Theorem negsunif 28213

Description: Uniformity property for surreal negation. If 𝐿 and 𝑅 are any cut that represents 𝐴, then they may be used instead of ( L ‘𝐴) and ( R ‘𝐴) in the definition of negation. (Contributed by Scott Fenton, 14-Feb-2025.)

Hypotheses

Ref	Expression
negsunif.1	⊢ (𝜑 → 𝐿 <<s 𝑅)
negsunif.2	⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))

Assertion

Ref	Expression
negsunif	⊢ (𝜑 → ( -us ‘𝐴) = (( -us “ 𝑅) |s ( -us “ 𝐿)))

Proof of Theorem negsunif

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		negsunif.2	. . . 4 ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
2		negsunif.1	. . . . 5 ⊢ (𝜑 → 𝐿 <<s 𝑅)
3	2	cutscld 27941	. . . 4 ⊢ (𝜑 → (𝐿 |s 𝑅) ∈ No )
4	1, 3	eqeltrd 2869	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
5		negsval 28183	. . 3 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
6	4, 5	syl 18	. 2 ⊢ (𝜑 → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
7		negcut2 28198	. . . 4 ⊢ (𝐴 ∈ No → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
8	4, 7	syl 18	. . 3 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
9	2, 1	cofcutr2d 28084	. . . . 5 ⊢ (𝜑 → ∀𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ 𝑅 𝑑 ≤s 𝑐)
10		negsfn 28181	. . . . . . . 8 ⊢ -us Fn No
11		sltsss2 27924	. . . . . . . . 9 ⊢ (𝐿 <<s 𝑅 → 𝑅 ⊆ No )
12	2, 11	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ⊆ No )
13		breq2 5117	. . . . . . . . 9 ⊢ (𝑏 = ( -us ‘𝑑) → (( -us ‘𝑐) ≤s 𝑏 ↔ ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
14	13	rexima 7237	. . . . . . . 8 ⊢ (( -us Fn No ∧ 𝑅 ⊆ No ) → (∃𝑏 ∈ ( -us “ 𝑅)( -us ‘𝑐) ≤s 𝑏 ↔ ∃𝑑 ∈ 𝑅 ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
15	10, 12, 14	sylancr 598	. . . . . . 7 ⊢ (𝜑 → (∃𝑏 ∈ ( -us “ 𝑅)( -us ‘𝑐) ≤s 𝑏 ↔ ∃𝑑 ∈ 𝑅 ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
16	15	ralbidv 3194	. . . . . 6 ⊢ (𝜑 → (∀𝑐 ∈ ( R ‘𝐴)∃𝑏 ∈ ( -us “ 𝑅)( -us ‘𝑐) ≤s 𝑏 ↔ ∀𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ 𝑅 ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
17	12	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → 𝑅 ⊆ No )
18	17	sselda 3945	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ 𝑑 ∈ 𝑅) → 𝑑 ∈ No )
19		rightssno 28032	. . . . . . . . . . 11 ⊢ ( R ‘𝐴) ⊆ No
20	19	sseli 3941	. . . . . . . . . 10 ⊢ (𝑐 ∈ ( R ‘𝐴) → 𝑐 ∈ No )
21	20	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ 𝑑 ∈ 𝑅) → 𝑐 ∈ No )
22	18, 21	lenegsd 28206	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ 𝑑 ∈ 𝑅) → (𝑑 ≤s 𝑐 ↔ ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
23	22	rexbidva 3193	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → (∃𝑑 ∈ 𝑅 𝑑 ≤s 𝑐 ↔ ∃𝑑 ∈ 𝑅 ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
24	23	ralbidva 3192	. . . . . 6 ⊢ (𝜑 → (∀𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ 𝑅 𝑑 ≤s 𝑐 ↔ ∀𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ 𝑅 ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
25	16, 24	bitr4d 285	. . . . 5 ⊢ (𝜑 → (∀𝑐 ∈ ( R ‘𝐴)∃𝑏 ∈ ( -us “ 𝑅)( -us ‘𝑐) ≤s 𝑏 ↔ ∀𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ 𝑅 𝑑 ≤s 𝑐))
26	9, 25	mpbird 260	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ ( R ‘𝐴)∃𝑏 ∈ ( -us “ 𝑅)( -us ‘𝑐) ≤s 𝑏)
27		breq1 5116	. . . . . . 7 ⊢ (𝑎 = ( -us ‘𝑐) → (𝑎 ≤s 𝑏 ↔ ( -us ‘𝑐) ≤s 𝑏))
28	27	rexbidv 3195	. . . . . 6 ⊢ (𝑎 = ( -us ‘𝑐) → (∃𝑏 ∈ ( -us “ 𝑅)𝑎 ≤s 𝑏 ↔ ∃𝑏 ∈ ( -us “ 𝑅)( -us ‘𝑐) ≤s 𝑏))
29	28	ralima 7236	. . . . 5 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ) → (∀𝑎 ∈ ( -us “ ( R ‘𝐴))∃𝑏 ∈ ( -us “ 𝑅)𝑎 ≤s 𝑏 ↔ ∀𝑐 ∈ ( R ‘𝐴)∃𝑏 ∈ ( -us “ 𝑅)( -us ‘𝑐) ≤s 𝑏))
30	10, 19, 29	mp2an 704	. . . 4 ⊢ (∀𝑎 ∈ ( -us “ ( R ‘𝐴))∃𝑏 ∈ ( -us “ 𝑅)𝑎 ≤s 𝑏 ↔ ∀𝑐 ∈ ( R ‘𝐴)∃𝑏 ∈ ( -us “ 𝑅)( -us ‘𝑐) ≤s 𝑏)
31	26, 30	sylibr 237	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ ( -us “ ( R ‘𝐴))∃𝑏 ∈ ( -us “ 𝑅)𝑎 ≤s 𝑏)
32	2, 1	cofcutr1d 28083	. . . . 5 ⊢ (𝜑 → ∀𝑐 ∈ ( L ‘𝐴)∃𝑑 ∈ 𝐿 𝑐 ≤s 𝑑)
33		sltsss1 27923	. . . . . . . . 9 ⊢ (𝐿 <<s 𝑅 → 𝐿 ⊆ No )
34	2, 33	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ⊆ No )
35		breq1 5116	. . . . . . . . 9 ⊢ (𝑏 = ( -us ‘𝑑) → (𝑏 ≤s ( -us ‘𝑐) ↔ ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
36	35	rexima 7237	. . . . . . . 8 ⊢ (( -us Fn No ∧ 𝐿 ⊆ No ) → (∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐) ↔ ∃𝑑 ∈ 𝐿 ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
37	10, 34, 36	sylancr 598	. . . . . . 7 ⊢ (𝜑 → (∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐) ↔ ∃𝑑 ∈ 𝐿 ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
38	37	ralbidv 3194	. . . . . 6 ⊢ (𝜑 → (∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐) ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑑 ∈ 𝐿 ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
39		leftssno 28031	. . . . . . . . . . 11 ⊢ ( L ‘𝐴) ⊆ No
40	39	sseli 3941	. . . . . . . . . 10 ⊢ (𝑐 ∈ ( L ‘𝐴) → 𝑐 ∈ No )
41	40	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ( L ‘𝐴)) ∧ 𝑑 ∈ 𝐿) → 𝑐 ∈ No )
42	34	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ( L ‘𝐴)) → 𝐿 ⊆ No )
43	42	sselda 3945	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ( L ‘𝐴)) ∧ 𝑑 ∈ 𝐿) → 𝑑 ∈ No )
44	41, 43	lenegsd 28206	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ( L ‘𝐴)) ∧ 𝑑 ∈ 𝐿) → (𝑐 ≤s 𝑑 ↔ ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
45	44	rexbidva 3193	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ( L ‘𝐴)) → (∃𝑑 ∈ 𝐿 𝑐 ≤s 𝑑 ↔ ∃𝑑 ∈ 𝐿 ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
46	45	ralbidva 3192	. . . . . 6 ⊢ (𝜑 → (∀𝑐 ∈ ( L ‘𝐴)∃𝑑 ∈ 𝐿 𝑐 ≤s 𝑑 ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑑 ∈ 𝐿 ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
47	38, 46	bitr4d 285	. . . . 5 ⊢ (𝜑 → (∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐) ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑑 ∈ 𝐿 𝑐 ≤s 𝑑))
48	32, 47	mpbird 260	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐))
49		breq2 5117	. . . . . . 7 ⊢ (𝑎 = ( -us ‘𝑐) → (𝑏 ≤s 𝑎 ↔ 𝑏 ≤s ( -us ‘𝑐)))
50	49	rexbidv 3195	. . . . . 6 ⊢ (𝑎 = ( -us ‘𝑐) → (∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s 𝑎 ↔ ∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐)))
51	50	ralima 7236	. . . . 5 ⊢ (( -us Fn No ∧ ( L ‘𝐴) ⊆ No ) → (∀𝑎 ∈ ( -us “ ( L ‘𝐴))∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s 𝑎 ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐)))
52	10, 39, 51	mp2an 704	. . . 4 ⊢ (∀𝑎 ∈ ( -us “ ( L ‘𝐴))∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s 𝑎 ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐))
53	48, 52	sylibr 237	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ ( -us “ ( L ‘𝐴))∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s 𝑎)
54		fnfun 6636	. . . . . . 7 ⊢ ( -us Fn No → Fun -us )
55	10, 54	ax-mp 5	. . . . . 6 ⊢ Fun -us
56		sltsex2 27922	. . . . . . 7 ⊢ (𝐿 <<s 𝑅 → 𝑅 ∈ V)
57	2, 56	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
58		funimaexg 6623	. . . . . 6 ⊢ ((Fun -us ∧ 𝑅 ∈ V) → ( -us “ 𝑅) ∈ V)
59	55, 57, 58	sylancr 598	. . . . 5 ⊢ (𝜑 → ( -us “ 𝑅) ∈ V)
60		snex 5411	. . . . . 6 ⊢ {( -us ‘𝐴)} ∈ V
61	60	a1i 11	. . . . 5 ⊢ (𝜑 → {( -us ‘𝐴)} ∈ V)
62		imassrn 6074	. . . . . . 7 ⊢ ( -us “ 𝑅) ⊆ ran -us
63		negsfo 28211	. . . . . . . 8 ⊢ -us : No –onto→ No
64		forn 6796	. . . . . . . 8 ⊢ ( -us : No –onto→ No → ran -us = No )
65	63, 64	ax-mp 5	. . . . . . 7 ⊢ ran -us = No
66	62, 65	sseqtri 3993	. . . . . 6 ⊢ ( -us “ 𝑅) ⊆ No
67	66	a1i 11	. . . . 5 ⊢ (𝜑 → ( -us “ 𝑅) ⊆ No )
68	4	negscld 28195	. . . . . 6 ⊢ (𝜑 → ( -us ‘𝐴) ∈ No )
69	68	snssd 4757	. . . . 5 ⊢ (𝜑 → {( -us ‘𝐴)} ⊆ No )
70		velsn 4610	. . . . . . . 8 ⊢ (𝑎 ∈ {( -us ‘𝐴)} ↔ 𝑎 = ( -us ‘𝐴))
71		fvelimab 6954	. . . . . . . . . . 11 ⊢ (( -us Fn No ∧ 𝑅 ⊆ No ) → (𝑏 ∈ ( -us “ 𝑅) ↔ ∃𝑑 ∈ 𝑅 ( -us ‘𝑑) = 𝑏))
72	10, 12, 71	sylancr 598	. . . . . . . . . 10 ⊢ (𝜑 → (𝑏 ∈ ( -us “ 𝑅) ↔ ∃𝑑 ∈ 𝑅 ( -us ‘𝑑) = 𝑏))
73	1	sneqd 4606	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝐴} = {(𝐿 |s 𝑅)})
74	73	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → {𝐴} = {(𝐿 |s 𝑅)})
75		cutcuts 27939	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 <<s 𝑅 → ((𝐿 |s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
76	2, 75	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐿 |s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
77	76	simp3d 1160	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {(𝐿 |s 𝑅)} <<s 𝑅)
78	77	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → {(𝐿 |s 𝑅)} <<s 𝑅)
79	74, 78	eqbrtrd 5137	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → {𝐴} <<s 𝑅)
80		snidg 4631	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ No → 𝐴 ∈ {𝐴})
81	4, 80	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
82	81	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → 𝐴 ∈ {𝐴})
83		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → 𝑑 ∈ 𝑅)
84	79, 82, 83	sltssepcd 27930	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → 𝐴 <s 𝑑)
85	4	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → 𝐴 ∈ No )
86	12	sselda 3945	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → 𝑑 ∈ No )
87	85, 86	ltnegsd 28205	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → (𝐴 <s 𝑑 ↔ ( -us ‘𝑑) <s ( -us ‘𝐴)))
88	84, 87	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → ( -us ‘𝑑) <s ( -us ‘𝐴))
89		breq1 5116	. . . . . . . . . . . 12 ⊢ (( -us ‘𝑑) = 𝑏 → (( -us ‘𝑑) <s ( -us ‘𝐴) ↔ 𝑏 <s ( -us ‘𝐴)))
90	88, 89	syl5ibcom 248	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → (( -us ‘𝑑) = 𝑏 → 𝑏 <s ( -us ‘𝐴)))
91	90	rexlimdva 3172	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑑 ∈ 𝑅 ( -us ‘𝑑) = 𝑏 → 𝑏 <s ( -us ‘𝐴)))
92	72, 91	sylbid 243	. . . . . . . . 9 ⊢ (𝜑 → (𝑏 ∈ ( -us “ 𝑅) → 𝑏 <s ( -us ‘𝐴)))
93		breq2 5117	. . . . . . . . . 10 ⊢ (𝑎 = ( -us ‘𝐴) → (𝑏 <s 𝑎 ↔ 𝑏 <s ( -us ‘𝐴)))
94	93	imbi2d 343	. . . . . . . . 9 ⊢ (𝑎 = ( -us ‘𝐴) → ((𝑏 ∈ ( -us “ 𝑅) → 𝑏 <s 𝑎) ↔ (𝑏 ∈ ( -us “ 𝑅) → 𝑏 <s ( -us ‘𝐴))))
95	92, 94	syl5ibrcom 250	. . . . . . . 8 ⊢ (𝜑 → (𝑎 = ( -us ‘𝐴) → (𝑏 ∈ ( -us “ 𝑅) → 𝑏 <s 𝑎)))
96	70, 95	biimtrid 245	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ {( -us ‘𝐴)} → (𝑏 ∈ ( -us “ 𝑅) → 𝑏 <s 𝑎)))
97	96	3imp 1126	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ {( -us ‘𝐴)} ∧ 𝑏 ∈ ( -us “ 𝑅)) → 𝑏 <s 𝑎)
98	97	3com23 1142	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ( -us “ 𝑅) ∧ 𝑎 ∈ {( -us ‘𝐴)}) → 𝑏 <s 𝑎)
99	59, 61, 67, 69, 98	sltsd 27926	. . . 4 ⊢ (𝜑 → ( -us “ 𝑅) <<s {( -us ‘𝐴)})
100	6	sneqd 4606	. . . 4 ⊢ (𝜑 → {( -us ‘𝐴)} = {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))})
101	99, 100	breqtrd 5141	. . 3 ⊢ (𝜑 → ( -us “ 𝑅) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))})
102		sltsex1 27921	. . . . . . 7 ⊢ (𝐿 <<s 𝑅 → 𝐿 ∈ V)
103	2, 102	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ V)
104		funimaexg 6623	. . . . . 6 ⊢ ((Fun -us ∧ 𝐿 ∈ V) → ( -us “ 𝐿) ∈ V)
105	55, 103, 104	sylancr 598	. . . . 5 ⊢ (𝜑 → ( -us “ 𝐿) ∈ V)
106		imassrn 6074	. . . . . . 7 ⊢ ( -us “ 𝐿) ⊆ ran -us
107	106, 65	sseqtri 3993	. . . . . 6 ⊢ ( -us “ 𝐿) ⊆ No
108	107	a1i 11	. . . . 5 ⊢ (𝜑 → ( -us “ 𝐿) ⊆ No )
109		fvelimab 6954	. . . . . . . . . 10 ⊢ (( -us Fn No ∧ 𝐿 ⊆ No ) → (𝑏 ∈ ( -us “ 𝐿) ↔ ∃𝑐 ∈ 𝐿 ( -us ‘𝑐) = 𝑏))
110	10, 34, 109	sylancr 598	. . . . . . . . 9 ⊢ (𝜑 → (𝑏 ∈ ( -us “ 𝐿) ↔ ∃𝑐 ∈ 𝐿 ( -us ‘𝑐) = 𝑏))
111	2	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝐿 <<s 𝑅)
112	111, 75	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → ((𝐿 |s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
113	112	simp2d 1159	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝐿 <<s {(𝐿 |s 𝑅)})
114	73	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → {𝐴} = {(𝐿 |s 𝑅)})
115	113, 114	breqtrrd 5143	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝐿 <<s {𝐴})
116		simpr 489	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝑐 ∈ 𝐿)
117	81	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝐴 ∈ {𝐴})
118	115, 116, 117	sltssepcd 27930	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝑐 <s 𝐴)
119	34	sselda 3945	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝑐 ∈ No )
120	4	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝐴 ∈ No )
121	119, 120	ltnegsd 28205	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → (𝑐 <s 𝐴 ↔ ( -us ‘𝐴) <s ( -us ‘𝑐)))
122	118, 121	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → ( -us ‘𝐴) <s ( -us ‘𝑐))
123		breq2 5117	. . . . . . . . . . 11 ⊢ (( -us ‘𝑐) = 𝑏 → (( -us ‘𝐴) <s ( -us ‘𝑐) ↔ ( -us ‘𝐴) <s 𝑏))
124	122, 123	syl5ibcom 248	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → (( -us ‘𝑐) = 𝑏 → ( -us ‘𝐴) <s 𝑏))
125	124	rexlimdva 3172	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑐 ∈ 𝐿 ( -us ‘𝑐) = 𝑏 → ( -us ‘𝐴) <s 𝑏))
126	110, 125	sylbid 243	. . . . . . . 8 ⊢ (𝜑 → (𝑏 ∈ ( -us “ 𝐿) → ( -us ‘𝐴) <s 𝑏))
127		breq1 5116	. . . . . . . . 9 ⊢ (𝑎 = ( -us ‘𝐴) → (𝑎 <s 𝑏 ↔ ( -us ‘𝐴) <s 𝑏))
128	127	imbi2d 343	. . . . . . . 8 ⊢ (𝑎 = ( -us ‘𝐴) → ((𝑏 ∈ ( -us “ 𝐿) → 𝑎 <s 𝑏) ↔ (𝑏 ∈ ( -us “ 𝐿) → ( -us ‘𝐴) <s 𝑏)))
129	126, 128	syl5ibrcom 250	. . . . . . 7 ⊢ (𝜑 → (𝑎 = ( -us ‘𝐴) → (𝑏 ∈ ( -us “ 𝐿) → 𝑎 <s 𝑏)))
130	70, 129	biimtrid 245	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ {( -us ‘𝐴)} → (𝑏 ∈ ( -us “ 𝐿) → 𝑎 <s 𝑏)))
131	130	3imp 1126	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ {( -us ‘𝐴)} ∧ 𝑏 ∈ ( -us “ 𝐿)) → 𝑎 <s 𝑏)
132	61, 105, 69, 108, 131	sltsd 27926	. . . 4 ⊢ (𝜑 → {( -us ‘𝐴)} <<s ( -us “ 𝐿))
133	100, 132	eqbrtrrd 5139	. . 3 ⊢ (𝜑 → {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} <<s ( -us “ 𝐿))
134	8, 31, 53, 101, 133	cofcut1d 28079	. 2 ⊢ (𝜑 → (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) = (( -us “ 𝑅) |s ( -us “ 𝐿)))
135	6, 134	eqtrd 2804	1 ⊢ (𝜑 → ( -us ‘𝐴) = (( -us “ 𝑅) |s ( -us “ 𝐿)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  {csn 4594   class class class wbr 5113  ran crn 5663   “ cima 5665  Fun wfun 6531   Fn wfn 6532  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7411   No csur 27769   <s clts 27770   ≤s cles 27873   <<s cslts 27915   |s ccuts 27917   L cleft 27983   R cright 27984   -u_s cnegs 28177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-1o 8452  df-2o 8453  df-nadd 8651  df-no 27772  df-lts 27773  df-bday 27774  df-les 27874  df-slts 27916  df-cuts 27918  df-0s 27965  df-made 27985  df-old 27986  df-left 27988  df-right 27989  df-norec 28096  df-norec2 28107  df-adds 28118  df-negs 28179

This theorem is referenced by:  zcuts  28565  renegscl  28656