Theorem negsproplem3 28324

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 28323	. . 3 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
4		cutcuts 28075	. . 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 18	. 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 28319	. . . . 5 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
7	2, 6	syl 18	. . . 4 ⊢ (𝜑 → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
8	7	eleq1d 2845	. . 3 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ↔ (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) ∈ No ))
9	7	sneqd 4596	. . . 4 ⊢ (𝜑 → {( -us ‘𝐴)} = {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))})
10	9	breq2d 5115	. . 3 ⊢ (𝜑 → (( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ↔ ( -us “ ( R ‘𝐴)) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))}))
11	9	breq1d 5113	. . 3 ⊢ (𝜑 → ({( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴)) ↔ {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} <<s ( -us “ ( L ‘𝐴))))
12	8, 10, 11	3anbi123d 1464	. 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 260	1 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∪ cun 3897  {csn 4584   class class class wbr 5103   “ cima 5658  ‘cfv 6534  (class class class)co 7415   No csur 27905   <s clts 27906   bday cbday 27907   <<s cslts 28051   |s ccuts 28053   L cleft 28119   R cright 28120   -u_s cnegs 28313

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6300  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7372  df-ov 7418  df-oprab 7419  df-mpo 7420  df-2nd 7989  df-frecs 8282  df-wrecs 8313  df-recs 8362  df-1o 8459  df-2o 8460  df-no 27908  df-lts 27909  df-bday 27910  df-slts 28052  df-cuts 28054  df-0s 28101  df-made 28121  df-old 28122  df-left 28124  df-right 28125  df-norec 28232  df-negs 28315

This theorem is used by:  negsproplem4  28325  negsproplem5  28326  negsproplem6  28327  negsprop  28329  negcut  28333