Theorem negsproplem2 28442

Description: Lemma for surreal negation. Show that the cut that defines negation is legitimate. (Contributed by Scott Fenton, 2-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)))))
negsproplem2.1	⊢ (𝜑 → 𝐴 ∈ No)

Assertion

Ref	Expression
negsproplem2	⊢ (𝜑 → ( -s “ (R‘𝐴)) <<s ( -s “ (L‘𝐴)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem negsproplem2

Dummy variables 𝑎 𝑏 𝑥_𝐿 𝑥_𝑅 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		negsfn 28436	. . . 4 ⊢ -s Fn No
2		fnfun 6658	. . . 4 ⊢ ( -s Fn No → Fun -s)
3	1, 2	ax-mp 5	. . 3 ⊢ Fun -s
4		fvex 6918	. . . 4 ⊢ (R‘𝐴) ∈ V
5	4	funimaex 6646	. . 3 ⊢ (Fun -s → ( -s “ (R‘𝐴)) ∈ V)
6	3, 5	mp1i 14	. 2 ⊢ (𝜑 → ( -s “ (R‘𝐴)) ∈ V)
7		fvex 6918	. . . 4 ⊢ (L‘𝐴) ∈ V
8	7	funimaex 6646	. . 3 ⊢ (Fun -s → ( -s “ (L‘𝐴)) ∈ V)
9	3, 8	mp1i 14	. 2 ⊢ (𝜑 → ( -s “ (L‘𝐴)) ∈ V)
10		rightssold 28285	. . . 4 ⊢ (R‘𝐴) ⊆ (O‘(bday‘𝐴))
11		imass2 6066	. . . 4 ⊢ ((R‘𝐴) ⊆ (O‘(bday‘𝐴)) → ( -s “ (R‘𝐴)) ⊆ ( -s “ (O‘(bday‘𝐴))))
12	10, 11	ax-mp 5	. . 3 ⊢ ( -s “ (R‘𝐴)) ⊆ ( -s “ (O‘(bday‘𝐴)))
13		negsproplem.1	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
14	13	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (O‘(bday‘𝐴))) → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
15		oldssno 28254	. . . . . . . . 9 ⊢ (O‘(bday‘𝐴)) ⊆ No
16	15	sseli 3909	. . . . . . . 8 ⊢ (𝑎 ∈ (O‘(bday‘𝐴)) → 𝑎 ∈ No)
17	16	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (O‘(bday‘𝐴))) → 𝑎 ∈ No)
18		0no 28222	. . . . . . . 8 ⊢ 0s ∈ No
19	18	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (O‘(bday‘𝐴))) → 0s ∈ No)
20		bday0 28224	. . . . . . . . . 10 ⊢ (bday‘ 0s ) = ∅
21	20	uneq2i 4167	. . . . . . . . 9 ⊢ ((bday‘𝑎) ∪ (bday‘ 0s )) = ((bday‘𝑎) ∪ ∅)
22		un0 4347	. . . . . . . . 9 ⊢ ((bday‘𝑎) ∪ ∅) = (bday‘𝑎)
23	21, 22	eqtri 2786	. . . . . . . 8 ⊢ ((bday‘𝑎) ∪ (bday‘ 0s )) = (bday‘𝑎)
24		oldbdayim 28302	. . . . . . . . . 10 ⊢ (𝑎 ∈ (O‘(bday‘𝐴)) → (bday‘𝑎) ∈ (bday‘𝐴))
25	24	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (O‘(bday‘𝐴))) → (bday‘𝑎) ∈ (bday‘𝐴))
26		elun1 4183	. . . . . . . . 9 ⊢ ((bday‘𝑎) ∈ (bday‘𝐴) → (bday‘𝑎) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)))
27	25, 26	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (O‘(bday‘𝐴))) → (bday‘𝑎) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)))
28	23, 27	eqeltrid 2868	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (O‘(bday‘𝐴))) → ((bday‘𝑎) ∪ (bday‘ 0s )) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)))
29	14, 17, 19, 28	negsproplem1 28441	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (O‘(bday‘𝐴))) → (( -s‘𝑎) ∈ No ∧ (𝑎 <s 0s → ( -s‘ 0s ) <s ( -s‘𝑎))))
30	29	simpld 500	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (O‘(bday‘𝐴))) → ( -s‘𝑎) ∈ No)
31	30	ralrimiva 3158	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ (O‘(bday‘𝐴))( -s‘𝑎) ∈ No)
32	1	fndmi 6662	. . . . . 6 ⊢ dom -s = No
33	15, 32	sseqtrri 3962	. . . . 5 ⊢ (O‘(bday‘𝐴)) ⊆ dom -s
34		funimass4 6969	. . . . 5 ⊢ ((Fun -s ∧ (O‘(bday‘𝐴)) ⊆ dom -s) → (( -s “ (O‘(bday‘𝐴))) ⊆ No ↔ ∀𝑎 ∈ (O‘(bday‘𝐴))( -s‘𝑎) ∈ No))
35	3, 33, 34	mp2an 705	. . . 4 ⊢ (( -s “ (O‘(bday‘𝐴))) ⊆ No ↔ ∀𝑎 ∈ (O‘(bday‘𝐴))( -s‘𝑎) ∈ No)
36	31, 35	sylibr 237	. . 3 ⊢ (𝜑 → ( -s “ (O‘(bday‘𝐴))) ⊆ No)
37	12, 36	sstrid 3924	. 2 ⊢ (𝜑 → ( -s “ (R‘𝐴)) ⊆ No)
38		leftssold 28284	. . . 4 ⊢ (L‘𝐴) ⊆ (O‘(bday‘𝐴))
39		imass2 6066	. . . 4 ⊢ ((L‘𝐴) ⊆ (O‘(bday‘𝐴)) → ( -s “ (L‘𝐴)) ⊆ ( -s “ (O‘(bday‘𝐴))))
40	38, 39	ax-mp 5	. . 3 ⊢ ( -s “ (L‘𝐴)) ⊆ ( -s “ (O‘(bday‘𝐴)))
41	40, 36	sstrid 3924	. 2 ⊢ (𝜑 → ( -s “ (L‘𝐴)) ⊆ No)
42		rightssno 28287	. . . . . . 7 ⊢ (R‘𝐴) ⊆ No
43		fvelimab 6977	. . . . . . 7 ⊢ (( -s Fn No ∧ (R‘𝐴) ⊆ No) → (𝑎 ∈ ( -s “ (R‘𝐴)) ↔ ∃𝑥𝑅 ∈ (R‘𝐴)( -s‘𝑥𝑅) = 𝑎))
44	1, 42, 43	mp2an 705	. . . . . 6 ⊢ (𝑎 ∈ ( -s “ (R‘𝐴)) ↔ ∃𝑥𝑅 ∈ (R‘𝐴)( -s‘𝑥𝑅) = 𝑎)
45		leftssno 28286	. . . . . . 7 ⊢ (L‘𝐴) ⊆ No
46		fvelimab 6977	. . . . . . 7 ⊢ (( -s Fn No ∧ (L‘𝐴) ⊆ No) → (𝑏 ∈ ( -s “ (L‘𝐴)) ↔ ∃𝑥𝐿 ∈ (L‘𝐴)( -s‘𝑥𝐿) = 𝑏))
47	1, 45, 46	mp2an 705	. . . . . 6 ⊢ (𝑏 ∈ ( -s “ (L‘𝐴)) ↔ ∃𝑥𝐿 ∈ (L‘𝐴)( -s‘𝑥𝐿) = 𝑏)
48	44, 47	anbi12i 640	. . . . 5 ⊢ ((𝑎 ∈ ( -s “ (R‘𝐴)) ∧ 𝑏 ∈ ( -s “ (L‘𝐴))) ↔ (∃𝑥𝑅 ∈ (R‘𝐴)( -s‘𝑥𝑅) = 𝑎 ∧ ∃𝑥𝐿 ∈ (L‘𝐴)( -s‘𝑥𝐿) = 𝑏))
49		reeanv 3238	. . . . 5 ⊢ (∃𝑥𝑅 ∈ (R‘𝐴)∃𝑥𝐿 ∈ (L‘𝐴)(( -s‘𝑥𝑅) = 𝑎 ∧ ( -s‘𝑥𝐿) = 𝑏) ↔ (∃𝑥𝑅 ∈ (R‘𝐴)( -s‘𝑥𝑅) = 𝑎 ∧ ∃𝑥𝐿 ∈ (L‘𝐴)( -s‘𝑥𝐿) = 𝑏))
50	48, 49	bitr4i 281	. . . 4 ⊢ ((𝑎 ∈ ( -s “ (R‘𝐴)) ∧ 𝑏 ∈ ( -s “ (L‘𝐴))) ↔ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑥𝐿 ∈ (L‘𝐴)(( -s‘𝑥𝑅) = 𝑎 ∧ ( -s‘𝑥𝐿) = 𝑏))
51		lltr 28275	. . . . . . . . 9 ⊢ (L‘𝐴) <<s (R‘𝐴)
52	51	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → (L‘𝐴) <<s (R‘𝐴))
53		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → 𝑥𝐿 ∈ (L‘𝐴))
54		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → 𝑥𝑅 ∈ (R‘𝐴))
55	52, 53, 54	sltssepcd 28185	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → 𝑥𝐿 <s 𝑥𝑅)
56	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
57	45	sseli 3909	. . . . . . . . . 10 ⊢ (𝑥𝐿 ∈ (L‘𝐴) → 𝑥𝐿 ∈ No)
58	57	ad2antll 742	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → 𝑥𝐿 ∈ No)
59	42	sseli 3909	. . . . . . . . . . 11 ⊢ (𝑥𝑅 ∈ (R‘𝐴) → 𝑥𝑅 ∈ No)
60	59	adantr 486	. . . . . . . . . 10 ⊢ ((𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴)) → 𝑥𝑅 ∈ No)
61	60	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → 𝑥𝑅 ∈ No)
62	38	sseli 3909	. . . . . . . . . . . . 13 ⊢ (𝑥𝐿 ∈ (L‘𝐴) → 𝑥𝐿 ∈ (O‘(bday‘𝐴)))
63	62	ad2antll 742	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → 𝑥𝐿 ∈ (O‘(bday‘𝐴)))
64		oldbdayim 28302	. . . . . . . . . . . 12 ⊢ (𝑥𝐿 ∈ (O‘(bday‘𝐴)) → (bday‘𝑥𝐿) ∈ (bday‘𝐴))
65	63, 64	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → (bday‘𝑥𝐿) ∈ (bday‘𝐴))
66	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (R‘𝐴) ⊆ (O‘(bday‘𝐴)))
67	66	sselda 3913	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥𝑅 ∈ (R‘𝐴)) → 𝑥𝑅 ∈ (O‘(bday‘𝐴)))
68	67	adantrr 730	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → 𝑥𝑅 ∈ (O‘(bday‘𝐴)))
69		oldbdayim 28302	. . . . . . . . . . . 12 ⊢ (𝑥𝑅 ∈ (O‘(bday‘𝐴)) → (bday‘𝑥𝑅) ∈ (bday‘𝐴))
70	68, 69	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → (bday‘𝑥𝑅) ∈ (bday‘𝐴))
71		bdayon 28165	. . . . . . . . . . . 12 ⊢ (bday‘𝑥𝐿) ∈ On
72		bdayon 28165	. . . . . . . . . . . 12 ⊢ (bday‘𝑥𝑅) ∈ On
73		bdayon 28165	. . . . . . . . . . . 12 ⊢ (bday‘𝐴) ∈ On
74		onunel 6485	. . . . . . . . . . . 12 ⊢ (((bday‘𝑥𝐿) ∈ On ∧ (bday‘𝑥𝑅) ∈ On ∧ (bday‘𝐴) ∈ On) → (((bday‘𝑥𝐿) ∪ (bday‘𝑥𝑅)) ∈ (bday‘𝐴) ↔ ((bday‘𝑥𝐿) ∈ (bday‘𝐴) ∧ (bday‘𝑥𝑅) ∈ (bday‘𝐴))))
75	71, 72, 73, 74	mp3an 1491	. . . . . . . . . . 11 ⊢ (((bday‘𝑥𝐿) ∪ (bday‘𝑥𝑅)) ∈ (bday‘𝐴) ↔ ((bday‘𝑥𝐿) ∈ (bday‘𝐴) ∧ (bday‘𝑥𝑅) ∈ (bday‘𝐴)))
76	65, 70, 75	sylanbrc 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → ((bday‘𝑥𝐿) ∪ (bday‘𝑥𝑅)) ∈ (bday‘𝐴))
77		elun1 4183	. . . . . . . . . 10 ⊢ (((bday‘𝑥𝐿) ∪ (bday‘𝑥𝑅)) ∈ (bday‘𝐴) → ((bday‘𝑥𝐿) ∪ (bday‘𝑥𝑅)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)))
78	76, 77	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → ((bday‘𝑥𝐿) ∪ (bday‘𝑥𝑅)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)))
79	56, 58, 61, 78	negsproplem1 28441	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → (( -s‘𝑥𝐿) ∈ No ∧ (𝑥𝐿 <s 𝑥𝑅 → ( -s‘𝑥𝑅) <s ( -s‘𝑥𝐿))))
80	79	simprd 501	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → (𝑥𝐿 <s 𝑥𝑅 → ( -s‘𝑥𝑅) <s ( -s‘𝑥𝐿)))
81	55, 80	mpd 16	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → ( -s‘𝑥𝑅) <s ( -s‘𝑥𝐿))
82		breq12 5111	. . . . . 6 ⊢ ((( -s‘𝑥𝑅) = 𝑎 ∧ ( -s‘𝑥𝐿) = 𝑏) → (( -s‘𝑥𝑅) <s ( -s‘𝑥𝐿) ↔ 𝑎 <s 𝑏))
83	81, 82	syl5ibcom 248	. . . . 5 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ (R‘𝐴) ∧ 𝑥𝐿 ∈ (L‘𝐴))) → ((( -s‘𝑥𝑅) = 𝑎 ∧ ( -s‘𝑥𝐿) = 𝑏) → 𝑎 <s 𝑏))
84	83	rexlimdvva 3223	. . . 4 ⊢ (𝜑 → (∃𝑥𝑅 ∈ (R‘𝐴)∃𝑥𝐿 ∈ (L‘𝐴)(( -s‘𝑥𝑅) = 𝑎 ∧ ( -s‘𝑥𝐿) = 𝑏) → 𝑎 <s 𝑏))
85	50, 84	biimtrid 245	. . 3 ⊢ (𝜑 → ((𝑎 ∈ ( -s “ (R‘𝐴)) ∧ 𝑏 ∈ ( -s “ (L‘𝐴))) → 𝑎 <s 𝑏))
86	85	3impib 1134	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ( -s “ (R‘𝐴)) ∧ 𝑏 ∈ ( -s “ (L‘𝐴))) → 𝑎 <s 𝑏)
87	6, 9, 37, 41, 86	sltsd 28181	1 ⊢ (𝜑 → ( -s “ (R‘𝐴)) <<s ( -s “ (L‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281   class class class wbr 5106  dom cdm 5655   “ cima 5658  Oncon0 6377  Fun wfun 6549   Fn wfn 6550  ‘cfv 6555  Nocsur 28024   <s clts 28025  bdaycbday 28026   <<s cslts 28170   0_s c0s 28218  Ocold 28236  Lcleft 28238  Rcright 28239   -_scnegs 28432

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-1o 8501  df-2o 8502  df-no 28027  df-lts 28028  df-bday 28029  df-slts 28171  df-cuts 28173  df-0s 28220  df-made 28240  df-old 28241  df-left 28243  df-right 28244  df-norec 28351  df-negs 28434

This theorem is used by:  negsproplem3  28443