Theorem negsproplem6 28202

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 27832	. . 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 851	. 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 4111	. . . . . . . . . 10 ⊢ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) = (( bday ‘𝐵) ∪ ( bday ‘𝐴))
9	8	eleq2i 2853	. . . . . . . . 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 3138	. . . . . . 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 28199	. . . . 5 ⊢ (𝜑 → (( -us ‘𝐵) ∈ No ∧ ( -us “ ( R ‘𝐵)) <<s {( -us ‘𝐵)} ∧ {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵))))
14	13	simp1d 1158	. . . 4 ⊢ (𝜑 → ( -us ‘𝐵) ∈ No )
15	14	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐵) ∈ No )
16	7	adantr 485	. . . . 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 782	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ No )
18		0no 27978	. . . . . 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 27980	. . . . . . . 8 ⊢ ( bday ‘ 0s ) = ∅
21	20	uneq2i 4118	. . . . . . 7 ⊢ (( bday ‘𝑑) ∪ ( bday ‘ 0s )) = (( bday ‘𝑑) ∪ ∅)
22		un0 4350	. . . . . . 7 ⊢ (( bday ‘𝑑) ∪ ∅) = ( bday ‘𝑑)
23	21, 22	eqtri 2784	. . . . . 6 ⊢ (( bday ‘𝑑) ∪ ( bday ‘ 0s )) = ( bday ‘𝑑)
24		simprr1 1238	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( bday ‘𝑑) ∈ ( bday ‘𝐴))
25		elun1 4134	. . . . . . 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 2865	. . . . 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 28197	. . . 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 499	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ No )
30	7, 1	negsproplem3 28199	. . . . 5 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))
31	30	simp1d 1158	. . . 4 ⊢ (𝜑 → ( -us ‘𝐴) ∈ No )
32	31	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐴) ∈ No )
33	13	simp3d 1160	. . . . 5 ⊢ (𝜑 → {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵)))
34	33	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵)))
35		fvex 6894	. . . . . 6 ⊢ ( -us ‘𝐵) ∈ V
36	35	snid 4627	. . . . 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 28192	. . . . 5 ⊢ -us Fn No
39		leftssno 28042	. . . . 5 ⊢ ( L ‘𝐵) ⊆ No
40		bdayon 27921	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
41		oldbday 28070	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝑑 ∈ No ) → (𝑑 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑑) ∈ ( bday ‘𝐴)))
42	40, 17, 41	sylancr 598	. . . . . . . 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 6885	. . . . . . . 8 ⊢ (𝜑 → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
45	44	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
46	43, 45	eleqtrd 2863	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( O ‘( bday ‘𝐵)))
47		simprr3 1240	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 <s 𝐵)
48		elleft 28020	. . . . . 6 ⊢ (𝑑 ∈ ( L ‘𝐵) ↔ (𝑑 ∈ ( O ‘( bday ‘𝐵)) ∧ 𝑑 <s 𝐵))
49	46, 47, 48	sylanbrc 594	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( L ‘𝐵))
50		fnfvima 7231	. . . . 5 ⊢ (( -us Fn No ∧ ( L ‘𝐵) ⊆ No ∧ 𝑑 ∈ ( L ‘𝐵)) → ( -us ‘𝑑) ∈ ( -us “ ( L ‘𝐵)))
51	38, 39, 49, 50	mp3an12i 1492	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ ( -us “ ( L ‘𝐵)))
52	34, 37, 51	sltssepcd 27941	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐵) <s ( -us ‘𝑑))
53	30	simp2d 1159	. . . . 5 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
54	53	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
55		rightssno 28043	. . . . 5 ⊢ ( R ‘𝐴) ⊆ No
56		simprr2 1239	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝐴 <s 𝑑)
57		elright 28021	. . . . . 6 ⊢ (𝑑 ∈ ( R ‘𝐴) ↔ (𝑑 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝑑))
58	43, 56, 57	sylanbrc 594	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ ( R ‘𝐴))
59		fnfvima 7231	. . . . 5 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ∧ 𝑑 ∈ ( R ‘𝐴)) → ( -us ‘𝑑) ∈ ( -us “ ( R ‘𝐴)))
60	38, 55, 58, 59	mp3an12i 1492	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ ( -us “ ( R ‘𝐴)))
61		fvex 6894	. . . . . 6 ⊢ ( -us ‘𝐴) ∈ V
62	61	snid 4627	. . . . 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 27941	. . 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 27903	. 2 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ (( bday ‘𝑑) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -us ‘𝐵) <s ( -us ‘𝐴))
66	6, 65	rexlimddv 3170	1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  ∀wral 3077  ∃wrex 3087   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  {csn 4588   class class class wbr 5108   “ cima 5664  Oncon0 6360   Fn wfn 6531  ‘cfv 6536   No csur 27780   <s clts 27781   bday cbday 27782   <<s cslts 27926   0_s c0s 27974   O cold 27992   L cleft 27994   R cright 27995   -u_s cnegs 28188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-1o 8452  df-2o 8453  df-no 27783  df-lts 27784  df-bday 27785  df-slts 27927  df-cuts 27929  df-0s 27976  df-made 27996  df-old 27997  df-left 27999  df-right 28000  df-norec 28107  df-negs 28190

This theorem is referenced by:  negsproplem7  28203