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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
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