Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sltrec Structured version   Visualization version   GIF version

Theorem sltrec 33352
Description: A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 11-Dec-2021.)
Assertion
Ref Expression
sltrec (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 <s 𝑌 ↔ (∃𝑐𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏𝐵 𝑏 ≤s 𝑌)))
Distinct variable groups:   𝐴,𝑏,𝑐   𝐵,𝑏,𝑐   𝐶,𝑏,𝑐   𝐷,𝑏,𝑐   𝑋,𝑏,𝑐   𝑌,𝑏,𝑐

Proof of Theorem sltrec
StepHypRef Expression
1 simplr 768 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 <<s 𝐷)
2 simpll 766 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐴 <<s 𝐵)
3 simprr 772 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 = (𝐶 |s 𝐷))
4 simprl 770 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑋 = (𝐴 |s 𝐵))
5 slerec 33351 . . . . 5 (((𝐶 <<s 𝐷𝐴 <<s 𝐵) ∧ (𝑌 = (𝐶 |s 𝐷) ∧ 𝑋 = (𝐴 |s 𝐵))) → (𝑌 ≤s 𝑋 ↔ (∀𝑏𝐵 𝑌 <s 𝑏 ∧ ∀𝑐𝐶 𝑐 <s 𝑋)))
61, 2, 3, 4, 5syl22anc 837 . . . 4 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑌 ≤s 𝑋 ↔ (∀𝑏𝐵 𝑌 <s 𝑏 ∧ ∀𝑐𝐶 𝑐 <s 𝑋)))
7 ancom 464 . . . 4 ((∀𝑏𝐵 𝑌 <s 𝑏 ∧ ∀𝑐𝐶 𝑐 <s 𝑋) ↔ (∀𝑐𝐶 𝑐 <s 𝑋 ∧ ∀𝑏𝐵 𝑌 <s 𝑏))
86, 7syl6bb 290 . . 3 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑌 ≤s 𝑋 ↔ (∀𝑐𝐶 𝑐 <s 𝑋 ∧ ∀𝑏𝐵 𝑌 <s 𝑏)))
9 scutcut 33340 . . . . . . 7 (𝐶 <<s 𝐷 → ((𝐶 |s 𝐷) ∈ No 𝐶 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐷))
109simp1d 1139 . . . . . 6 (𝐶 <<s 𝐷 → (𝐶 |s 𝐷) ∈ No )
1110ad2antlr 726 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐶 |s 𝐷) ∈ No )
123, 11eqeltrd 2914 . . . 4 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 No )
13 scutcut 33340 . . . . . . 7 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
1413simp1d 1139 . . . . . 6 (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
1514ad2antrr 725 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ∈ No )
164, 15eqeltrd 2914 . . . 4 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑋 No )
17 slenlt 33305 . . . 4 ((𝑌 No 𝑋 No ) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
1812, 16, 17syl2anc 587 . . 3 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
19 ssltss1 33331 . . . . . . . . 9 (𝐶 <<s 𝐷𝐶 No )
2019ad2antlr 726 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 No )
2120sselda 3942 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐𝐶) → 𝑐 No )
2216adantr 484 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐𝐶) → 𝑋 No )
23 sltnle 33306 . . . . . . 7 ((𝑐 No 𝑋 No ) → (𝑐 <s 𝑋 ↔ ¬ 𝑋 ≤s 𝑐))
2421, 22, 23syl2anc 587 . . . . . 6 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐𝐶) → (𝑐 <s 𝑋 ↔ ¬ 𝑋 ≤s 𝑐))
2524ralbidva 3186 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (∀𝑐𝐶 𝑐 <s 𝑋 ↔ ∀𝑐𝐶 ¬ 𝑋 ≤s 𝑐))
2612adantr 484 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏𝐵) → 𝑌 No )
27 ssltss2 33332 . . . . . . . . 9 (𝐴 <<s 𝐵𝐵 No )
2827ad2antrr 725 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐵 No )
2928sselda 3942 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏𝐵) → 𝑏 No )
30 sltnle 33306 . . . . . . 7 ((𝑌 No 𝑏 No ) → (𝑌 <s 𝑏 ↔ ¬ 𝑏 ≤s 𝑌))
3126, 29, 30syl2anc 587 . . . . . 6 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏𝐵) → (𝑌 <s 𝑏 ↔ ¬ 𝑏 ≤s 𝑌))
3231ralbidva 3186 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (∀𝑏𝐵 𝑌 <s 𝑏 ↔ ∀𝑏𝐵 ¬ 𝑏 ≤s 𝑌))
3325, 32anbi12d 633 . . . 4 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → ((∀𝑐𝐶 𝑐 <s 𝑋 ∧ ∀𝑏𝐵 𝑌 <s 𝑏) ↔ (∀𝑐𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏𝐵 ¬ 𝑏 ≤s 𝑌)))
34 ralnex 3224 . . . . . 6 (∀𝑐𝐶 ¬ 𝑋 ≤s 𝑐 ↔ ¬ ∃𝑐𝐶 𝑋 ≤s 𝑐)
35 ralnex 3224 . . . . . 6 (∀𝑏𝐵 ¬ 𝑏 ≤s 𝑌 ↔ ¬ ∃𝑏𝐵 𝑏 ≤s 𝑌)
3634, 35anbi12i 629 . . . . 5 ((∀𝑐𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏𝐵 ¬ 𝑏 ≤s 𝑌) ↔ (¬ ∃𝑐𝐶 𝑋 ≤s 𝑐 ∧ ¬ ∃𝑏𝐵 𝑏 ≤s 𝑌))
37 ioran 981 . . . . 5 (¬ (∃𝑐𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏𝐵 𝑏 ≤s 𝑌) ↔ (¬ ∃𝑐𝐶 𝑋 ≤s 𝑐 ∧ ¬ ∃𝑏𝐵 𝑏 ≤s 𝑌))
3836, 37bitr4i 281 . . . 4 ((∀𝑐𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏𝐵 ¬ 𝑏 ≤s 𝑌) ↔ ¬ (∃𝑐𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏𝐵 𝑏 ≤s 𝑌))
3933, 38syl6bb 290 . . 3 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → ((∀𝑐𝐶 𝑐 <s 𝑋 ∧ ∀𝑏𝐵 𝑌 <s 𝑏) ↔ ¬ (∃𝑐𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏𝐵 𝑏 ≤s 𝑌)))
408, 18, 393bitr3d 312 . 2 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (¬ 𝑋 <s 𝑌 ↔ ¬ (∃𝑐𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏𝐵 𝑏 ≤s 𝑌)))
4140con4bid 320 1 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 <s 𝑌 ↔ (∃𝑐𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏𝐵 𝑏 ≤s 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2114  wral 3130  wrex 3131  wss 3908  {csn 4539   class class class wbr 5042  (class class class)co 7140   No csur 33221   <s cslt 33222   ≤s csle 33297   <<s csslt 33324   |s cscut 33326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-ord 6172  df-on 6173  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1o 8089  df-2o 8090  df-no 33224  df-slt 33225  df-bday 33226  df-sle 33298  df-sslt 33325  df-scut 33327
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator