Theorem negsproplem6 28083

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 27755	. . 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 4181	. . . . . . . . . 10 ⊢ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) = (( bday ‘𝐵) ∪ ( bday ‘𝐴))
9	8	eleq2i 2836	. . . . . . . . 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 3134	. . . . . . 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 28080	. . . . 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 27889	. . . . . 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 27891	. . . . . . . 8 ⊢ ( bday ‘ 0s ) = ∅
21	20	uneq2i 4188	. . . . . . 7 ⊢ (( bday ‘𝑑) ∪ ( bday ‘ 0s )) = (( bday ‘𝑑) ∪ ∅)
22		un0 4417	. . . . . . 7 ⊢ (( bday ‘𝑑) ∪ ∅) = ( bday ‘𝑑)
23	21, 22	eqtri 2768	. . . . . 6 ⊢ (( bday ‘𝑑) ∪ ( bday ‘ 0s )) = ( bday ‘𝑑)
24		simprr1 1221	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( bday ‘𝑑) ∈ ( bday ‘𝐴))
25		elun1 4205	. . . . . . 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 2848	. . . . 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 28078	. . . 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 28080	. . . . 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 6933	. . . . . 6 ⊢ ( -us ‘𝐵) ∈ V
36	35	snid 4684	. . . . 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 28073	. . . . 5 ⊢ -us Fn No
39		leftssno 27937	. . . . 5 ⊢ ( L ‘𝐵) ⊆ No
40		bdayelon 27839	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
41		oldbday 27957	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝑑 ∈ No ) → (𝑑 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑑) ∈ ( bday ‘𝐴)))
42	40, 17, 41	sylancr 586	. . . . . . . 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 6924	. . . . . . . 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 2846	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( O ‘( bday ‘𝐵)))
47		simprr3 1223	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 <s 𝐵)
48		leftval 27920	. . . . . . 7 ⊢ ( L ‘𝐵) = {𝑑 ∈ ( O ‘( bday ‘𝐵)) ∣ 𝑑 <s 𝐵}
49	48	reqabi 3467	. . . . . 6 ⊢ (𝑑 ∈ ( L ‘𝐵) ↔ (𝑑 ∈ ( O ‘( bday ‘𝐵)) ∧ 𝑑 <s 𝐵))
50	46, 47, 49	sylanbrc 582	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( L ‘𝐵))
51		fnfvima 7270	. . . . 5 ⊢ (( -us Fn No ∧ ( L ‘𝐵) ⊆ No ∧ 𝑑 ∈ ( L ‘𝐵)) → ( -us ‘𝑑) ∈ ( -us “ ( L ‘𝐵)))
52	38, 39, 50, 51	mp3an12i 1465	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ ( -us “ ( L ‘𝐵)))
53	34, 37, 52	ssltsepcd 27857	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐵) <s ( -us ‘𝑑))
54	30	simp2d 1143	. . . . 5 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
55	54	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
56		rightssno 27938	. . . . 5 ⊢ ( R ‘𝐴) ⊆ No
57		simprr2 1222	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝐴 <s 𝑑)
58		rightval 27921	. . . . . . 7 ⊢ ( R ‘𝐴) = {𝑑 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑑}
59	58	reqabi 3467	. . . . . 6 ⊢ (𝑑 ∈ ( R ‘𝐴) ↔ (𝑑 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝑑))
60	43, 57, 59	sylanbrc 582	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( R ‘𝐴))
61		fnfvima 7270	. . . . 5 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ∧ 𝑑 ∈ ( R ‘𝐴)) → ( -us ‘𝑑) ∈ ( -us “ ( R ‘𝐴)))
62	38, 56, 60, 61	mp3an12i 1465	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ ( -us “ ( R ‘𝐴)))
63		fvex 6933	. . . . . 6 ⊢ ( -us ‘𝐴) ∈ V
64	63	snid 4684	. . . . 5 ⊢ ( -us ‘𝐴) ∈ {( -us ‘𝐴)}
65	64	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐴) ∈ {( -us ‘𝐴)})
66	55, 62, 65	ssltsepcd 27857	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝑑) <s ( -us ‘𝐴))
67	15, 29, 32, 53, 66	slttrd 27822	. 2 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐵) <s ( -us ‘𝐴))
68	6, 67	rexlimddv 3167	1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ∪ cun 3974   ⊆ wss 3976  ∅c0 4352  {csn 4648   class class class wbr 5166   “ cima 5703  Oncon0 6395   Fn wfn 6568  ‘cfv 6573   No csur 27702   <s cslt 27703   bday cbday 27704   <<s csslt 27843   0_s c0s 27885   O cold 27900   L cleft 27902   R cright 27903   -u_s cnegs 28069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-negs 28071

This theorem is referenced by:  negsproplem7  28084