Theorem lesrecd 27792

Description: A comparison law for surreals considered as cuts of sets of surreals. Definition from [Conway] p. 4. Theorem 4 of [Alling] p. 186. Theorem 2.5 of [Gonshor] p. 9. (Contributed by Scott Fenton, 5-Dec-2025.)

Hypotheses

Ref	Expression
lesrecd.1	⊢ (𝜑 → 𝐴 <<s 𝐵)
lesrecd.2	⊢ (𝜑 → 𝐶 <<s 𝐷)
lesrecd.3	⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))
lesrecd.4	⊢ (𝜑 → 𝑌 = (𝐶 |s 𝐷))

Assertion

Ref	Expression
lesrecd	⊢ (𝜑 → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌)))

Distinct variable groups:   𝐴,𝑎,𝑑   𝐵,𝑎,𝑑   𝐶,𝑎,𝑑   𝐷,𝑎,𝑑   𝑋,𝑎,𝑑   𝑌,𝑎,𝑑

Allowed substitution hints:   𝜑(𝑎,𝑑)

Proof of Theorem lesrecd

Step	Hyp	Ref	Expression
1		lesrecd.1	. 2 ⊢ (𝜑 → 𝐴 <<s 𝐵)
2		lesrecd.2	. 2 ⊢ (𝜑 → 𝐶 <<s 𝐷)
3		lesrecd.3	. 2 ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))
4		lesrecd.4	. 2 ⊢ (𝜑 → 𝑌 = (𝐶 |s 𝐷))
5		lesrec 27791	. 2 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌)))
6	1, 2, 3, 4, 5	syl22anc 839	1 ⊢ (𝜑 → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∀wral 3051   class class class wbr 5085  (class class class)co 7367   <s clts 27604   ≤s cles 27708   <<s cslts 27749   |s ccuts 27751

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752

This theorem is referenced by:  ltsrec  27793  eqcuts3  27796  rightge0  27813  onsbnd  28273