Theorem negsproplem6 28352

Description: Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐴 is the same age as 𝐵. (Contributed by Scott Fenton, 3-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)))))
negsproplem4.1	⊢ (𝜑 → 𝐴 ∈ No )
negsproplem4.2	⊢ (𝜑 → 𝐵 ∈ No )
negsproplem4.3	⊢ (𝜑 → 𝐴 <s 𝐵)
negsproplem6.4	⊢ (𝜑 → ( bday ‘𝐴) = ( bday ‘𝐵))

Assertion

Ref	Expression
negsproplem6	⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

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

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem negsproplem6

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		negsproplem4.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
2		negsproplem4.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ No )
3		negsproplem6.4	. . 3 ⊢ (𝜑 → ( bday ‘𝐴) = ( bday ‘𝐵))
4		negsproplem4.3	. . 3 ⊢ (𝜑 → 𝐴 <s 𝐵)
5		nodense 27982	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑑 ∈ No (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))
6	1, 2, 3, 4, 5	syl22anc 852	. 2 ⊢ (𝜑 → ∃𝑑 ∈ No (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))
7		negsproplem.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
8		uncom 4104	. . . . . . . . . 10 ⊢ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) = (( bday ‘𝐵) ∪ ( bday ‘𝐴))
9	8	eleq2i 2852	. . . . . . . . 9 ⊢ ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) ↔ (( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐵) ∪ ( bday ‘𝐴)))
10	9	imbi1i 352	. . . . . . . 8 ⊢ (((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐵) ∪ ( bday ‘𝐴)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
11	10	2ralbii 3137	. . . . . . 7 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐵) ∪ ( bday ‘𝐴)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
12	7, 11	sylib 221	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐵) ∪ ( bday ‘𝐴)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
13	12, 2	negsproplem3 28349	. . . . 5 ⊢ (𝜑 → (( -us ‘𝐵) ∈ No ∧ ( -us “ ( R ‘𝐵)) <<s {( -us ‘𝐵)} ∧ {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵))))
14	13	simp1d 1160	. . . 4 ⊢ (𝜑 → ( -us ‘𝐵) ∈ No )
15	14	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐵) ∈ No )
16	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
17		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ No )
18		0no 28128	. . . . . 6 ⊢ 0s ∈ No
19	18	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 0s ∈ No )
20		bday0 28130	. . . . . . . 8 ⊢ ( bday ‘ 0s ) = ∅
21	20	uneq2i 4111	. . . . . . 7 ⊢ (( bday ‘𝑑) ∪ ( bday ‘ 0s )) = (( bday ‘𝑑) ∪ ∅)
22		un0 4343	. . . . . . 7 ⊢ (( bday ‘𝑑) ∪ ∅) = ( bday ‘𝑑)
23	21, 22	eqtri 2783	. . . . . 6 ⊢ (( bday ‘𝑑) ∪ ( bday ‘ 0s )) = ( bday ‘𝑑)
24		simprr1 1240	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( bday ‘𝑑) ∈ ( bday ‘𝐴))
25		elun1 4127	. . . . . . 7 ⊢ (( bday ‘𝑑) ∈ ( bday ‘𝐴) → ( bday ‘𝑑) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
26	24, 25	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( bday ‘𝑑) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
27	23, 26	eqeltrid 2864	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → (( bday ‘𝑑) ∪ ( bday ‘ 0s )) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
28	16, 17, 19, 27	negsproplem1 28347	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → (( -us ‘𝑑) ∈ No ∧ (𝑑 <s 0s → ( -us ‘ 0s ) <s ( -us ‘𝑑))))
29	28	simpld 500	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ No )
30	7, 1	negsproplem3 28349	. . . . 5 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))
31	30	simp1d 1160	. . . 4 ⊢ (𝜑 → ( -us ‘𝐴) ∈ No )
32	31	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐴) ∈ No )
33	13	simp3d 1162	. . . . 5 ⊢ (𝜑 → {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵)))
34	33	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵)))
35		fvex 6886	. . . . . 6 ⊢ ( -us ‘𝐵) ∈ V
36	35	snid 4622	. . . . 5 ⊢ ( -us ‘𝐵) ∈ {( -us ‘𝐵)}
37	36	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐵) ∈ {( -us ‘𝐵)})
38		negsfn 28342	. . . . 5 ⊢ -us Fn No
39		leftssno 28192	. . . . 5 ⊢ ( L ‘𝐵) ⊆ No
40		bdayon 28071	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
41		oldbday 28220	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝑑 ∈ No ) → (𝑑 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑑) ∈ ( bday ‘𝐴)))
42	40, 17, 41	sylancr 599	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → (𝑑 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑑) ∈ ( bday ‘𝐴)))
43	24, 42	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( O ‘( bday ‘𝐴)))
44	3	fveq2d 6877	. . . . . . . 8 ⊢ (𝜑 → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
45	44	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
46	43, 45	eleqtrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( O ‘( bday ‘𝐵)))
47		simprr3 1242	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 <s 𝐵)
48		elleft 28170	. . . . . 6 ⊢ (𝑑 ∈ ( L ‘𝐵) ↔ (𝑑 ∈ ( O ‘( bday ‘𝐵)) ∧ 𝑑 <s 𝐵))
49	46, 47, 48	sylanbrc 595	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( L ‘𝐵))
50		fnfvima 7227	. . . . 5 ⊢ (( -us Fn No ∧ ( L ‘𝐵) ⊆ No ∧ 𝑑 ∈ ( L ‘𝐵)) → ( -us ‘𝑑) ∈ ( -us “ ( L ‘𝐵)))
51	38, 39, 49, 50	mp3an12i 1494	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ ( -us “ ( L ‘𝐵)))
52	34, 37, 51	sltssepcd 28091	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐵) <s ( -us ‘𝑑))
53	30	simp2d 1161	. . . . 5 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
54	53	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
55		rightssno 28193	. . . . 5 ⊢ ( R ‘𝐴) ⊆ No
56		simprr2 1241	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝐴 <s 𝑑)
57		elright 28171	. . . . . 6 ⊢ (𝑑 ∈ ( R ‘𝐴) ↔ (𝑑 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝑑))
58	43, 56, 57	sylanbrc 595	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( R ‘𝐴))
59		fnfvima 7227	. . . . 5 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ∧ 𝑑 ∈ ( R ‘𝐴)) → ( -us ‘𝑑) ∈ ( -us “ ( R ‘𝐴)))
60	38, 55, 58, 59	mp3an12i 1494	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ ( -us “ ( R ‘𝐴)))
61		fvex 6886	. . . . . 6 ⊢ ( -us ‘𝐴) ∈ V
62	61	snid 4622	. . . . 5 ⊢ ( -us ‘𝐴) ∈ {( -us ‘𝐴)}
63	62	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐴) ∈ {( -us ‘𝐴)})
64	54, 60, 63	sltssepcd 28091	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝑑) <s ( -us ‘𝐴))
65	15, 29, 32, 52, 64	ltstrd 28053	. 2 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐵) <s ( -us ‘𝐴))
66	6, 65	rexlimddv 3169	1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ∪ cun 3896   ⊆ wss 3898  ∅c0 4278  {csn 4583   class class class wbr 5102   “ cima 5650  Oncon0 6351   Fn wfn 6522  ‘cfv 6527   No csur 27930   <s clts 27931   bday cbday 27932   <<s cslts 28076   0_s c0s 28124   O cold 28142   L cleft 28144   R cright 28145   -u_s cnegs 28338

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 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-1o 8454  df-2o 8455  df-no 27933  df-lts 27934  df-bday 27935  df-slts 28077  df-cuts 28079  df-0s 28126  df-made 28146  df-old 28147  df-left 28149  df-right 28150  df-norec 28257  df-negs 28340

This theorem is used by:  negsproplem7  28353