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Theorem sltrec 32250
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 776 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 <<s 𝐷)
2 simpll 774 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐴 <<s 𝐵)
3 simprr 780 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 = (𝐶 |s 𝐷))
4 simprl 778 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑋 = (𝐴 |s 𝐵))
5 slerec 32249 . . . . 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 858 . . . 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 450 . . . 4 ((∀𝑏𝐵 𝑌 <s 𝑏 ∧ ∀𝑐𝐶 𝑐 <s 𝑋) ↔ (∀𝑐𝐶 𝑐 <s 𝑋 ∧ ∀𝑏𝐵 𝑌 <s 𝑏))
86, 7syl6bb 278 . . 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 32238 . . . . . . 7 (𝐶 <<s 𝐷 → ((𝐶 |s 𝐷) ∈ No 𝐶 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐷))
109simp1d 1165 . . . . . 6 (𝐶 <<s 𝐷 → (𝐶 |s 𝐷) ∈ No )
1110ad2antlr 709 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐶 |s 𝐷) ∈ No )
123, 11eqeltrd 2892 . . . 4 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 No )
13 scutcut 32238 . . . . . . 7 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
1413simp1d 1165 . . . . . 6 (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
1514ad2antrr 708 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ∈ No )
164, 15eqeltrd 2892 . . . 4 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑋 No )
17 slenlt 32203 . . . 4 ((𝑌 No 𝑋 No ) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
1812, 16, 17syl2anc 575 . . 3 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
19 ssltss1 32229 . . . . . . . . 9 (𝐶 <<s 𝐷𝐶 No )
2019ad2antlr 709 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 No )
2120sselda 3805 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐𝐶) → 𝑐 No )
2216adantr 468 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐𝐶) → 𝑋 No )
23 sltnle 32204 . . . . . . 7 ((𝑐 No 𝑋 No ) → (𝑐 <s 𝑋 ↔ ¬ 𝑋 ≤s 𝑐))
2421, 22, 23syl2anc 575 . . . . . 6 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐𝐶) → (𝑐 <s 𝑋 ↔ ¬ 𝑋 ≤s 𝑐))
2524ralbidva 3180 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (∀𝑐𝐶 𝑐 <s 𝑋 ↔ ∀𝑐𝐶 ¬ 𝑋 ≤s 𝑐))
2612adantr 468 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏𝐵) → 𝑌 No )
27 ssltss2 32230 . . . . . . . . 9 (𝐴 <<s 𝐵𝐵 No )
2827ad2antrr 708 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐵 No )
2928sselda 3805 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏𝐵) → 𝑏 No )
30 sltnle 32204 . . . . . . 7 ((𝑌 No 𝑏 No ) → (𝑌 <s 𝑏 ↔ ¬ 𝑏 ≤s 𝑌))
3126, 29, 30syl2anc 575 . . . . . 6 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏𝐵) → (𝑌 <s 𝑏 ↔ ¬ 𝑏 ≤s 𝑌))
3231ralbidva 3180 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (∀𝑏𝐵 𝑌 <s 𝑏 ↔ ∀𝑏𝐵 ¬ 𝑏 ≤s 𝑌))
3325, 32anbi12d 618 . . . 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 3187 . . . . . 6 (∀𝑐𝐶 ¬ 𝑋 ≤s 𝑐 ↔ ¬ ∃𝑐𝐶 𝑋 ≤s 𝑐)
35 ralnex 3187 . . . . . 6 (∀𝑏𝐵 ¬ 𝑏 ≤s 𝑌 ↔ ¬ ∃𝑏𝐵 𝑏 ≤s 𝑌)
3634, 35anbi12i 614 . . . . 5 ((∀𝑐𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏𝐵 ¬ 𝑏 ≤s 𝑌) ↔ (¬ ∃𝑐𝐶 𝑋 ≤s 𝑐 ∧ ¬ ∃𝑏𝐵 𝑏 ≤s 𝑌))
37 ioran 997 . . . . 5 (¬ (∃𝑐𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏𝐵 𝑏 ≤s 𝑌) ↔ (¬ ∃𝑐𝐶 𝑋 ≤s 𝑐 ∧ ¬ ∃𝑏𝐵 𝑏 ≤s 𝑌))
3836, 37bitr4i 269 . . . 4 ((∀𝑐𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏𝐵 ¬ 𝑏 ≤s 𝑌) ↔ ¬ (∃𝑐𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏𝐵 𝑏 ≤s 𝑌))
3933, 38syl6bb 278 . . 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 300 . 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 308 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 197  wa 384  wo 865   = wceq 1637  wcel 2157  wral 3103  wrex 3104  wss 3776  {csn 4377   class class class wbr 4851  (class class class)co 6877   No csur 32119   <s cslt 32120   ≤s csle 32195   <<s csslt 32222   |s cscut 32224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-ord 5946  df-on 5947  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-1o 7799  df-2o 7800  df-no 32122  df-slt 32123  df-bday 32124  df-sle 32196  df-sslt 32223  df-scut 32225
This theorem is referenced by: (None)
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