Theorem negsproplem3 27943

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 27942	. . 3 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
4		scutcut 27720	. . 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 27938	. . . . 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 2814	. . 3 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ↔ (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) ∈ No ))
9	7	sneqd 4604	. . . 4 ⊢ (𝜑 → {( -us ‘𝐴)} = {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))})
10	9	breq2d 5122	. . 3 ⊢ (𝜑 → (( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ↔ ( -us “ ( R ‘𝐴)) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))}))
11	9	breq1d 5120	. . 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 3045   ∪ cun 3915  {csn 4592   class class class wbr 5110   “ cima 5644  ‘cfv 6514  (class class class)co 7390   No csur 27558   <s cslt 27559   bday cbday 27560   <<s csslt 27699   |s cscut 27701   L cleft 27760   R cright 27761   -u_s cnegs 27932

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-1o 8437  df-2o 8438  df-no 27561  df-slt 27562  df-bday 27563  df-sslt 27700  df-scut 27702  df-0s 27743  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec 27852  df-negs 27934

This theorem is referenced by:  negsproplem4  27944  negsproplem5  27945  negsproplem6  27946  negsprop  27948  negscut  27952