Theorem negsunif 28087

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 27848	. . . 4 ⊢ (𝜑 → (𝐿 |s 𝑅) ∈ No )
4	1, 3	eqeltrd 2841	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
5		negsval 28057	. . 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 28072	. . . 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 27960	. . . . 5 ⊢ (𝜑 → ∀𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ 𝑅 𝑑 ≤s 𝑐)
10		negsfn 28055	. . . . . . . 8 ⊢ -us Fn No
11		ssltss2 27834	. . . . . . . . 9 ⊢ (𝐿 <<s 𝑅 → 𝑅 ⊆ No )
12	2, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ⊆ No )
13		breq2 5147	. . . . . . . . 9 ⊢ (𝑏 = ( -us ‘𝑑) → (( -us ‘𝑐) ≤s 𝑏 ↔ ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
14	13	rexima 7258	. . . . . . . 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 3178	. . . . . 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 3983	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ 𝑑 ∈ 𝑅) → 𝑑 ∈ No )
19		rightssno 27920	. . . . . . . . . . 11 ⊢ ( R ‘𝐴) ⊆ No
20	19	sseli 3979	. . . . . . . . . 10 ⊢ (𝑐 ∈ ( R ‘𝐴) → 𝑐 ∈ No )
21	20	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ 𝑑 ∈ 𝑅) → 𝑐 ∈ No )
22	18, 21	slenegd 28080	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ 𝑑 ∈ 𝑅) → (𝑑 ≤s 𝑐 ↔ ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
23	22	rexbidva 3177	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → (∃𝑑 ∈ 𝑅 𝑑 ≤s 𝑐 ↔ ∃𝑑 ∈ 𝑅 ( -us ‘𝑐) ≤s ( -us ‘𝑑)))
24	23	ralbidva 3176	. . . . . 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 5146	. . . . . . 7 ⊢ (𝑎 = ( -us ‘𝑐) → (𝑎 ≤s 𝑏 ↔ ( -us ‘𝑐) ≤s 𝑏))
28	27	rexbidv 3179	. . . . . 6 ⊢ (𝑎 = ( -us ‘𝑐) → (∃𝑏 ∈ ( -us “ 𝑅)𝑎 ≤s 𝑏 ↔ ∃𝑏 ∈ ( -us “ 𝑅)( -us ‘𝑐) ≤s 𝑏))
29	28	ralima 7257	. . . . 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 27959	. . . . 5 ⊢ (𝜑 → ∀𝑐 ∈ ( L ‘𝐴)∃𝑑 ∈ 𝐿 𝑐 ≤s 𝑑)
33		ssltss1 27833	. . . . . . . . 9 ⊢ (𝐿 <<s 𝑅 → 𝐿 ⊆ No )
34	2, 33	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ⊆ No )
35		breq1 5146	. . . . . . . . 9 ⊢ (𝑏 = ( -us ‘𝑑) → (𝑏 ≤s ( -us ‘𝑐) ↔ ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
36	35	rexima 7258	. . . . . . . 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 3178	. . . . . 6 ⊢ (𝜑 → (∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐) ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑑 ∈ 𝐿 ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
39		leftssno 27919	. . . . . . . . . . 11 ⊢ ( L ‘𝐴) ⊆ No
40	39	sseli 3979	. . . . . . . . . 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 3983	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ( L ‘𝐴)) ∧ 𝑑 ∈ 𝐿) → 𝑑 ∈ No )
44	41, 43	slenegd 28080	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ( L ‘𝐴)) ∧ 𝑑 ∈ 𝐿) → (𝑐 ≤s 𝑑 ↔ ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
45	44	rexbidva 3177	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ( L ‘𝐴)) → (∃𝑑 ∈ 𝐿 𝑐 ≤s 𝑑 ↔ ∃𝑑 ∈ 𝐿 ( -us ‘𝑑) ≤s ( -us ‘𝑐)))
46	45	ralbidva 3176	. . . . . 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 5147	. . . . . . 7 ⊢ (𝑎 = ( -us ‘𝑐) → (𝑏 ≤s 𝑎 ↔ 𝑏 ≤s ( -us ‘𝑐)))
50	49	rexbidv 3179	. . . . . 6 ⊢ (𝑎 = ( -us ‘𝑐) → (∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s 𝑎 ↔ ∃𝑏 ∈ ( -us “ 𝐿)𝑏 ≤s ( -us ‘𝑐)))
51	50	ralima 7257	. . . . 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 6668	. . . . . . 7 ⊢ ( -us Fn No → Fun -us )
55	10, 54	ax-mp 5	. . . . . 6 ⊢ Fun -us
56		ssltex2 27832	. . . . . . 7 ⊢ (𝐿 <<s 𝑅 → 𝑅 ∈ V)
57	2, 56	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
58		funimaexg 6653	. . . . . 6 ⊢ ((Fun -us ∧ 𝑅 ∈ V) → ( -us “ 𝑅) ∈ V)
59	55, 57, 58	sylancr 587	. . . . 5 ⊢ (𝜑 → ( -us “ 𝑅) ∈ V)
60		snex 5436	. . . . . 6 ⊢ {( -us ‘𝐴)} ∈ V
61	60	a1i 11	. . . . 5 ⊢ (𝜑 → {( -us ‘𝐴)} ∈ V)
62		imassrn 6089	. . . . . . 7 ⊢ ( -us “ 𝑅) ⊆ ran -us
63		negsfo 28085	. . . . . . . 8 ⊢ -us : No –onto→ No
64		forn 6823	. . . . . . . 8 ⊢ ( -us : No –onto→ No → ran -us = No )
65	63, 64	ax-mp 5	. . . . . . 7 ⊢ ran -us = No
66	62, 65	sseqtri 4032	. . . . . 6 ⊢ ( -us “ 𝑅) ⊆ No
67	66	a1i 11	. . . . 5 ⊢ (𝜑 → ( -us “ 𝑅) ⊆ No )
68	4	negscld 28069	. . . . . 6 ⊢ (𝜑 → ( -us ‘𝐴) ∈ No )
69	68	snssd 4809	. . . . 5 ⊢ (𝜑 → {( -us ‘𝐴)} ⊆ No )
70		velsn 4642	. . . . . . . 8 ⊢ (𝑎 ∈ {( -us ‘𝐴)} ↔ 𝑎 = ( -us ‘𝐴))
71		fvelimab 6981	. . . . . . . . . . 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 4638	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝐴} = {(𝐿 |s 𝑅)})
74	73	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → {𝐴} = {(𝐿 |s 𝑅)})
75		scutcut 27846	. . . . . . . . . . . . . . . . . 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 1145	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {(𝐿 |s 𝑅)} <<s 𝑅)
78	77	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → {(𝐿 |s 𝑅)} <<s 𝑅)
79	74, 78	eqbrtrd 5165	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → {𝐴} <<s 𝑅)
80		snidg 4660	. . . . . . . . . . . . . . . 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 27839	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → 𝐴 <s 𝑑)
85	4	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → 𝐴 ∈ No )
86	12	sselda 3983	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑅) → 𝑑 ∈ No )
87	85, 86	sltnegd 28079	. . . . . . . . . . . . 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 5146	. . . . . . . . . . . 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 3155	. . . . . . . . . 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 5147	. . . . . . . . . 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 1111	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ {( -us ‘𝐴)} ∧ 𝑏 ∈ ( -us “ 𝑅)) → 𝑏 <s 𝑎)
98	97	3com23 1127	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ( -us “ 𝑅) ∧ 𝑎 ∈ {( -us ‘𝐴)}) → 𝑏 <s 𝑎)
99	59, 61, 67, 69, 98	ssltd 27836	. . . 4 ⊢ (𝜑 → ( -us “ 𝑅) <<s {( -us ‘𝐴)})
100	6	sneqd 4638	. . . 4 ⊢ (𝜑 → {( -us ‘𝐴)} = {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))})
101	99, 100	breqtrd 5169	. . 3 ⊢ (𝜑 → ( -us “ 𝑅) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))})
102		ssltex1 27831	. . . . . . 7 ⊢ (𝐿 <<s 𝑅 → 𝐿 ∈ V)
103	2, 102	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ V)
104		funimaexg 6653	. . . . . 6 ⊢ ((Fun -us ∧ 𝐿 ∈ V) → ( -us “ 𝐿) ∈ V)
105	55, 103, 104	sylancr 587	. . . . 5 ⊢ (𝜑 → ( -us “ 𝐿) ∈ V)
106		imassrn 6089	. . . . . . 7 ⊢ ( -us “ 𝐿) ⊆ ran -us
107	106, 65	sseqtri 4032	. . . . . 6 ⊢ ( -us “ 𝐿) ⊆ No
108	107	a1i 11	. . . . 5 ⊢ (𝜑 → ( -us “ 𝐿) ⊆ No )
109		fvelimab 6981	. . . . . . . . . 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 1144	. . . . . . . . . . . . . 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 5171	. . . . . . . . . . . . 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 27839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝑐 <s 𝐴)
119	34	sselda 3983	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝑐 ∈ No )
120	4	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐿) → 𝐴 ∈ No )
121	119, 120	sltnegd 28079	. . . . . . . . . . . 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 5147	. . . . . . . . . . 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 3155	. . . . . . . . 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 5146	. . . . . . . . 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 1111	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ {( -us ‘𝐴)} ∧ 𝑏 ∈ ( -us “ 𝐿)) → 𝑎 <s 𝑏)
132	61, 105, 69, 108, 131	ssltd 27836	. . . 4 ⊢ (𝜑 → {( -us ‘𝐴)} <<s ( -us “ 𝐿))
133	100, 132	eqbrtrrd 5167	. . 3 ⊢ (𝜑 → {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} <<s ( -us “ 𝐿))
134	8, 31, 53, 101, 133	cofcut1d 27955	. 2 ⊢ (𝜑 → (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) = (( -us “ 𝑅) |s ( -us “ 𝐿)))
135	6, 134	eqtrd 2777	1 ⊢ (𝜑 → ( -us ‘𝐴) = (( -us “ 𝑅) |s ( -us “ 𝐿)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  {csn 4626   class class class wbr 5143  ran crn 5686   “ cima 5688  Fun wfun 6555   Fn wfn 6556  –onto→wfo 6559  ‘cfv 6561  (class class class)co 7431   No csur 27684   <s cslt 27685   ≤s csle 27789   <<s csslt 27825   |s cscut 27827   L cleft 27884   R cright 27885   -u_s cnegs 28051

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053

This theorem is referenced by:  zscut  28393  renegscl  28430