔ (∀𝑎𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏)))
Distinct variable groups:   𝐴,𝑎,𝑔,𝑢,𝑣,𝑥,𝑦   𝑊,𝑎,𝑥   𝑆,𝑎   𝑇,𝑏   𝑊,𝑏,𝑔   𝑆,𝑏,𝑔,𝑥   𝑢,𝐵,𝑦   𝑇,𝑎,𝑔,𝑥   𝐵,𝑏,𝑔,𝑣,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑏)   𝐵(𝑎)   𝑆(𝑦,𝑣,𝑢)   𝑇(𝑦,𝑣,𝑢)   𝑊(𝑦,𝑣,𝑢)

Proof of Theorem nosupinfsep
StepHypRef Expression
1 ssun1 4178 . . . . . . . 8 dom 𝑆 ⊆ (dom 𝑆 ∪ dom 𝑇)
2 resabs1 6024 . . . . . . . 8 (dom 𝑆 ⊆ (dom 𝑆 ∪ dom 𝑇) → ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) = (𝑊 ↾ dom 𝑆))
31, 2ax-mp 5 . . . . . . 7 ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) = (𝑊 ↾ dom 𝑆)
43breq1i 5150 . . . . . 6 (((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ↔ (𝑊 ↾ dom 𝑆) <s 𝑆)
54notbii 320 . . . . 5 (¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆)
6 ssun2 4179 . . . . . . . 8 dom 𝑇 ⊆ (dom 𝑆 ∪ dom 𝑇)
7 resabs1 6024 . . . . . . . 8 (dom 𝑇 ⊆ (dom 𝑆 ∪ dom 𝑇) → ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇) = (𝑊 ↾ dom 𝑇))
86, 7ax-mp 5 . . . . . . 7 ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇) = (𝑊 ↾ dom 𝑇)
98breq2i 5151 . . . . . 6 (𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇) ↔ 𝑇 <s (𝑊 ↾ dom 𝑇))
109notbii 320 . . . . 5 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇) ↔ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))
115, 10anbi12i 628 . . . 4 ((¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇)) ↔ (¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇)))
1211bicomi 224 . . 3 ((¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇)) ↔ (¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇)))
1312a1i 11 . 2 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → ((¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇)) ↔ (¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇))))
14 simp1l 1198 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → 𝐴 No )
15 simp1r 1199 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → 𝐴 ∈ V)
16 simp3 1139 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → 𝑊 No )
17 nosupinfsep.1 . . . . 5 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
1817nosupbnd2 27761 . . . 4 ((𝐴 No 𝐴 ∈ V ∧ 𝑊 No ) → (∀𝑎𝐴 𝑎 <s 𝑊 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆))
1914, 15, 16, 18syl3anc 1373 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → (∀𝑎𝐴 𝑎 <s 𝑊 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆))
20 simp2l 1200 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → 𝐵 No )
21 simp2r 1201 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → 𝐵 ∈ V)
22 nosupinfsep.2 . . . . 5 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
2322noinfbnd2 27776 . . . 4 ((𝐵 No 𝐵 ∈ V ∧ 𝑊 No ) → (∀𝑏𝐵 𝑊 <s 𝑏 ↔ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇)))
2420, 21, 16, 23syl3anc 1373 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → (∀𝑏𝐵 𝑊 <s 𝑏 ↔ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇)))
2519, 24anbi12d 632 . 2 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → ((∀𝑎𝐴 𝑎 <s 𝑊 ∧ ∀𝑏𝐵 𝑊 <s 𝑏) ↔ (¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))))
2617nosupno 27748 . . . . . . . 8 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
27 nodmon 27695 . . . . . . . 8 (𝑆 No → dom 𝑆 ∈ On)
2826, 27syl 17 . . . . . . 7 ((𝐴 No 𝐴 ∈ V) → dom 𝑆 ∈ On)
29283ad2ant1 1134 . . . . . 6 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → dom 𝑆 ∈ On)
3022noinfno 27763 . . . . . . . 8 ((𝐵 No 𝐵 ∈ V) → 𝑇 No )
31 nodmon 27695 . . . . . . . 8 (𝑇 No → dom 𝑇 ∈ On)
3230, 31syl 17 . . . . . . 7 ((𝐵 No 𝐵 ∈ V) → dom 𝑇 ∈ On)
33323ad2ant2 1135 . . . . . 6 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → dom 𝑇 ∈ On)
34 onun2 6492 . . . . . 6 ((dom 𝑆 ∈ On ∧ dom 𝑇 ∈ On) → (dom 𝑆 ∪ dom 𝑇) ∈ On)
3529, 33, 34syl2anc 584 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → (dom 𝑆 ∪ dom 𝑇) ∈ On)
36 noreson 27705 . . . . 5 ((𝑊 No ∧ (dom 𝑆 ∪ dom 𝑇) ∈ On) → (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No )
3716, 35, 36syl2anc 584 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No )
3817nosupbnd2 27761 . . . 4 ((𝐴 No 𝐴 ∈ V ∧ (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No ) → (∀𝑎𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↔ ¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆))
3914, 15, 37, 38syl3anc 1373 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → (∀𝑎𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↔ ¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆))
4020, 21, 373jca 1129 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → (𝐵 No 𝐵 ∈ V ∧ (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No ))
4122noinfbnd2 27776 . . . 4 ((𝐵 No 𝐵 ∈ V ∧ (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No ) → (∀𝑏𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏 ↔ ¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇)))
4240, 41syl 17 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → (∀𝑏𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏 ↔ ¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇)))
4339, 42anbi12d 632 . 2 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → ((∀𝑎𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏) ↔ (¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇))))
4413, 25, 433bitr4d 311 1 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ 𝑊 No ) → ((∀𝑎𝐴 𝑎 <s 𝑊 ∧ ∀𝑏𝐵 𝑊 <s 𝑏) ↔ (∀𝑎𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  cun 3949  wss 3951  ifcif 4525  {csn 4626  cop 4632   class class class wbr 5143  cmpt 5225  dom cdm 5685  cres 5687  Oncon0 6384  suc csuc 6386  cio 6512  cfv 6561  crio 7387  1oc1o 8499  2oc2o 8500   No csur 27684   <s cslt 27685
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
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-uni 4908  df-int 4947  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-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-ord 6387  df-on 6388  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-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689
This theorem is referenced by:  noetalem1  27786
  Copyright terms: Public domain W3C validator