Theorem negsid 28091

Description: Surreal addition of a number and its negative. Theorem 4(iii) of [Conway] p. 17. (Contributed by Scott Fenton, 3-Feb-2025.)

Assertion

Ref	Expression
negsid	⊢ (𝐴 ∈ No → (𝐴 +s ( -us ‘𝐴)) = 0s )

Proof of Theorem negsid

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑝 𝑞 𝑥 𝑥_𝐿 𝑥_𝑅 𝑥_𝑂 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → 𝑥 = 𝑥𝑂)
2		fveq2 6920	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → ( -us ‘𝑥) = ( -us ‘𝑥𝑂))
3	1, 2	oveq12d 7466	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s ( -us ‘𝑥)) = (𝑥𝑂 +s ( -us ‘𝑥𝑂)))
4	3	eqeq1d 2742	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s ( -us ‘𝑥)) = 0s ↔ (𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ))
5		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
6		fveq2 6920	. . . 4 ⊢ (𝑥 = 𝐴 → ( -us ‘𝑥) = ( -us ‘𝐴))
7	5, 6	oveq12d 7466	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s ( -us ‘𝑥)) = (𝐴 +s ( -us ‘𝐴)))
8	7	eqeq1d 2742	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s ( -us ‘𝑥)) = 0s ↔ (𝐴 +s ( -us ‘𝐴)) = 0s ))
9		lltropt 27929	. . . . . 6 ⊢ ( L ‘𝑥) <<s ( R ‘𝑥)
10	9	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → ( L ‘𝑥) <<s ( R ‘𝑥))
11		negscut2 28090	. . . . . 6 ⊢ (𝑥 ∈ No → ( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)))
12	11	adantr 480	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → ( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)))
13		lrcut 27959	. . . . . . 7 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
14	13	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
15	14	eqcomd 2746	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → 𝑥 = (( L ‘𝑥) |s ( R ‘𝑥)))
16		negsval 28075	. . . . . 6 ⊢ (𝑥 ∈ No → ( -us ‘𝑥) = (( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥))))
17	16	adantr 480	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → ( -us ‘𝑥) = (( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥))))
18	10, 12, 15, 17	addsunif 28053	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (𝑥 +s ( -us ‘𝑥)) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑝 ∈ ( -us “ ( R ‘𝑥))𝑏 = (𝑥 +s 𝑝)}) |s ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑞 ∈ ( -us “ ( L ‘𝑥))𝑑 = (𝑥 +s 𝑞)})))
19		negsfn 28073	. . . . . . . . 9 ⊢ -us Fn No
20		rightssno 27938	. . . . . . . . 9 ⊢ ( R ‘𝑥) ⊆ No
21		oveq2 7456	. . . . . . . . . . 11 ⊢ (𝑝 = ( -us ‘𝑥𝑅) → (𝑥 +s 𝑝) = (𝑥 +s ( -us ‘𝑥𝑅)))
22	21	eqeq2d 2751	. . . . . . . . . 10 ⊢ (𝑝 = ( -us ‘𝑥𝑅) → (𝑏 = (𝑥 +s 𝑝) ↔ 𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))))
23	22	rexima 7275	. . . . . . . . 9 ⊢ (( -us Fn No ∧ ( R ‘𝑥) ⊆ No ) → (∃𝑝 ∈ ( -us “ ( R ‘𝑥))𝑏 = (𝑥 +s 𝑝) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))))
24	19, 20, 23	mp2an 691	. . . . . . . 8 ⊢ (∃𝑝 ∈ ( -us “ ( R ‘𝑥))𝑏 = (𝑥 +s 𝑝) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅)))
25	24	abbii 2812	. . . . . . 7 ⊢ {𝑏 ∣ ∃𝑝 ∈ ( -us “ ( R ‘𝑥))𝑏 = (𝑥 +s 𝑝)} = {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}
26	25	uneq2i 4188	. . . . . 6 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑝 ∈ ( -us “ ( R ‘𝑥))𝑏 = (𝑥 +s 𝑝)}) = ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))})
27		leftssno 27937	. . . . . . . . 9 ⊢ ( L ‘𝑥) ⊆ No
28		oveq2 7456	. . . . . . . . . . 11 ⊢ (𝑞 = ( -us ‘𝑥𝐿) → (𝑥 +s 𝑞) = (𝑥 +s ( -us ‘𝑥𝐿)))
29	28	eqeq2d 2751	. . . . . . . . . 10 ⊢ (𝑞 = ( -us ‘𝑥𝐿) → (𝑑 = (𝑥 +s 𝑞) ↔ 𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))))
30	29	rexima 7275	. . . . . . . . 9 ⊢ (( -us Fn No ∧ ( L ‘𝑥) ⊆ No ) → (∃𝑞 ∈ ( -us “ ( L ‘𝑥))𝑑 = (𝑥 +s 𝑞) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))))
31	19, 27, 30	mp2an 691	. . . . . . . 8 ⊢ (∃𝑞 ∈ ( -us “ ( L ‘𝑥))𝑑 = (𝑥 +s 𝑞) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿)))
32	31	abbii 2812	. . . . . . 7 ⊢ {𝑑 ∣ ∃𝑞 ∈ ( -us “ ( L ‘𝑥))𝑑 = (𝑥 +s 𝑞)} = {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))}
33	32	uneq2i 4188	. . . . . 6 ⊢ ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑞 ∈ ( -us “ ( L ‘𝑥))𝑑 = (𝑥 +s 𝑞)}) = ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))})
34	26, 33	oveq12i 7460	. . . . 5 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑝 ∈ ( -us “ ( R ‘𝑥))𝑏 = (𝑥 +s 𝑝)}) |s ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑞 ∈ ( -us “ ( L ‘𝑥))𝑑 = (𝑥 +s 𝑞)})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}) |s ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))}))
35		fvex 6933	. . . . . . . . . 10 ⊢ ( L ‘𝑥) ∈ V
36	35	abrexex 8003	. . . . . . . . 9 ⊢ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∈ V
37		fvex 6933	. . . . . . . . . 10 ⊢ ( R ‘𝑥) ∈ V
38	37	abrexex 8003	. . . . . . . . 9 ⊢ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))} ∈ V
39	36, 38	unex 7779	. . . . . . . 8 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}) ∈ V
40	39	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}) ∈ V)
41		snex 5451	. . . . . . . 8 ⊢ { 0s } ∈ V
42	41	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → { 0s } ∈ V)
43	27	sseli 4004	. . . . . . . . . . . . 13 ⊢ (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ No )
44	43	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑥𝐿 ∈ No )
45		simpll 766	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑥 ∈ No )
46	45	negscld 28087	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → ( -us ‘𝑥) ∈ No )
47	44, 46	addscld 28031	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥𝐿 +s ( -us ‘𝑥)) ∈ No )
48		eleq1 2832	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑥𝐿 +s ( -us ‘𝑥)) → (𝑎 ∈ No ↔ (𝑥𝐿 +s ( -us ‘𝑥)) ∈ No ))
49	47, 48	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑎 = (𝑥𝐿 +s ( -us ‘𝑥)) → 𝑎 ∈ No ))
50	49	rexlimdva 3161	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥)) → 𝑎 ∈ No ))
51	50	abssdv 4091	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ⊆ No )
52		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑥 ∈ No )
53	20	sseli 4004	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥𝑅 ∈ No )
54	53	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑥𝑅 ∈ No )
55	54	negscld 28087	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → ( -us ‘𝑥𝑅) ∈ No )
56	52, 55	addscld 28031	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥 +s ( -us ‘𝑥𝑅)) ∈ No )
57		eleq1 2832	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑥 +s ( -us ‘𝑥𝑅)) → (𝑏 ∈ No ↔ (𝑥 +s ( -us ‘𝑥𝑅)) ∈ No ))
58	56, 57	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑏 = (𝑥 +s ( -us ‘𝑥𝑅)) → 𝑏 ∈ No ))
59	58	rexlimdva 3161	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅)) → 𝑏 ∈ No ))
60	59	abssdv 4091	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))} ⊆ No )
61	51, 60	unssd 4215	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}) ⊆ No )
62		0sno 27889	. . . . . . . 8 ⊢ 0s ∈ No
63		snssi 4833	. . . . . . . 8 ⊢ ( 0s ∈ No → { 0s } ⊆ No )
64	62, 63	mp1i 13	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → { 0s } ⊆ No )
65		elun 4176	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}) ↔ (𝑝 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∨ 𝑝 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}))
66		vex 3492	. . . . . . . . . . . . 13 ⊢ 𝑝 ∈ V
67		eqeq1 2744	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑝 → (𝑎 = (𝑥𝐿 +s ( -us ‘𝑥)) ↔ 𝑝 = (𝑥𝐿 +s ( -us ‘𝑥))))
68	67	rexbidv 3185	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑝 → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s ( -us ‘𝑥))))
69	66, 68	elab 3694	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)))
70		eqeq1 2744	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑝 → (𝑏 = (𝑥 +s ( -us ‘𝑥𝑅)) ↔ 𝑝 = (𝑥 +s ( -us ‘𝑥𝑅))))
71	70	rexbidv 3185	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑝 → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑝 = (𝑥 +s ( -us ‘𝑥𝑅))))
72	66, 71	elab 3694	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))} ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑝 = (𝑥 +s ( -us ‘𝑥𝑅)))
73	69, 72	orbi12i 913	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∨ 𝑝 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}) ↔ (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑝 = (𝑥 +s ( -us ‘𝑥𝑅))))
74	65, 73	bitri 275	. . . . . . . . . 10 ⊢ (𝑝 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}) ↔ (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑝 = (𝑥 +s ( -us ‘𝑥𝑅))))
75		velsn 4664	. . . . . . . . . 10 ⊢ (𝑞 ∈ { 0s } ↔ 𝑞 = 0s )
76	74, 75	anbi12i 627	. . . . . . . . 9 ⊢ ((𝑝 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}) ∧ 𝑞 ∈ { 0s }) ↔ ((∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑝 = (𝑥 +s ( -us ‘𝑥𝑅))) ∧ 𝑞 = 0s ))
77		leftval 27920	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( L ‘𝑥) = {𝑥𝐿 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥𝐿 <s 𝑥}
78	77	reqabi 3467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥𝐿 ∈ ( L ‘𝑥) ↔ (𝑥𝐿 ∈ ( O ‘( bday ‘𝑥)) ∧ 𝑥𝐿 <s 𝑥))
79	78	simprbi 496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 <s 𝑥)
80	79	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑥𝐿 <s 𝑥)
81		sltnegim 28088	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥𝐿 ∈ No ∧ 𝑥 ∈ No ) → (𝑥𝐿 <s 𝑥 → ( -us ‘𝑥) <s ( -us ‘𝑥𝐿)))
82	44, 45, 81	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥𝐿 <s 𝑥 → ( -us ‘𝑥) <s ( -us ‘𝑥𝐿)))
83	80, 82	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → ( -us ‘𝑥) <s ( -us ‘𝑥𝐿))
84	44	negscld 28087	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → ( -us ‘𝑥𝐿) ∈ No )
85	46, 84, 44	sltadd2d 28048	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (( -us ‘𝑥) <s ( -us ‘𝑥𝐿) ↔ (𝑥𝐿 +s ( -us ‘𝑥)) <s (𝑥𝐿 +s ( -us ‘𝑥𝐿))))
86	83, 85	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥𝐿 +s ( -us ‘𝑥)) <s (𝑥𝐿 +s ( -us ‘𝑥𝐿)))
87		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝐿 → 𝑥𝑂 = 𝑥𝐿)
88		fveq2 6920	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝐿 → ( -us ‘𝑥𝑂) = ( -us ‘𝑥𝐿))
89	87, 88	oveq12d 7466	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 +s ( -us ‘𝑥𝑂)) = (𝑥𝐿 +s ( -us ‘𝑥𝐿)))
90	89	eqeq1d 2742	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 = 𝑥𝐿 → ((𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ↔ (𝑥𝐿 +s ( -us ‘𝑥𝐿)) = 0s ))
91		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s )
92		elun1 4205	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
93	92	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
94	90, 91, 93	rspcdva 3636	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥𝐿 +s ( -us ‘𝑥𝐿)) = 0s )
95	86, 94	breqtrd 5192	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥𝐿 +s ( -us ‘𝑥)) <s 0s )
96		breq1 5169	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)) → (𝑝 <s 0s ↔ (𝑥𝐿 +s ( -us ‘𝑥)) <s 0s ))
97	95, 96	syl5ibrcom 247	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)) → 𝑝 <s 0s ))
98	97	rexlimdva 3161	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)) → 𝑝 <s 0s ))
99	98	imp 406	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s ( -us ‘𝑥))) → 𝑝 <s 0s )
100		rightval 27921	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( R ‘𝑥) = {𝑥𝑅 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑥𝑅}
101	100	reqabi 3467	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑅 ∈ ( R ‘𝑥) ↔ (𝑥𝑅 ∈ ( O ‘( bday ‘𝑥)) ∧ 𝑥 <s 𝑥𝑅))
102	101	simprbi 496	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥 <s 𝑥𝑅)
103	102	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑥 <s 𝑥𝑅)
104	52, 54, 55	sltadd1d 28049	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥 <s 𝑥𝑅 ↔ (𝑥 +s ( -us ‘𝑥𝑅)) <s (𝑥𝑅 +s ( -us ‘𝑥𝑅))))
105	103, 104	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥 +s ( -us ‘𝑥𝑅)) <s (𝑥𝑅 +s ( -us ‘𝑥𝑅)))
106		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝑅 → 𝑥𝑂 = 𝑥𝑅)
107		fveq2 6920	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝑅 → ( -us ‘𝑥𝑂) = ( -us ‘𝑥𝑅))
108	106, 107	oveq12d 7466	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = 𝑥𝑅 → (𝑥𝑂 +s ( -us ‘𝑥𝑂)) = (𝑥𝑅 +s ( -us ‘𝑥𝑅)))
109	108	eqeq1d 2742	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 = 𝑥𝑅 → ((𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ↔ (𝑥𝑅 +s ( -us ‘𝑥𝑅)) = 0s ))
110		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s )
111		elun2 4206	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
112	111	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
113	109, 110, 112	rspcdva 3636	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥𝑅 +s ( -us ‘𝑥𝑅)) = 0s )
114	105, 113	breqtrd 5192	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥 +s ( -us ‘𝑥𝑅)) <s 0s )
115		breq1 5169	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = (𝑥 +s ( -us ‘𝑥𝑅)) → (𝑝 <s 0s ↔ (𝑥 +s ( -us ‘𝑥𝑅)) <s 0s ))
116	114, 115	syl5ibrcom 247	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑝 = (𝑥 +s ( -us ‘𝑥𝑅)) → 𝑝 <s 0s ))
117	116	rexlimdva 3161	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑝 = (𝑥 +s ( -us ‘𝑥𝑅)) → 𝑝 <s 0s ))
118	117	imp 406	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑝 = (𝑥 +s ( -us ‘𝑥𝑅))) → 𝑝 <s 0s )
119	99, 118	jaodan 958	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑝 = (𝑥 +s ( -us ‘𝑥𝑅)))) → 𝑝 <s 0s )
120		breq2 5170	. . . . . . . . . . 11 ⊢ (𝑞 = 0s → (𝑝 <s 𝑞 ↔ 𝑝 <s 0s ))
121	119, 120	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑝 = (𝑥 +s ( -us ‘𝑥𝑅)))) → (𝑞 = 0s → 𝑝 <s 𝑞))
122	121	expimpd 453	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (((∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑝 = (𝑥 +s ( -us ‘𝑥𝑅))) ∧ 𝑞 = 0s ) → 𝑝 <s 𝑞))
123	76, 122	biimtrid 242	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → ((𝑝 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}) ∧ 𝑞 ∈ { 0s }) → 𝑝 <s 𝑞))
124	123	3impib 1116	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑝 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}) ∧ 𝑞 ∈ { 0s }) → 𝑝 <s 𝑞)
125	40, 42, 61, 64, 124	ssltd 27854	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}) <<s { 0s })
126	37	abrexex 8003	. . . . . . . . 9 ⊢ {𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∈ V
127	35	abrexex 8003	. . . . . . . . 9 ⊢ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))} ∈ V
128	126, 127	unex 7779	. . . . . . . 8 ⊢ ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))}) ∈ V
129	128	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))}) ∈ V)
130	52	negscld 28087	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → ( -us ‘𝑥) ∈ No )
131	54, 130	addscld 28031	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥𝑅 +s ( -us ‘𝑥)) ∈ No )
132		eleq1 2832	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑥𝑅 +s ( -us ‘𝑥)) → (𝑐 ∈ No ↔ (𝑥𝑅 +s ( -us ‘𝑥)) ∈ No ))
133	131, 132	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑐 = (𝑥𝑅 +s ( -us ‘𝑥)) → 𝑐 ∈ No ))
134	133	rexlimdva 3161	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥)) → 𝑐 ∈ No ))
135	134	abssdv 4091	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → {𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ⊆ No )
136	45, 84	addscld 28031	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥 +s ( -us ‘𝑥𝐿)) ∈ No )
137		eleq1 2832	. . . . . . . . . . 11 ⊢ (𝑑 = (𝑥 +s ( -us ‘𝑥𝐿)) → (𝑑 ∈ No ↔ (𝑥 +s ( -us ‘𝑥𝐿)) ∈ No ))
138	136, 137	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑑 = (𝑥 +s ( -us ‘𝑥𝐿)) → 𝑑 ∈ No ))
139	138	rexlimdva 3161	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿)) → 𝑑 ∈ No ))
140	139	abssdv 4091	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))} ⊆ No )
141	135, 140	unssd 4215	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))}) ⊆ No )
142		velsn 4664	. . . . . . . . . 10 ⊢ (𝑝 ∈ { 0s } ↔ 𝑝 = 0s )
143		elun 4176	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))}) ↔ (𝑞 ∈ {𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∨ 𝑞 ∈ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))}))
144		vex 3492	. . . . . . . . . . . . 13 ⊢ 𝑞 ∈ V
145		eqeq1 2744	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑞 → (𝑐 = (𝑥𝑅 +s ( -us ‘𝑥)) ↔ 𝑞 = (𝑥𝑅 +s ( -us ‘𝑥))))
146	145	rexbidv 3185	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑞 → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥))))
147	144, 146	elab 3694	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ {𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)))
148		eqeq1 2744	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑞 → (𝑑 = (𝑥 +s ( -us ‘𝑥𝐿)) ↔ 𝑞 = (𝑥 +s ( -us ‘𝑥𝐿))))
149	148	rexbidv 3185	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑞 → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿))))
150	144, 149	elab 3694	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))} ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿)))
151	147, 150	orbi12i 913	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ {𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∨ 𝑞 ∈ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))}) ↔ (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)) ∨ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿))))
152	143, 151	bitri 275	. . . . . . . . . 10 ⊢ (𝑞 ∈ ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))}) ↔ (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)) ∨ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿))))
153	142, 152	anbi12i 627	. . . . . . . . 9 ⊢ ((𝑝 ∈ { 0s } ∧ 𝑞 ∈ ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))})) ↔ (𝑝 = 0s ∧ (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)) ∨ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿)))))
154		sltnegim 28088	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑅 ∈ No ) → (𝑥 <s 𝑥𝑅 → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥)))
155	52, 54, 154	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥 <s 𝑥𝑅 → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥)))
156	103, 155	mpd 15	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥))
157	55, 130, 54	sltadd2d 28048	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (( -us ‘𝑥𝑅) <s ( -us ‘𝑥) ↔ (𝑥𝑅 +s ( -us ‘𝑥𝑅)) <s (𝑥𝑅 +s ( -us ‘𝑥))))
158	156, 157	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥𝑅 +s ( -us ‘𝑥𝑅)) <s (𝑥𝑅 +s ( -us ‘𝑥)))
159	113, 158	eqbrtrrd 5190	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 0s <s (𝑥𝑅 +s ( -us ‘𝑥)))
160		breq2 5170	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)) → ( 0s <s 𝑞 ↔ 0s <s (𝑥𝑅 +s ( -us ‘𝑥))))
161	159, 160	syl5ibrcom 247	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)) → 0s <s 𝑞))
162	161	rexlimdva 3161	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)) → 0s <s 𝑞))
163	162	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥))) → 0s <s 𝑞)
164	44, 45, 84	sltadd1d 28049	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥𝐿 <s 𝑥 ↔ (𝑥𝐿 +s ( -us ‘𝑥𝐿)) <s (𝑥 +s ( -us ‘𝑥𝐿))))
165	80, 164	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥𝐿 +s ( -us ‘𝑥𝐿)) <s (𝑥 +s ( -us ‘𝑥𝐿)))
166	94, 165	eqbrtrrd 5190	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 0s <s (𝑥 +s ( -us ‘𝑥𝐿)))
167		breq2 5170	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = (𝑥 +s ( -us ‘𝑥𝐿)) → ( 0s <s 𝑞 ↔ 0s <s (𝑥 +s ( -us ‘𝑥𝐿))))
168	166, 167	syl5ibrcom 247	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑞 = (𝑥 +s ( -us ‘𝑥𝐿)) → 0s <s 𝑞))
169	168	rexlimdva 3161	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿)) → 0s <s 𝑞))
170	169	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿))) → 0s <s 𝑞)
171	163, 170	jaodan 958	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)) ∨ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿)))) → 0s <s 𝑞)
172		breq1 5169	. . . . . . . . . . . 12 ⊢ (𝑝 = 0s → (𝑝 <s 𝑞 ↔ 0s <s 𝑞))
173	171, 172	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)) ∨ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿)))) → (𝑝 = 0s → 𝑝 <s 𝑞))
174	173	ex 412	. . . . . . . . . 10 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → ((∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)) ∨ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿))) → (𝑝 = 0s → 𝑝 <s 𝑞)))
175	174	impcomd 411	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → ((𝑝 = 0s ∧ (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)) ∨ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿)))) → 𝑝 <s 𝑞))
176	153, 175	biimtrid 242	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → ((𝑝 ∈ { 0s } ∧ 𝑞 ∈ ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))})) → 𝑝 <s 𝑞))
177	176	3impib 1116	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑝 ∈ { 0s } ∧ 𝑞 ∈ ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))})) → 𝑝 <s 𝑞)
178	42, 129, 64, 141, 177	ssltd 27854	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → { 0s } <<s ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))}))
179	125, 178	cuteq0 27895	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}) |s ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))})) = 0s )
180	34, 179	eqtrid 2792	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∪ {𝑏 ∣ ∃𝑝 ∈ ( -us “ ( R ‘𝑥))𝑏 = (𝑥 +s 𝑝)}) |s ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑞 ∈ ( -us “ ( L ‘𝑥))𝑑 = (𝑥 +s 𝑞)})) = 0s )
181	18, 180	eqtrd 2780	. . 3 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (𝑥 +s ( -us ‘𝑥)) = 0s )
182	181	ex 412	. 2 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s → (𝑥 +s ( -us ‘𝑥)) = 0s ))
183	4, 8, 182	noinds 27996	1 ⊢ (𝐴 ∈ No → (𝐴 +s ( -us ‘𝐴)) = 0s )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976  {csn 4648   class class class wbr 5166   “ cima 5703   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448   No csur 27702   <s cslt 27703   bday cbday 27704   <<s csslt 27843   |s cscut 27845   0_s c0s 27885   O cold 27900   L cleft 27902   R cright 27903   +_s cadds 28010   -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-ot 4657  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-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  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-norec2 28000  df-adds 28011  df-negs 28071

This theorem is referenced by:  negsidd  28092  negsex  28093  negnegs  28094  negsdi  28100  subsid  28117  subadds  28118