Theorem negsproplem3 27936

Description: Lemma for surreal negation. Give the cut properties of surreal negation. (Contributed by Scott Fenton, 2-Feb-2025.)

Hypotheses

Ref	Expression
negsproplem.1	⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
negsproplem2.1	⊢ (𝜑 → 𝐴 ∈ No )

Assertion

Ref	Expression
negsproplem3	⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem negsproplem3

Step	Hyp	Ref	Expression
1		negsproplem.1	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
2		negsproplem2.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ No )
3	1, 2	negsproplem2 27935	. . 3 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
4		scutcut 27713	. . 3 ⊢ (( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)) → ((( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} ∧ {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} <<s ( -us “ ( L ‘𝐴))))
5	3, 4	syl 17	. 2 ⊢ (𝜑 → ((( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} ∧ {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} <<s ( -us “ ( L ‘𝐴))))
6		negsval 27931	. . . . 5 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
7	2, 6	syl 17	. . . 4 ⊢ (𝜑 → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
8	7	eleq1d 2813	. . 3 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ↔ (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) ∈ No ))
9	7	sneqd 4601	. . . 4 ⊢ (𝜑 → {( -us ‘𝐴)} = {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))})
10	9	breq2d 5119	. . 3 ⊢ (𝜑 → (( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ↔ ( -us “ ( R ‘𝐴)) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))}))
11	9	breq1d 5117	. . 3 ⊢ (𝜑 → ({( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴)) ↔ {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} <<s ( -us “ ( L ‘𝐴))))
12	8, 10, 11	3anbi123d 1438	. 2 ⊢ (𝜑 → ((( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))) ↔ ((( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} ∧ {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} <<s ( -us “ ( L ‘𝐴)))))
13	5, 12	mpbird 257	1 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∪ cun 3912  {csn 4589   class class class wbr 5107   “ cima 5641  ‘cfv 6511  (class class class)co 7387   No csur 27551   <s cslt 27552   bday cbday 27553   <<s csslt 27692   |s cscut 27694   L cleft 27753   R cright 27754   -u_s cnegs 27925

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-1o 8434  df-2o 8435  df-no 27554  df-slt 27555  df-bday 27556  df-sslt 27693  df-scut 27695  df-0s 27736  df-made 27755  df-old 27756  df-left 27758  df-right 27759  df-norec 27845  df-negs 27927

This theorem is referenced by:  negsproplem4  27937  negsproplem5  27938  negsproplem6  27939  negsprop  27941  negscut  27945