Theorem negsproplem6 28296

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 27926	. . 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 4108	. . . . . . . . . 10 ⊢ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) = (( bday ‘𝐵) ∪ ( bday ‘𝐴))
9	8	eleq2i 2854	. . . . . . . . 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 3139	. . . . . . 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 28293	. . . . 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 28072	. . . . . 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 28074	. . . . . . . 8 ⊢ ( bday ‘ 0s ) = ∅
21	20	uneq2i 4115	. . . . . . 7 ⊢ (( bday ‘𝑑) ∪ ( bday ‘ 0s )) = (( bday ‘𝑑) ∪ ∅)
22		un0 4347	. . . . . . 7 ⊢ (( bday ‘𝑑) ∪ ∅) = ( bday ‘𝑑)
23	21, 22	eqtri 2785	. . . . . 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 4131	. . . . . . 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 2866	. . . . 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 28291	. . . 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 28293	. . . . 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 6895	. . . . . 6 ⊢ ( -us ‘𝐵) ∈ V
36	35	snid 4626	. . . . 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 28286	. . . . 5 ⊢ -us Fn No
39		leftssno 28136	. . . . 5 ⊢ ( L ‘𝐵) ⊆ No
40		bdayon 28015	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
41		oldbday 28164	. . . . . . . . 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 6886	. . . . . . . 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 2864	. . . . . 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 28114	. . . . . 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 7235	. . . . 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 28035	. . 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 28137	. . . . 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 28115	. . . . . 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 7235	. . . . 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 6895	. . . . . 6 ⊢ ( -us ‘𝐴) ∈ V
62	61	snid 4626	. . . . 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 28035	. . 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 27997	. 2 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐵) <s ( -us ‘𝐴))
66	6, 65	rexlimddv 3171	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 3078  ∃wrex 3088   ∪ cun 3900   ⊆ wss 3902  ∅c0 4282  {csn 4587   class class class wbr 5107   “ cima 5662  Oncon0 6361   Fn wfn 6532  ‘cfv 6537   No csur 27874   <s clts 27875   bday cbday 27876   <<s cslts 28020   0_s c0s 28068   O cold 28086   L cleft 28088   R cright 28089   -u_s cnegs 28282

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8458  df-2o 8459  df-no 27877  df-lts 27878  df-bday 27879  df-slts 28021  df-cuts 28023  df-0s 28070  df-made 28090  df-old 28091  df-left 28093  df-right 28094  df-norec 28201  df-negs 28284

This theorem is used by:  negsproplem7  28297