Theorem negsid 28121

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 6861	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → ( -us ‘𝑥) = ( -us ‘𝑥𝑂))
3	1, 2	oveq12d 7408	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s ( -us ‘𝑥)) = (𝑥𝑂 +s ( -us ‘𝑥𝑂)))
4	3	eqeq1d 2763	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s ( -us ‘𝑥)) = 0s ↔ (𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ))
5		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
6		fveq2 6861	. . . 4 ⊢ (𝑥 = 𝐴 → ( -us ‘𝑥) = ( -us ‘𝐴))
7	5, 6	oveq12d 7408	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s ( -us ‘𝑥)) = (𝐴 +s ( -us ‘𝐴)))
8	7	eqeq1d 2763	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s ( -us ‘𝑥)) = 0s ↔ (𝐴 +s ( -us ‘𝐴)) = 0s ))
9		lltr 27942	. . . . . 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		negcut2 28120	. . . . . 6 ⊢ (𝑥 ∈ No → ( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)))
12	11	adantr 484	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → ( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)))
13		lrcut 27984	. . . . . . 7 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
14	13	adantr 484	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
15	14	eqcomd 2767	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → 𝑥 = (( L ‘𝑥) |s ( R ‘𝑥)))
16		negsval 28105	. . . . . 6 ⊢ (𝑥 ∈ No → ( -us ‘𝑥) = (( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥))))
17	16	adantr 484	. . . . 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 28082	. . . 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 28103	. . . . . . . . 9 ⊢ -us Fn No
20		rightssno 27954	. . . . . . . . 9 ⊢ ( R ‘𝑥) ⊆ No
21		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑝 = ( -us ‘𝑥𝑅) → (𝑥 +s 𝑝) = (𝑥 +s ( -us ‘𝑥𝑅)))
22	21	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑝 = ( -us ‘𝑥𝑅) → (𝑏 = (𝑥 +s 𝑝) ↔ 𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))))
23	22	rexima 7216	. . . . . . . . 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 702	. . . . . . . 8 ⊢ (∃𝑝 ∈ ( -us “ ( R ‘𝑥))𝑏 = (𝑥 +s 𝑝) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅)))
25	24	abbii 2828	. . . . . . 7 ⊢ {𝑏 ∣ ∃𝑝 ∈ ( -us “ ( R ‘𝑥))𝑏 = (𝑥 +s 𝑝)} = {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))}
26	25	uneq2i 4116	. . . . . 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 27953	. . . . . . . . 9 ⊢ ( L ‘𝑥) ⊆ No
28		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑞 = ( -us ‘𝑥𝐿) → (𝑥 +s 𝑞) = (𝑥 +s ( -us ‘𝑥𝐿)))
29	28	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑞 = ( -us ‘𝑥𝐿) → (𝑑 = (𝑥 +s 𝑞) ↔ 𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))))
30	29	rexima 7216	. . . . . . . . 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 702	. . . . . . . 8 ⊢ (∃𝑞 ∈ ( -us “ ( L ‘𝑥))𝑑 = (𝑥 +s 𝑞) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿)))
32	31	abbii 2828	. . . . . . 7 ⊢ {𝑑 ∣ ∃𝑞 ∈ ( -us “ ( L ‘𝑥))𝑑 = (𝑥 +s 𝑞)} = {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))}
33	32	uneq2i 4116	. . . . . 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 7402	. . . . 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 6874	. . . . . . . . . 10 ⊢ ( L ‘𝑥) ∈ V
36	35	abrexex 7937	. . . . . . . . 9 ⊢ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ∈ V
37		fvex 6874	. . . . . . . . . 10 ⊢ ( R ‘𝑥) ∈ V
38	37	abrexex 7937	. . . . . . . . 9 ⊢ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))} ∈ V
39	36, 38	unex 7721	. . . . . . . 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 5393	. . . . . . . 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 3930	. . . . . . . . . . . . 13 ⊢ (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ No )
44	43	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑥𝐿 ∈ No )
45		simpll 776	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑥 ∈ No )
46	45	negscld 28117	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → ( -us ‘𝑥) ∈ No )
47	44, 46	addscld 28060	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥𝐿 +s ( -us ‘𝑥)) ∈ No )
48		eleq1 2849	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑥𝐿 +s ( -us ‘𝑥)) → (𝑎 ∈ No ↔ (𝑥𝐿 +s ( -us ‘𝑥)) ∈ No ))
49	47, 48	syl5ibrcom 249	. . . . . . . . . 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 3162	. . . . . . . . 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 4018	. . . . . . . 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 776	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑥 ∈ No )
53	20	sseli 3930	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥𝑅 ∈ No )
54	53	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑥𝑅 ∈ No )
55	54	negscld 28117	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → ( -us ‘𝑥𝑅) ∈ No )
56	52, 55	addscld 28060	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥 +s ( -us ‘𝑥𝑅)) ∈ No )
57		eleq1 2849	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑥 +s ( -us ‘𝑥𝑅)) → (𝑏 ∈ No ↔ (𝑥 +s ( -us ‘𝑥𝑅)) ∈ No ))
58	56, 57	syl5ibrcom 249	. . . . . . . . . 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 3162	. . . . . . . . 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 4018	. . . . . . . 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 4142	. . . . . . 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		0no 27889	. . . . . . . 8 ⊢ 0s ∈ No
63		snssi 4741	. . . . . . . 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 4104	. . . . . . . . . . 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 3457	. . . . . . . . . . . . 13 ⊢ 𝑝 ∈ V
67		eqeq1 2765	. . . . . . . . . . . . . 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 3637	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s ( -us ‘𝑥))} ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)))
70		eqeq1 2765	. . . . . . . . . . . . . 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 3637	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑏 = (𝑥 +s ( -us ‘𝑥𝑅))} ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑝 = (𝑥 +s ( -us ‘𝑥𝑅)))
73	69, 72	orbi12i 925	. . . . . . . . . . 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 277	. . . . . . . . . 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 4595	. . . . . . . . . 10 ⊢ (𝑞 ∈ { 0s } ↔ 𝑞 = 0s )
76	74, 75	anbi12i 637	. . . . . . . . 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		leftlt 27933	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 <s 𝑥)
78	77	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑥𝐿 <s 𝑥)
79		ltnegsim 28118	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥𝐿 ∈ No ∧ 𝑥 ∈ No ) → (𝑥𝐿 <s 𝑥 → ( -us ‘𝑥) <s ( -us ‘𝑥𝐿)))
80	44, 45, 79	syl2anc 593	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥𝐿 <s 𝑥 → ( -us ‘𝑥) <s ( -us ‘𝑥𝐿)))
81	78, 80	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → ( -us ‘𝑥) <s ( -us ‘𝑥𝐿))
82	44	negscld 28117	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → ( -us ‘𝑥𝐿) ∈ No )
83	46, 82, 44	ltadds2d 28077	. . . . . . . . . . . . . . . . 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))))
84	81, 83	mpbid 234	. . . . . . . . . . . . . . . 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)))
85		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝐿 → 𝑥𝑂 = 𝑥𝐿)
86		fveq2 6861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝐿 → ( -us ‘𝑥𝑂) = ( -us ‘𝑥𝐿))
87	85, 86	oveq12d 7408	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 +s ( -us ‘𝑥𝑂)) = (𝑥𝐿 +s ( -us ‘𝑥𝐿)))
88	87	eqeq1d 2763	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 = 𝑥𝐿 → ((𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ↔ (𝑥𝐿 +s ( -us ‘𝑥𝐿)) = 0s ))
89		simplr 778	. . . . . . . . . . . . . . . . 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 )
90		elun1 4132	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
91	90	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
92	88, 89, 91	rspcdva 3581	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥𝐿 +s ( -us ‘𝑥𝐿)) = 0s )
93	84, 92	breqtrd 5123	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥𝐿 +s ( -us ‘𝑥)) <s 0s )
94		breq1 5100	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)) → (𝑝 <s 0s ↔ (𝑥𝐿 +s ( -us ‘𝑥)) <s 0s ))
95	93, 94	syl5ibrcom 249	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)) → 𝑝 <s 0s ))
96	95	rexlimdva 3162	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s ( -us ‘𝑥)) → 𝑝 <s 0s ))
97	96	imp 410	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s ( -us ‘𝑥))) → 𝑝 <s 0s )
98		rightgt 27934	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥 <s 𝑥𝑅)
99	98	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑥 <s 𝑥𝑅)
100	52, 54, 55	ltadds1d 28078	. . . . . . . . . . . . . . . . 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))))
101	99, 100	mpbid 234	. . . . . . . . . . . . . . . 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)))
102		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝑅 → 𝑥𝑂 = 𝑥𝑅)
103		fveq2 6861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝑅 → ( -us ‘𝑥𝑂) = ( -us ‘𝑥𝑅))
104	102, 103	oveq12d 7408	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = 𝑥𝑅 → (𝑥𝑂 +s ( -us ‘𝑥𝑂)) = (𝑥𝑅 +s ( -us ‘𝑥𝑅)))
105	104	eqeq1d 2763	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 = 𝑥𝑅 → ((𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ↔ (𝑥𝑅 +s ( -us ‘𝑥𝑅)) = 0s ))
106		simplr 778	. . . . . . . . . . . . . . . . 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 )
107		elun2 4133	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
108	107	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
109	105, 106, 108	rspcdva 3581	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥𝑅 +s ( -us ‘𝑥𝑅)) = 0s )
110	101, 109	breqtrd 5123	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥 +s ( -us ‘𝑥𝑅)) <s 0s )
111		breq1 5100	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = (𝑥 +s ( -us ‘𝑥𝑅)) → (𝑝 <s 0s ↔ (𝑥 +s ( -us ‘𝑥𝑅)) <s 0s ))
112	110, 111	syl5ibrcom 249	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑝 = (𝑥 +s ( -us ‘𝑥𝑅)) → 𝑝 <s 0s ))
113	112	rexlimdva 3162	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑝 = (𝑥 +s ( -us ‘𝑥𝑅)) → 𝑝 <s 0s ))
114	113	imp 410	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑝 = (𝑥 +s ( -us ‘𝑥𝑅))) → 𝑝 <s 0s )
115	97, 114	jaodan 970	. . . . . . . . . . 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 )
116		breq2 5101	. . . . . . . . . . 11 ⊢ (𝑞 = 0s → (𝑝 <s 𝑞 ↔ 𝑝 <s 0s ))
117	115, 116	syl5ibrcom 249	. . . . . . . . . 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))
118	117	expimpd 457	. . . . . . . . 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))
119	76, 118	biimtrid 244	. . . . . . . 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))
120	119	3impib 1128	. . . . . . 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)
121	40, 42, 61, 64, 120	sltsd 27848	. . . . . 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 })
122	37	abrexex 7937	. . . . . . . . 9 ⊢ {𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∈ V
123	35	abrexex 7937	. . . . . . . . 9 ⊢ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))} ∈ V
124	122, 123	unex 7721	. . . . . . . 8 ⊢ ({𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))}) ∈ V
125	124	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)
126	52	negscld 28117	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → ( -us ‘𝑥) ∈ No )
127	54, 126	addscld 28060	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥𝑅 +s ( -us ‘𝑥)) ∈ No )
128		eleq1 2849	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑥𝑅 +s ( -us ‘𝑥)) → (𝑐 ∈ No ↔ (𝑥𝑅 +s ( -us ‘𝑥)) ∈ No ))
129	127, 128	syl5ibrcom 249	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑐 = (𝑥𝑅 +s ( -us ‘𝑥)) → 𝑐 ∈ No ))
130	129	rexlimdva 3162	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥)) → 𝑐 ∈ No ))
131	130	abssdv 4018	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → {𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ⊆ No )
132	45, 82	addscld 28060	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥 +s ( -us ‘𝑥𝐿)) ∈ No )
133		eleq1 2849	. . . . . . . . . . 11 ⊢ (𝑑 = (𝑥 +s ( -us ‘𝑥𝐿)) → (𝑑 ∈ No ↔ (𝑥 +s ( -us ‘𝑥𝐿)) ∈ No ))
134	132, 133	syl5ibrcom 249	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑑 = (𝑥 +s ( -us ‘𝑥𝐿)) → 𝑑 ∈ No ))
135	134	rexlimdva 3162	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿)) → 𝑑 ∈ No ))
136	135	abssdv 4018	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))} ⊆ No )
137	131, 136	unssd 4142	. . . . . . 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 )
138		velsn 4595	. . . . . . . . . 10 ⊢ (𝑝 ∈ { 0s } ↔ 𝑝 = 0s )
139		elun 4104	. . . . . . . . . . 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))}))
140		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑞 ∈ V
141		eqeq1 2765	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑞 → (𝑐 = (𝑥𝑅 +s ( -us ‘𝑥)) ↔ 𝑞 = (𝑥𝑅 +s ( -us ‘𝑥))))
142	141	rexbidv 3185	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑞 → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥))))
143	140, 142	elab 3637	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ {𝑐 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑐 = (𝑥𝑅 +s ( -us ‘𝑥))} ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)))
144		eqeq1 2765	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑞 → (𝑑 = (𝑥 +s ( -us ‘𝑥𝐿)) ↔ 𝑞 = (𝑥 +s ( -us ‘𝑥𝐿))))
145	144	rexbidv 3185	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑞 → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿))))
146	140, 145	elab 3637	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ {𝑑 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑑 = (𝑥 +s ( -us ‘𝑥𝐿))} ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿)))
147	143, 146	orbi12i 925	. . . . . . . . . . 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))))
148	139, 147	bitri 277	. . . . . . . . . 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))))
149	138, 148	anbi12i 637	. . . . . . . . 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)))))
150		ltnegsim 28118	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑅 ∈ No ) → (𝑥 <s 𝑥𝑅 → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥)))
151	52, 54, 150	syl2anc 593	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥 <s 𝑥𝑅 → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥)))
152	99, 151	mpd 15	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥))
153	55, 126, 54	ltadds2d 28077	. . . . . . . . . . . . . . . . . 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))))
154	152, 153	mpbid 234	. . . . . . . . . . . . . . . . 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)))
155	109, 154	eqbrtrrd 5121	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 0s <s (𝑥𝑅 +s ( -us ‘𝑥)))
156		breq2 5101	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)) → ( 0s <s 𝑞 ↔ 0s <s (𝑥𝑅 +s ( -us ‘𝑥))))
157	155, 156	syl5ibrcom 249	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)) → 0s <s 𝑞))
158	157	rexlimdva 3162	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥)) → 0s <s 𝑞))
159	158	imp 410	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑞 = (𝑥𝑅 +s ( -us ‘𝑥))) → 0s <s 𝑞)
160	44, 45, 82	ltadds1d 28078	. . . . . . . . . . . . . . . . . 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))))
161	78, 160	mpbid 234	. . . . . . . . . . . . . . . . 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)))
162	92, 161	eqbrtrrd 5121	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 0s <s (𝑥 +s ( -us ‘𝑥𝐿)))
163		breq2 5101	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = (𝑥 +s ( -us ‘𝑥𝐿)) → ( 0s <s 𝑞 ↔ 0s <s (𝑥 +s ( -us ‘𝑥𝐿))))
164	162, 163	syl5ibrcom 249	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑞 = (𝑥 +s ( -us ‘𝑥𝐿)) → 0s <s 𝑞))
165	164	rexlimdva 3162	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿)) → 0s <s 𝑞))
166	165	imp 410	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) ∧ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑞 = (𝑥 +s ( -us ‘𝑥𝐿))) → 0s <s 𝑞)
167	159, 166	jaodan 970	. . . . . . . . . . . 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)
168		breq1 5100	. . . . . . . . . . . 12 ⊢ (𝑝 = 0s → (𝑝 <s 𝑞 ↔ 0s <s 𝑞))
169	167, 168	syl5ibrcom 249	. . . . . . . . . . 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))
170	169	ex 416	. . . . . . . . . 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)))
171	170	impcomd 415	. . . . . . . . 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))
172	149, 171	biimtrid 244	. . . . . . . 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))
173	172	3impib 1128	. . . . . . 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)
174	42, 125, 64, 137, 173	sltsd 27848	. . . . . 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))}))
175	121, 174	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 )
176	34, 175	eqtrid 2808	. . . 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 )
177	18, 176	eqtrd 2796	. . 3 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s ) → (𝑥 +s ( -us ‘𝑥)) = 0s )
178	177	ex 416	. 2 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s ( -us ‘𝑥𝑂)) = 0s → (𝑥 +s ( -us ‘𝑥)) = 0s ))
179	4, 8, 178	noinds 28025	1 ⊢ (𝐴 ∈ No → (𝐴 +s ( -us ‘𝐴)) = 0s )

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1559   ∈ wcel 2141  {cab 2739  ∀wral 3075  ∃wrex 3085  Vcvv 3453   ∪ cun 3900   ⊆ wss 3902  {csn 4579   class class class wbr 5097   “ cima 5646   Fn wfn 6510  ‘cfv 6515  (class class class)co 7390   No csur 27691   <s clts 27692   <<s cslts 27837   |s ccuts 27839   0_s c0s 27885   L cleft 27905   R cright 27906   +_s cadds 28039   -u_s cnegs 28099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-1o 8430  df-2o 8431  df-nadd 8629  df-no 27694  df-lts 27695  df-bday 27696  df-les 27796  df-slts 27838  df-cuts 27840  df-0s 27887  df-made 27907  df-old 27908  df-left 27910  df-right 27911  df-norec 28018  df-norec2 28029  df-adds 28040  df-negs 28101

This theorem is referenced by:  negsidd  28122  negsex  28123  negnegs  28124  negsdi  28130  subsid  28149  subadds  28150