Theorem negsproplem6 28419

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)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘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 → ( -s‘𝐵) <s ( -s‘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 28049	. . 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)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
8		uncom 4157	. . . . . . . . . 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)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐵) ∪ (bday‘𝐴)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
11	10	2ralbii 3137	. . . . . . 7 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐵) ∪ (bday‘𝐴)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
12	7, 11	sylib 221	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐵) ∪ (bday‘𝐴)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
13	12, 2	negsproplem3 28416	. . . . 5 ⊢ (𝜑 → (( -s‘𝐵) ∈ No ∧ ( -s “ (R‘𝐵)) <<s {( -s‘𝐵)} ∧ {( -s‘𝐵)} <<s ( -s “ (L‘𝐵))))
14	13	simp1d 1160	. . . 4 ⊢ (𝜑 → ( -s‘𝐵) ∈ No)
15	14	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -s‘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)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
17		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → 𝑑 ∈ No)
18		0no 28195	. . . . . 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 28197	. . . . . . . 8 ⊢ (bday‘ 0s ) = ∅
21	20	uneq2i 4164	. . . . . . 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 4180	. . . . . . 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 28414	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → (( -s‘𝑑) ∈ No ∧ (𝑑 <s 0s → ( -s‘ 0s ) <s ( -s‘𝑑))))
29	28	simpld 500	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -s‘𝑑) ∈ No)
30	7, 1	negsproplem3 28416	. . . . 5 ⊢ (𝜑 → (( -s‘𝐴) ∈ No ∧ ( -s “ (R‘𝐴)) <<s {( -s‘𝐴)} ∧ {( -s‘𝐴)} <<s ( -s “ (L‘𝐴))))
31	30	simp1d 1160	. . . 4 ⊢ (𝜑 → ( -s‘𝐴) ∈ No)
32	31	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -s‘𝐴) ∈ No)
33	13	simp3d 1162	. . . . 5 ⊢ (𝜑 → {( -s‘𝐵)} <<s ( -s “ (L‘𝐵)))
34	33	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → {( -s‘𝐵)} <<s ( -s “ (L‘𝐵)))
35		fvex 6898	. . . . . 6 ⊢ ( -s‘𝐵) ∈ V
36	35	snid 4622	. . . . 5 ⊢ ( -s‘𝐵) ∈ {( -s‘𝐵)}
37	36	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -s‘𝐵) ∈ {( -s‘𝐵)})
38		negsfn 28409	. . . . 5 ⊢ -s Fn No
39		leftssno 28259	. . . . 5 ⊢ (L‘𝐵) ⊆ No
40		bdayon 28138	. . . . . . . . 9 ⊢ (bday‘𝐴) ∈ On
41		oldbday 28287	. . . . . . . . 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 6889	. . . . . . . 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 28237	. . . . . 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 7239	. . . . 5 ⊢ (( -s Fn No ∧ (L‘𝐵) ⊆ No ∧ 𝑑 ∈ (L‘𝐵)) → ( -s‘𝑑) ∈ ( -s “ (L‘𝐵)))
51	38, 39, 49, 50	mp3an12i 1494	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -s‘𝑑) ∈ ( -s “ (L‘𝐵)))
52	34, 37, 51	sltssepcd 28158	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -s‘𝐵) <s ( -s‘𝑑))
53	30	simp2d 1161	. . . . 5 ⊢ (𝜑 → ( -s “ (R‘𝐴)) <<s {( -s‘𝐴)})
54	53	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -s “ (R‘𝐴)) <<s {( -s‘𝐴)})
55		rightssno 28260	. . . . 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 28238	. . . . . 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 7239	. . . . 5 ⊢ (( -s Fn No ∧ (R‘𝐴) ⊆ No ∧ 𝑑 ∈ (R‘𝐴)) → ( -s‘𝑑) ∈ ( -s “ (R‘𝐴)))
60	38, 55, 58, 59	mp3an12i 1494	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -s‘𝑑) ∈ ( -s “ (R‘𝐴)))
61		fvex 6898	. . . . . 6 ⊢ ( -s‘𝐴) ∈ V
62	61	snid 4622	. . . . 5 ⊢ ( -s‘𝐴) ∈ {( -s‘𝐴)}
63	62	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -s‘𝐴) ∈ {( -s‘𝐴)})
64	54, 60, 63	sltssepcd 28158	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -s‘𝑑) <s ( -s‘𝐴))
65	15, 29, 32, 52, 64	ltstrd 28120	. 2 ⊢ ((𝜑 ∧ (𝑑 ∈ No ∧ ((bday‘𝑑) ∈ (bday‘𝐴) ∧ 𝐴 <s 𝑑 ∧ 𝑑 <s 𝐵))) → ( -s‘𝐵) <s ( -s‘𝐴))
66	6, 65	rexlimddv 3169	1 ⊢ (𝜑 → ( -s‘𝐵) <s ( -s‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  {csn 4583   class class class wbr 5102   “ cima 5653  Oncon0 6362   Fn wfn 6533  ‘cfv 6538  Nocsur 27997   <s clts 27998  bdaycbday 27999   <<s cslts 28143   0_s c0s 28191  Ocold 28209  Lcleft 28211  Rcright 28212   -_scnegs 28405

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  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 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-slts 28144  df-cuts 28146  df-0s 28193  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-negs 28407

This theorem is used by:  negsproplem7  28420