Theorem negsproplem3 28062

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 28061	. . 3 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
4		scutcut 27846	. . 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 28057	. . . . 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 2826	. . 3 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ↔ (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) ∈ No ))
9	7	sneqd 4638	. . . 4 ⊢ (𝜑 → {( -us ‘𝐴)} = {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))})
10	9	breq2d 5155	. . 3 ⊢ (𝜑 → (( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ↔ ( -us “ ( R ‘𝐴)) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))}))
11	9	breq1d 5153	. . 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 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ∪ cun 3949  {csn 4626   class class class wbr 5143   “ cima 5688  ‘cfv 6561  (class class class)co 7431   No csur 27684   <s cslt 27685   bday cbday 27686   <<s csslt 27825   |s cscut 27827   L cleft 27884   R cright 27885   -u_s cnegs 28051

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-negs 28053

This theorem is referenced by:  negsproplem4  28063  negsproplem5  28064  negsproplem6  28065  negsprop  28067  negscut  28071