Theorem negsproplem6 28002

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 27658	. . 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 838	. 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 2826	. . . . . . . . 9 ⊢ ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) ↔ (( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐵) ∪ ( bday ‘𝐴)))
10	9	imbi1i 349	. . . . . . . 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 3109	. . . . . . 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 218	. . . . . 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 27999	. . . . 5 ⊢ (𝜑 → (( -us ‘𝐵) ∈ No ∧ ( -us “ ( R ‘𝐵)) <<s {( -us ‘𝐵)} ∧ {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵))))
14	13	simp1d 1142	. . . 4 ⊢ (𝜑 → ( -us ‘𝐵) ∈ No )
15	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐵) ∈ No )
16	7	adantr 480	. . . . 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 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ No )
18		0sno 27797	. . . . . 6 ⊢ 0s ∈ No
19	18	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 0s ∈ No )
20		bday0s 27799	. . . . . . . 8 ⊢ ( bday ‘ 0s ) = ∅
21	20	uneq2i 4115	. . . . . . 7 ⊢ (( bday ‘𝑑) ∪ ( bday ‘ 0s )) = (( bday ‘𝑑) ∪ ∅)
22		un0 4344	. . . . . . 7 ⊢ (( bday ‘𝑑) ∪ ∅) = ( bday ‘𝑑)
23	21, 22	eqtri 2757	. . . . . 6 ⊢ (( bday ‘𝑑) ∪ ( bday ‘ 0s )) = ( bday ‘𝑑)
24		simprr1 1222	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( bday ‘𝑑) ∈ ( bday ‘𝐴))
25		elun1 4132	. . . . . . 7 ⊢ (( bday ‘𝑑) ∈ ( bday ‘𝐴) → ( bday ‘𝑑) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
26	24, 25	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( bday ‘𝑑) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
27	23, 26	eqeltrid 2838	. . . . 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 27997	. . . 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 494	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ No )
30	7, 1	negsproplem3 27999	. . . . 5 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))
31	30	simp1d 1142	. . . 4 ⊢ (𝜑 → ( -us ‘𝐴) ∈ No )
32	31	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐴) ∈ No )
33	13	simp3d 1144	. . . . 5 ⊢ (𝜑 → {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵)))
34	33	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵)))
35		fvex 6845	. . . . . 6 ⊢ ( -us ‘𝐵) ∈ V
36	35	snid 4617	. . . . 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 27992	. . . . 5 ⊢ -us Fn No
39		leftssno 27853	. . . . 5 ⊢ ( L ‘𝐵) ⊆ No
40		bdayelon 27742	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
41		oldbday 27873	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝑑 ∈ No ) → (𝑑 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑑) ∈ ( bday ‘𝐴)))
42	40, 17, 41	sylancr 587	. . . . . . . 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 257	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( O ‘( bday ‘𝐴)))
44	3	fveq2d 6836	. . . . . . . 8 ⊢ (𝜑 → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
45	44	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
46	43, 45	eleqtrd 2836	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( O ‘( bday ‘𝐵)))
47		simprr3 1224	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 <s 𝐵)
48		elleft 27833	. . . . . 6 ⊢ (𝑑 ∈ ( L ‘𝐵) ↔ (𝑑 ∈ ( O ‘( bday ‘𝐵)) ∧ 𝑑 <s 𝐵))
49	46, 47, 48	sylanbrc 583	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( L ‘𝐵))
50		fnfvima 7177	. . . . 5 ⊢ (( -us Fn No ∧ ( L ‘𝐵) ⊆ No ∧ 𝑑 ∈ ( L ‘𝐵)) → ( -us ‘𝑑) ∈ ( -us “ ( L ‘𝐵)))
51	38, 39, 49, 50	mp3an12i 1467	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ ( -us “ ( L ‘𝐵)))
52	34, 37, 51	ssltsepcd 27762	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐵) <s ( -us ‘𝑑))
53	30	simp2d 1143	. . . . 5 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
54	53	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
55		rightssno 27854	. . . . 5 ⊢ ( R ‘𝐴) ⊆ No
56		simprr2 1223	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝐴 <s 𝑑)
57		elright 27834	. . . . . 6 ⊢ (𝑑 ∈ ( R ‘𝐴) ↔ (𝑑 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝑑))
58	43, 56, 57	sylanbrc 583	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( R ‘𝐴))
59		fnfvima 7177	. . . . 5 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ∧ 𝑑 ∈ ( R ‘𝐴)) → ( -us ‘𝑑) ∈ ( -us “ ( R ‘𝐴)))
60	38, 55, 58, 59	mp3an12i 1467	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ ( -us “ ( R ‘𝐴)))
61		fvex 6845	. . . . . 6 ⊢ ( -us ‘𝐴) ∈ V
62	61	snid 4617	. . . . 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	ssltsepcd 27762	. . 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	slttrd 27725	. 2 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐵) <s ( -us ‘𝐴))
66	6, 65	rexlimddv 3141	1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058   ∪ cun 3897   ⊆ wss 3899  ∅c0 4283  {csn 4578   class class class wbr 5096   “ cima 5625  Oncon0 6315   Fn wfn 6485  ‘cfv 6490   No csur 27605   <s cslt 27606   bday cbday 27607   <<s csslt 27747   0_s c0s 27793   O cold 27811   L cleft 27813   R cright 27814   -u_s cnegs 27988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-2o 8396  df-no 27608  df-slt 27609  df-bday 27610  df-sslt 27748  df-scut 27750  df-0s 27795  df-made 27815  df-old 27816  df-left 27818  df-right 27819  df-norec 27908  df-negs 27990

This theorem is referenced by:  negsproplem7  28003