Theorem negsunif 27968

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	scutcld 27722	. . . 4 ⊢ (𝜑 → (𝐿 |s 𝑅) ∈ No )
4	1, 3	eqeltrd 2829	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
5		negsval 27938	. . 3 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
6	4, 5	syl 17	. 2 ⊢ (𝜑 → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
7		negscut2 27953	. . . 4 ⊢ (𝐴 ∈ No → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
8	4, 7	syl 17	. . 3 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
9	2, 1	cofcutr2d 27841	. . . . 5 ⊢ (𝜑 → ∀𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ 𝑅 𝑑 ≤s 𝑐)
10		negsfn 27936	. . . . . . . 8 ⊢ -us Fn No
11		ssltss2 27708	. . . . . . . . 9 ⊢ (𝐿 <<s 𝑅 → 𝑅 ⊆ No )
12	2, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ⊆ No )
13		breq2 5114	. . . . . . . . 9 ⊢ (𝑏 = ( -us ‘𝑑) → (( -us ‘𝑐) ≤s 𝑏 ↔ ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
14	13	rexima 7215	. . . . . . . 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 587	. . . . . . 7 ⊢ (𝜑 → (∃𝑏 ∈ ( -us “ 𝑅)( -us ‘𝑐) ≤s 𝑏 ↔ ∃𝑑 ∈ 𝑅 ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
16	15	ralbidv 3157	. . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → 𝑅 ⊆ No )
18	17	sselda 3949	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ 𝑑 ∈ 𝑅) → 𝑑 ∈ No )
19		rightssno 27800	. . . . . . . . . . 11 ⊢ ( R ‘𝐴) ⊆ No
20	19	sseli 3945	. . . . . . . . . 10 ⊢ (𝑐 ∈ ( R ‘𝐴) → 𝑐 ∈ No )
21	20	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ 𝑑 ∈ 𝑅) → 𝑐 ∈ No )
22	18, 21	slenegd 27961	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ 𝑑 ∈ 𝑅) → (𝑑 ≤s 𝑐 ↔ ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
23	22	rexbidva 3156	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → (∃𝑑 ∈ 𝑅 𝑑 ≤s 𝑐 ↔ ∃𝑑 ∈ 𝑅 ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
24	23	ralbidva 3155	. . . . . 6 ⊢ (𝜑 → (∀𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ 𝑅 𝑑 ≤s 𝑐 ↔ ∀𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ 𝑅 ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
25	16, 24	bitr4d 282	. . . . 5 ⊢ (𝜑 → (∀𝑐 ∈ ( R ‘𝐴)∃𝑏 ∈ ( -us “ 𝑅)( -us ‘𝑐) ≤s 𝑏 ↔ ∀𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ 𝑅 𝑑 ≤s 𝑐))
26	9, 25	mpbird 257	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ ( R ‘𝐴)∃𝑏 ∈ ( -us “ 𝑅)( -us ‘𝑐) ≤s 𝑏)
27		breq1 5113	. . . . . . 7 ⊢ (𝑎 = ( -us ‘𝑐) → (𝑎 ≤s 𝑏 ↔ ( -us ‘𝑐) ≤s 𝑏))
28	27	rexbidv 3158	. . . . . 6 ⊢ (𝑎 = ( -us ‘𝑐) → (∃𝑏 ∈ ( -us “ 𝑅)𝑎 ≤s 𝑏 ↔ ∃𝑏 ∈ ( -us “ 𝑅)( -us ‘𝑐) ≤s 𝑏))
29	28	ralima 7214	. . . . 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 692	. . . 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 234	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ ( -us “ ( R ‘𝐴))∃𝑏 ∈ ( -us “ 𝑅)𝑎 ≤s 𝑏)
32	2, 1	cofcutr1d 27840	. . . . 5 ⊢ (𝜑 → ∀𝑐 ∈ ( L ‘𝐴)∃𝑑 ∈ 𝐿 𝑐 ≤s 𝑑)
33		ssltss1 27707	. . . . . . . . 9 ⊢ (𝐿 <<s 𝑅 → 𝐿 ⊆ No )
34	2, 33	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ⊆ No )
35		breq1 5113	. . . . . . . . 9 ⊢ (𝑏 = ( -us ‘𝑑) → (𝑏 ≤s ( -us ‘𝑐) ↔ ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
36	35	rexima 7215	. . . . . . . 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 587	. . . . . . 7 ⊢ (𝜑 → (∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐) ↔ ∃𝑑 ∈ 𝐿 ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
38	37	ralbidv 3157	. . . . . 6 ⊢ (𝜑 → (∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐) ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑑 ∈ 𝐿 ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
39		leftssno 27799	. . . . . . . . . . 11 ⊢ ( L ‘𝐴) ⊆ No
40	39	sseli 3945	. . . . . . . . . 10 ⊢ (𝑐 ∈ ( L ‘𝐴) → 𝑐 ∈ No )
41	40	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ( L ‘𝐴)) ∧ 𝑑 ∈ 𝐿) → 𝑐 ∈ No )
42	34	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ( L ‘𝐴)) → 𝐿 ⊆ No )
43	42	sselda 3949	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ( L ‘𝐴)) ∧ 𝑑 ∈ 𝐿) → 𝑑 ∈ No )
44	41, 43	slenegd 27961	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ( L ‘𝐴)) ∧ 𝑑 ∈ 𝐿) → (𝑐 ≤s 𝑑 ↔ ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
45	44	rexbidva 3156	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ( L ‘𝐴)) → (∃𝑑 ∈ 𝐿 𝑐 ≤s 𝑑 ↔ ∃𝑑 ∈ 𝐿 ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
46	45	ralbidva 3155	. . . . . 6 ⊢ (𝜑 → (∀𝑐 ∈ ( L ‘𝐴)∃𝑑 ∈ 𝐿 𝑐 ≤s 𝑑 ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑑 ∈ 𝐿 ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
47	38, 46	bitr4d 282	. . . . 5 ⊢ (𝜑 → (∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐) ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑑 ∈ 𝐿 𝑐 ≤s 𝑑))
48	32, 47	mpbird 257	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐))
49		breq2 5114	. . . . . . 7 ⊢ (𝑎 = ( -us ‘𝑐) → (𝑏 ≤s 𝑎 ↔ 𝑏 ≤s ( -us ‘𝑐)))
50	49	rexbidv 3158	. . . . . 6 ⊢ (𝑎 = ( -us ‘𝑐) → (∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s 𝑎 ↔ ∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐)))
51	50	ralima 7214	. . . . 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 692	. . . 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 234	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ ( -us “ ( L ‘𝐴))∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s 𝑎)
54		fnfun 6621	. . . . . . 7 ⊢ ( -us Fn No → Fun -us )
55	10, 54	ax-mp 5	. . . . . 6 ⊢ Fun -us
56		ssltex2 27706	. . . . . . 7 ⊢ (𝐿 <<s 𝑅 → 𝑅 ∈ V)
57	2, 56	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
58		funimaexg 6606	. . . . . 6 ⊢ ((Fun -us ∧ 𝑅 ∈ V) → ( -us “ 𝑅) ∈ V)
59	55, 57, 58	sylancr 587	. . . . 5 ⊢ (𝜑 → ( -us “ 𝑅) ∈ V)
60		snex 5394	. . . . . 6 ⊢ {( -us ‘𝐴)} ∈ V
61	60	a1i 11	. . . . 5 ⊢ (𝜑 → {( -us ‘𝐴)} ∈ V)
62		imassrn 6045	. . . . . . 7 ⊢ ( -us “ 𝑅) ⊆ ran -us
63		negsfo 27966	. . . . . . . 8 ⊢ -us : No –onto→ No
64		forn 6778	. . . . . . . 8 ⊢ ( -us : No –onto→ No → ran -us = No )
65	63, 64	ax-mp 5	. . . . . . 7 ⊢ ran -us = No
66	62, 65	sseqtri 3998	. . . . . 6 ⊢ ( -us “ 𝑅) ⊆ No
67	66	a1i 11	. . . . 5 ⊢ (𝜑 → ( -us “ 𝑅) ⊆ No )
68	4	negscld 27950	. . . . . 6 ⊢ (𝜑 → ( -us ‘𝐴) ∈ No )
69	68	snssd 4776	. . . . 5 ⊢ (𝜑 → {( -us ‘𝐴)} ⊆ No )
70		velsn 4608	. . . . . . . 8 ⊢ (𝑎 ∈ {( -us ‘𝐴)} ↔ 𝑎 = ( -us ‘𝐴))
71		fvelimab 6936	. . . . . . . . . . 11 ⊢ (( -us Fn No ∧ 𝑅 ⊆ No ) → (𝑏 ∈ ( -us “ 𝑅) ↔ ∃𝑑 ∈ 𝑅 ( -us ‘𝑑) = 𝑏))
72	10, 12, 71	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (𝑏 ∈ ( -us “ 𝑅) ↔ ∃𝑑 ∈ 𝑅 ( -us ‘𝑑) = 𝑏))
73	1	sneqd 4604	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝐴} = {(𝐿 |s 𝑅)})
74	73	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → {𝐴} = {(𝐿 |s 𝑅)})
75		scutcut 27720	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 <<s 𝑅 → ((𝐿 |s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
76	2, 75	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐿 |s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
77	76	simp3d 1144	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {(𝐿 |s 𝑅)} <<s 𝑅)
78	77	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → {(𝐿 |s 𝑅)} <<s 𝑅)
79	74, 78	eqbrtrd 5132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → {𝐴} <<s 𝑅)
80		snidg 4627	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ No → 𝐴 ∈ {𝐴})
81	4, 80	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
82	81	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → 𝐴 ∈ {𝐴})
83		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → 𝑑 ∈ 𝑅)
84	79, 82, 83	ssltsepcd 27713	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → 𝐴 <s 𝑑)
85	4	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → 𝐴 ∈ No )
86	12	sselda 3949	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → 𝑑 ∈ No )
87	85, 86	sltnegd 27960	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → (𝐴 <s 𝑑 ↔ ( -us ‘𝑑) <s ( -us ‘𝐴)))
88	84, 87	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → ( -us ‘𝑑) <s ( -us ‘𝐴))
89		breq1 5113	. . . . . . . . . . . 12 ⊢ (( -us ‘𝑑) = 𝑏 → (( -us ‘𝑑) <s ( -us ‘𝐴) ↔ 𝑏 <s ( -us ‘𝐴)))
90	88, 89	syl5ibcom 245	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → (( -us ‘𝑑) = 𝑏 → 𝑏 <s ( -us ‘𝐴)))
91	90	rexlimdva 3135	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑑 ∈ 𝑅 ( -us ‘𝑑) = 𝑏 → 𝑏 <s ( -us ‘𝐴)))
92	72, 91	sylbid 240	. . . . . . . . 9 ⊢ (𝜑 → (𝑏 ∈ ( -us “ 𝑅) → 𝑏 <s ( -us ‘𝐴)))
93		breq2 5114	. . . . . . . . . 10 ⊢ (𝑎 = ( -us ‘𝐴) → (𝑏 <s 𝑎 ↔ 𝑏 <s ( -us ‘𝐴)))
94	93	imbi2d 340	. . . . . . . . 9 ⊢ (𝑎 = ( -us ‘𝐴) → ((𝑏 ∈ ( -us “ 𝑅) → 𝑏 <s 𝑎) ↔ (𝑏 ∈ ( -us “ 𝑅) → 𝑏 <s ( -us ‘𝐴))))
95	92, 94	syl5ibrcom 247	. . . . . . . 8 ⊢ (𝜑 → (𝑎 = ( -us ‘𝐴) → (𝑏 ∈ ( -us “ 𝑅) → 𝑏 <s 𝑎)))
96	70, 95	biimtrid 242	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ {( -us ‘𝐴)} → (𝑏 ∈ ( -us “ 𝑅) → 𝑏 <s 𝑎)))
97	96	3imp 1110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ {( -us ‘𝐴)} ∧ 𝑏 ∈ ( -us “ 𝑅)) → 𝑏 <s 𝑎)
98	97	3com23 1126	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ( -us “ 𝑅) ∧ 𝑎 ∈ {( -us ‘𝐴)}) → 𝑏 <s 𝑎)
99	59, 61, 67, 69, 98	ssltd 27710	. . . 4 ⊢ (𝜑 → ( -us “ 𝑅) <<s {( -us ‘𝐴)})
100	6	sneqd 4604	. . . 4 ⊢ (𝜑 → {( -us ‘𝐴)} = {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))})
101	99, 100	breqtrd 5136	. . 3 ⊢ (𝜑 → ( -us “ 𝑅) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))})
102		ssltex1 27705	. . . . . . 7 ⊢ (𝐿 <<s 𝑅 → 𝐿 ∈ V)
103	2, 102	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ V)
104		funimaexg 6606	. . . . . 6 ⊢ ((Fun -us ∧ 𝐿 ∈ V) → ( -us “ 𝐿) ∈ V)
105	55, 103, 104	sylancr 587	. . . . 5 ⊢ (𝜑 → ( -us “ 𝐿) ∈ V)
106		imassrn 6045	. . . . . . 7 ⊢ ( -us “ 𝐿) ⊆ ran -us
107	106, 65	sseqtri 3998	. . . . . 6 ⊢ ( -us “ 𝐿) ⊆ No
108	107	a1i 11	. . . . 5 ⊢ (𝜑 → ( -us “ 𝐿) ⊆ No )
109		fvelimab 6936	. . . . . . . . . 10 ⊢ (( -us Fn No ∧ 𝐿 ⊆ No ) → (𝑏 ∈ ( -us “ 𝐿) ↔ ∃𝑐 ∈ 𝐿 ( -us ‘𝑐) = 𝑏))
110	10, 34, 109	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (𝑏 ∈ ( -us “ 𝐿) ↔ ∃𝑐 ∈ 𝐿 ( -us ‘𝑐) = 𝑏))
111	2	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝐿 <<s 𝑅)
112	111, 75	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → ((𝐿 |s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
113	112	simp2d 1143	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝐿 <<s {(𝐿 |s 𝑅)})
114	73	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → {𝐴} = {(𝐿 |s 𝑅)})
115	113, 114	breqtrrd 5138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝐿 <<s {𝐴})
116		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝑐 ∈ 𝐿)
117	81	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝐴 ∈ {𝐴})
118	115, 116, 117	ssltsepcd 27713	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝑐 <s 𝐴)
119	34	sselda 3949	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝑐 ∈ No )
120	4	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝐴 ∈ No )
121	119, 120	sltnegd 27960	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → (𝑐 <s 𝐴 ↔ ( -us ‘𝐴) <s ( -us ‘𝑐)))
122	118, 121	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → ( -us ‘𝐴) <s ( -us ‘𝑐))
123		breq2 5114	. . . . . . . . . . 11 ⊢ (( -us ‘𝑐) = 𝑏 → (( -us ‘𝐴) <s ( -us ‘𝑐) ↔ ( -us ‘𝐴) <s 𝑏))
124	122, 123	syl5ibcom 245	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → (( -us ‘𝑐) = 𝑏 → ( -us ‘𝐴) <s 𝑏))
125	124	rexlimdva 3135	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑐 ∈ 𝐿 ( -us ‘𝑐) = 𝑏 → ( -us ‘𝐴) <s 𝑏))
126	110, 125	sylbid 240	. . . . . . . 8 ⊢ (𝜑 → (𝑏 ∈ ( -us “ 𝐿) → ( -us ‘𝐴) <s 𝑏))
127		breq1 5113	. . . . . . . . 9 ⊢ (𝑎 = ( -us ‘𝐴) → (𝑎 <s 𝑏 ↔ ( -us ‘𝐴) <s 𝑏))
128	127	imbi2d 340	. . . . . . . 8 ⊢ (𝑎 = ( -us ‘𝐴) → ((𝑏 ∈ ( -us “ 𝐿) → 𝑎 <s 𝑏) ↔ (𝑏 ∈ ( -us “ 𝐿) → ( -us ‘𝐴) <s 𝑏)))
129	126, 128	syl5ibrcom 247	. . . . . . 7 ⊢ (𝜑 → (𝑎 = ( -us ‘𝐴) → (𝑏 ∈ ( -us “ 𝐿) → 𝑎 <s 𝑏)))
130	70, 129	biimtrid 242	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ {( -us ‘𝐴)} → (𝑏 ∈ ( -us “ 𝐿) → 𝑎 <s 𝑏)))
131	130	3imp 1110	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ {( -us ‘𝐴)} ∧ 𝑏 ∈ ( -us “ 𝐿)) → 𝑎 <s 𝑏)
132	61, 105, 69, 108, 131	ssltd 27710	. . . 4 ⊢ (𝜑 → {( -us ‘𝐴)} <<s ( -us “ 𝐿))
133	100, 132	eqbrtrrd 5134	. . 3 ⊢ (𝜑 → {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} <<s ( -us “ 𝐿))
134	8, 31, 53, 101, 133	cofcut1d 27836	. 2 ⊢ (𝜑 → (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) = (( -us “ 𝑅) |s ( -us “ 𝐿)))
135	6, 134	eqtrd 2765	1 ⊢ (𝜑 → ( -us ‘𝐴) = (( -us “ 𝑅) |s ( -us “ 𝐿)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ⊆ wss 3917  {csn 4592   class class class wbr 5110  ran crn 5642   “ cima 5644  Fun wfun 6508   Fn wfn 6509  –onto→wfo 6512  ‘cfv 6514  (class class class)co 7390   No csur 27558   <s cslt 27559   ≤s csle 27663   <<s csslt 27699   |s cscut 27701   L cleft 27760   R cright 27761   -u_s cnegs 27932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-1o 8437  df-2o 8438  df-nadd 8633  df-no 27561  df-slt 27562  df-bday 27563  df-sle 27664  df-sslt 27700  df-scut 27702  df-0s 27743  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec 27852  df-norec2 27863  df-adds 27874  df-negs 27934

This theorem is referenced by:  zscut  28302  renegscl  28356