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Theorem negsunif 27443
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 (𝜑𝐿 <<s 𝑅)
negsunif.2 (𝜑𝐴 = (𝐿 |s 𝑅))
Assertion
Ref Expression
negsunif (𝜑 → ( -us𝐴) = (( -us𝑅) |s ( -us𝐿)))

Proof of Theorem negsunif
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 negsunif.2 . . . 4 (𝜑𝐴 = (𝐿 |s 𝑅))
2 negsunif.1 . . . . 5 (𝜑𝐿 <<s 𝑅)
32scutcld 27230 . . . 4 (𝜑 → (𝐿 |s 𝑅) ∈ No )
41, 3eqeltrd 2832 . . 3 (𝜑𝐴 No )
5 negsval 27416 . . 3 (𝐴 No → ( -us𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
64, 5syl 17 . 2 (𝜑 → ( -us𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
7 negscut2 27430 . . . 4 (𝐴 No → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
84, 7syl 17 . . 3 (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
92, 1cofcutr2d 27333 . . . . 5 (𝜑 → ∀𝑐 ∈ ( R ‘𝐴)∃𝑑𝑅 𝑑 ≤s 𝑐)
10 negsfn 27414 . . . . . . . 8 -us Fn No
11 ssltss2 27217 . . . . . . . . 9 (𝐿 <<s 𝑅𝑅 No )
122, 11syl 17 . . . . . . . 8 (𝜑𝑅 No )
13 breq2 5145 . . . . . . . . 9 (𝑏 = ( -us𝑑) → (( -us𝑐) ≤s 𝑏 ↔ ( -us𝑐) ≤s ( -us𝑑)))
1413imaeqsexv 7344 . . . . . . . 8 (( -us Fn No 𝑅 No ) → (∃𝑏 ∈ ( -us𝑅)( -us𝑐) ≤s 𝑏 ↔ ∃𝑑𝑅 ( -us𝑐) ≤s ( -us𝑑)))
1510, 12, 14sylancr 587 . . . . . . 7 (𝜑 → (∃𝑏 ∈ ( -us𝑅)( -us𝑐) ≤s 𝑏 ↔ ∃𝑑𝑅 ( -us𝑐) ≤s ( -us𝑑)))
1615ralbidv 3176 . . . . . 6 (𝜑 → (∀𝑐 ∈ ( R ‘𝐴)∃𝑏 ∈ ( -us𝑅)( -us𝑐) ≤s 𝑏 ↔ ∀𝑐 ∈ ( R ‘𝐴)∃𝑑𝑅 ( -us𝑐) ≤s ( -us𝑑)))
1712adantr 481 . . . . . . . . . 10 ((𝜑𝑐 ∈ ( R ‘𝐴)) → 𝑅 No )
1817sselda 3978 . . . . . . . . 9 (((𝜑𝑐 ∈ ( R ‘𝐴)) ∧ 𝑑𝑅) → 𝑑 No )
19 rightssno 27299 . . . . . . . . . . 11 ( R ‘𝐴) ⊆ No
2019sseli 3974 . . . . . . . . . 10 (𝑐 ∈ ( R ‘𝐴) → 𝑐 No )
2120ad2antlr 725 . . . . . . . . 9 (((𝜑𝑐 ∈ ( R ‘𝐴)) ∧ 𝑑𝑅) → 𝑐 No )
22 sleneg 27436 . . . . . . . . 9 ((𝑑 No 𝑐 No ) → (𝑑 ≤s 𝑐 ↔ ( -us𝑐) ≤s ( -us𝑑)))
2318, 21, 22syl2anc 584 . . . . . . . 8 (((𝜑𝑐 ∈ ( R ‘𝐴)) ∧ 𝑑𝑅) → (𝑑 ≤s 𝑐 ↔ ( -us𝑐) ≤s ( -us𝑑)))
2423rexbidva 3175 . . . . . . 7 ((𝜑𝑐 ∈ ( R ‘𝐴)) → (∃𝑑𝑅 𝑑 ≤s 𝑐 ↔ ∃𝑑𝑅 ( -us𝑐) ≤s ( -us𝑑)))
2524ralbidva 3174 . . . . . 6 (𝜑 → (∀𝑐 ∈ ( R ‘𝐴)∃𝑑𝑅 𝑑 ≤s 𝑐 ↔ ∀𝑐 ∈ ( R ‘𝐴)∃𝑑𝑅 ( -us𝑐) ≤s ( -us𝑑)))
2616, 25bitr4d 281 . . . . 5 (𝜑 → (∀𝑐 ∈ ( R ‘𝐴)∃𝑏 ∈ ( -us𝑅)( -us𝑐) ≤s 𝑏 ↔ ∀𝑐 ∈ ( R ‘𝐴)∃𝑑𝑅 𝑑 ≤s 𝑐))
279, 26mpbird 256 . . . 4 (𝜑 → ∀𝑐 ∈ ( R ‘𝐴)∃𝑏 ∈ ( -us𝑅)( -us𝑐) ≤s 𝑏)
28 breq1 5144 . . . . . . 7 (𝑎 = ( -us𝑐) → (𝑎 ≤s 𝑏 ↔ ( -us𝑐) ≤s 𝑏))
2928rexbidv 3177 . . . . . 6 (𝑎 = ( -us𝑐) → (∃𝑏 ∈ ( -us𝑅)𝑎 ≤s 𝑏 ↔ ∃𝑏 ∈ ( -us𝑅)( -us𝑐) ≤s 𝑏))
3029imaeqsalv 7345 . . . . 5 (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ) → (∀𝑎 ∈ ( -us “ ( R ‘𝐴))∃𝑏 ∈ ( -us𝑅)𝑎 ≤s 𝑏 ↔ ∀𝑐 ∈ ( R ‘𝐴)∃𝑏 ∈ ( -us𝑅)( -us𝑐) ≤s 𝑏))
3110, 19, 30mp2an 690 . . . 4 (∀𝑎 ∈ ( -us “ ( R ‘𝐴))∃𝑏 ∈ ( -us𝑅)𝑎 ≤s 𝑏 ↔ ∀𝑐 ∈ ( R ‘𝐴)∃𝑏 ∈ ( -us𝑅)( -us𝑐) ≤s 𝑏)
3227, 31sylibr 233 . . 3 (𝜑 → ∀𝑎 ∈ ( -us “ ( R ‘𝐴))∃𝑏 ∈ ( -us𝑅)𝑎 ≤s 𝑏)
332, 1cofcutr1d 27332 . . . . 5 (𝜑 → ∀𝑐 ∈ ( L ‘𝐴)∃𝑑𝐿 𝑐 ≤s 𝑑)
34 ssltss1 27216 . . . . . . . . 9 (𝐿 <<s 𝑅𝐿 No )
352, 34syl 17 . . . . . . . 8 (𝜑𝐿 No )
36 breq1 5144 . . . . . . . . 9 (𝑏 = ( -us𝑑) → (𝑏 ≤s ( -us𝑐) ↔ ( -us𝑑) ≤s ( -us𝑐)))
3736imaeqsexv 7344 . . . . . . . 8 (( -us Fn No 𝐿 No ) → (∃𝑏 ∈ ( -us𝐿)𝑏 ≤s ( -us𝑐) ↔ ∃𝑑𝐿 ( -us𝑑) ≤s ( -us𝑐)))
3810, 35, 37sylancr 587 . . . . . . 7 (𝜑 → (∃𝑏 ∈ ( -us𝐿)𝑏 ≤s ( -us𝑐) ↔ ∃𝑑𝐿 ( -us𝑑) ≤s ( -us𝑐)))
3938ralbidv 3176 . . . . . 6 (𝜑 → (∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us𝐿)𝑏 ≤s ( -us𝑐) ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑑𝐿 ( -us𝑑) ≤s ( -us𝑐)))
40 leftssno 27298 . . . . . . . . . . 11 ( L ‘𝐴) ⊆ No
4140sseli 3974 . . . . . . . . . 10 (𝑐 ∈ ( L ‘𝐴) → 𝑐 No )
4241ad2antlr 725 . . . . . . . . 9 (((𝜑𝑐 ∈ ( L ‘𝐴)) ∧ 𝑑𝐿) → 𝑐 No )
4335adantr 481 . . . . . . . . . 10 ((𝜑𝑐 ∈ ( L ‘𝐴)) → 𝐿 No )
4443sselda 3978 . . . . . . . . 9 (((𝜑𝑐 ∈ ( L ‘𝐴)) ∧ 𝑑𝐿) → 𝑑 No )
45 sleneg 27436 . . . . . . . . 9 ((𝑐 No 𝑑 No ) → (𝑐 ≤s 𝑑 ↔ ( -us𝑑) ≤s ( -us𝑐)))
4642, 44, 45syl2anc 584 . . . . . . . 8 (((𝜑𝑐 ∈ ( L ‘𝐴)) ∧ 𝑑𝐿) → (𝑐 ≤s 𝑑 ↔ ( -us𝑑) ≤s ( -us𝑐)))
4746rexbidva 3175 . . . . . . 7 ((𝜑𝑐 ∈ ( L ‘𝐴)) → (∃𝑑𝐿 𝑐 ≤s 𝑑 ↔ ∃𝑑𝐿 ( -us𝑑) ≤s ( -us𝑐)))
4847ralbidva 3174 . . . . . 6 (𝜑 → (∀𝑐 ∈ ( L ‘𝐴)∃𝑑𝐿 𝑐 ≤s 𝑑 ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑑𝐿 ( -us𝑑) ≤s ( -us𝑐)))
4939, 48bitr4d 281 . . . . 5 (𝜑 → (∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us𝐿)𝑏 ≤s ( -us𝑐) ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑑𝐿 𝑐 ≤s 𝑑))
5033, 49mpbird 256 . . . 4 (𝜑 → ∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us𝐿)𝑏 ≤s ( -us𝑐))
51 breq2 5145 . . . . . . 7 (𝑎 = ( -us𝑐) → (𝑏 ≤s 𝑎𝑏 ≤s ( -us𝑐)))
5251rexbidv 3177 . . . . . 6 (𝑎 = ( -us𝑐) → (∃𝑏 ∈ ( -us𝐿)𝑏 ≤s 𝑎 ↔ ∃𝑏 ∈ ( -us𝐿)𝑏 ≤s ( -us𝑐)))
5352imaeqsalv 7345 . . . . 5 (( -us Fn No ∧ ( L ‘𝐴) ⊆ No ) → (∀𝑎 ∈ ( -us “ ( L ‘𝐴))∃𝑏 ∈ ( -us𝐿)𝑏 ≤s 𝑎 ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us𝐿)𝑏 ≤s ( -us𝑐)))
5410, 40, 53mp2an 690 . . . 4 (∀𝑎 ∈ ( -us “ ( L ‘𝐴))∃𝑏 ∈ ( -us𝐿)𝑏 ≤s 𝑎 ↔ ∀𝑐 ∈ ( L ‘𝐴)∃𝑏 ∈ ( -us𝐿)𝑏 ≤s ( -us𝑐))
5550, 54sylibr 233 . . 3 (𝜑 → ∀𝑎 ∈ ( -us “ ( L ‘𝐴))∃𝑏 ∈ ( -us𝐿)𝑏 ≤s 𝑎)
56 fnfun 6638 . . . . . . 7 ( -us Fn No → Fun -us )
5710, 56ax-mp 5 . . . . . 6 Fun -us
58 ssltex2 27215 . . . . . . 7 (𝐿 <<s 𝑅𝑅 ∈ V)
592, 58syl 17 . . . . . 6 (𝜑𝑅 ∈ V)
60 funimaexg 6623 . . . . . 6 ((Fun -us𝑅 ∈ V) → ( -us𝑅) ∈ V)
6157, 59, 60sylancr 587 . . . . 5 (𝜑 → ( -us𝑅) ∈ V)
62 snex 5424 . . . . . 6 {( -us𝐴)} ∈ V
6362a1i 11 . . . . 5 (𝜑 → {( -us𝐴)} ∈ V)
64 imassrn 6060 . . . . . . 7 ( -us𝑅) ⊆ ran -us
65 negsfo 27441 . . . . . . . 8 -us : No onto No
66 forn 6795 . . . . . . . 8 ( -us : No onto No → ran -us = No )
6765, 66ax-mp 5 . . . . . . 7 ran -us = No
6864, 67sseqtri 4014 . . . . . 6 ( -us𝑅) ⊆ No
6968a1i 11 . . . . 5 (𝜑 → ( -us𝑅) ⊆ No )
704negscld 27427 . . . . . 6 (𝜑 → ( -us𝐴) ∈ No )
7170snssd 4805 . . . . 5 (𝜑 → {( -us𝐴)} ⊆ No )
72 velsn 4638 . . . . . . . 8 (𝑎 ∈ {( -us𝐴)} ↔ 𝑎 = ( -us𝐴))
73 fvelimab 6950 . . . . . . . . . . 11 (( -us Fn No 𝑅 No ) → (𝑏 ∈ ( -us𝑅) ↔ ∃𝑑𝑅 ( -us𝑑) = 𝑏))
7410, 12, 73sylancr 587 . . . . . . . . . 10 (𝜑 → (𝑏 ∈ ( -us𝑅) ↔ ∃𝑑𝑅 ( -us𝑑) = 𝑏))
751sneqd 4634 . . . . . . . . . . . . . . . 16 (𝜑 → {𝐴} = {(𝐿 |s 𝑅)})
7675adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑑𝑅) → {𝐴} = {(𝐿 |s 𝑅)})
77 scutcut 27228 . . . . . . . . . . . . . . . . . 18 (𝐿 <<s 𝑅 → ((𝐿 |s 𝑅) ∈ No 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
782, 77syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐿 |s 𝑅) ∈ No 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
7978simp3d 1144 . . . . . . . . . . . . . . . 16 (𝜑 → {(𝐿 |s 𝑅)} <<s 𝑅)
8079adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑑𝑅) → {(𝐿 |s 𝑅)} <<s 𝑅)
8176, 80eqbrtrd 5163 . . . . . . . . . . . . . 14 ((𝜑𝑑𝑅) → {𝐴} <<s 𝑅)
82 snidg 4656 . . . . . . . . . . . . . . . 16 (𝐴 No 𝐴 ∈ {𝐴})
834, 82syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ {𝐴})
8483adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑑𝑅) → 𝐴 ∈ {𝐴})
85 simpr 485 . . . . . . . . . . . . . 14 ((𝜑𝑑𝑅) → 𝑑𝑅)
8681, 84, 85ssltsepcd 27221 . . . . . . . . . . . . 13 ((𝜑𝑑𝑅) → 𝐴 <s 𝑑)
874adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑑𝑅) → 𝐴 No )
8812sselda 3978 . . . . . . . . . . . . . 14 ((𝜑𝑑𝑅) → 𝑑 No )
89 sltneg 27435 . . . . . . . . . . . . . 14 ((𝐴 No 𝑑 No ) → (𝐴 <s 𝑑 ↔ ( -us𝑑) <s ( -us𝐴)))
9087, 88, 89syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑑𝑅) → (𝐴 <s 𝑑 ↔ ( -us𝑑) <s ( -us𝐴)))
9186, 90mpbid 231 . . . . . . . . . . . 12 ((𝜑𝑑𝑅) → ( -us𝑑) <s ( -us𝐴))
92 breq1 5144 . . . . . . . . . . . 12 (( -us𝑑) = 𝑏 → (( -us𝑑) <s ( -us𝐴) ↔ 𝑏 <s ( -us𝐴)))
9391, 92syl5ibcom 244 . . . . . . . . . . 11 ((𝜑𝑑𝑅) → (( -us𝑑) = 𝑏𝑏 <s ( -us𝐴)))
9493rexlimdva 3154 . . . . . . . . . 10 (𝜑 → (∃𝑑𝑅 ( -us𝑑) = 𝑏𝑏 <s ( -us𝐴)))
9574, 94sylbid 239 . . . . . . . . 9 (𝜑 → (𝑏 ∈ ( -us𝑅) → 𝑏 <s ( -us𝐴)))
96 breq2 5145 . . . . . . . . . 10 (𝑎 = ( -us𝐴) → (𝑏 <s 𝑎𝑏 <s ( -us𝐴)))
9796imbi2d 340 . . . . . . . . 9 (𝑎 = ( -us𝐴) → ((𝑏 ∈ ( -us𝑅) → 𝑏 <s 𝑎) ↔ (𝑏 ∈ ( -us𝑅) → 𝑏 <s ( -us𝐴))))
9895, 97syl5ibrcom 246 . . . . . . . 8 (𝜑 → (𝑎 = ( -us𝐴) → (𝑏 ∈ ( -us𝑅) → 𝑏 <s 𝑎)))
9972, 98biimtrid 241 . . . . . . 7 (𝜑 → (𝑎 ∈ {( -us𝐴)} → (𝑏 ∈ ( -us𝑅) → 𝑏 <s 𝑎)))
100993imp 1111 . . . . . 6 ((𝜑𝑎 ∈ {( -us𝐴)} ∧ 𝑏 ∈ ( -us𝑅)) → 𝑏 <s 𝑎)
1011003com23 1126 . . . . 5 ((𝜑𝑏 ∈ ( -us𝑅) ∧ 𝑎 ∈ {( -us𝐴)}) → 𝑏 <s 𝑎)
10261, 63, 69, 71, 101ssltd 27219 . . . 4 (𝜑 → ( -us𝑅) <<s {( -us𝐴)})
1036sneqd 4634 . . . 4 (𝜑 → {( -us𝐴)} = {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))})
104102, 103breqtrd 5167 . . 3 (𝜑 → ( -us𝑅) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))})
105 ssltex1 27214 . . . . . . 7 (𝐿 <<s 𝑅𝐿 ∈ V)
1062, 105syl 17 . . . . . 6 (𝜑𝐿 ∈ V)
107 funimaexg 6623 . . . . . 6 ((Fun -us𝐿 ∈ V) → ( -us𝐿) ∈ V)
10857, 106, 107sylancr 587 . . . . 5 (𝜑 → ( -us𝐿) ∈ V)
109 imassrn 6060 . . . . . . 7 ( -us𝐿) ⊆ ran -us
110109, 67sseqtri 4014 . . . . . 6 ( -us𝐿) ⊆ No
111110a1i 11 . . . . 5 (𝜑 → ( -us𝐿) ⊆ No )
112 fvelimab 6950 . . . . . . . . . 10 (( -us Fn No 𝐿 No ) → (𝑏 ∈ ( -us𝐿) ↔ ∃𝑐𝐿 ( -us𝑐) = 𝑏))
11310, 35, 112sylancr 587 . . . . . . . . 9 (𝜑 → (𝑏 ∈ ( -us𝐿) ↔ ∃𝑐𝐿 ( -us𝑐) = 𝑏))
1142adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑐𝐿) → 𝐿 <<s 𝑅)
115114, 77syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑐𝐿) → ((𝐿 |s 𝑅) ∈ No 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
116115simp2d 1143 . . . . . . . . . . . . . 14 ((𝜑𝑐𝐿) → 𝐿 <<s {(𝐿 |s 𝑅)})
11775adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑐𝐿) → {𝐴} = {(𝐿 |s 𝑅)})
118116, 117breqtrrd 5169 . . . . . . . . . . . . 13 ((𝜑𝑐𝐿) → 𝐿 <<s {𝐴})
119 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑐𝐿) → 𝑐𝐿)
12083adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑐𝐿) → 𝐴 ∈ {𝐴})
121118, 119, 120ssltsepcd 27221 . . . . . . . . . . . 12 ((𝜑𝑐𝐿) → 𝑐 <s 𝐴)
12235sselda 3978 . . . . . . . . . . . . 13 ((𝜑𝑐𝐿) → 𝑐 No )
1234adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑐𝐿) → 𝐴 No )
124 sltneg 27435 . . . . . . . . . . . . 13 ((𝑐 No 𝐴 No ) → (𝑐 <s 𝐴 ↔ ( -us𝐴) <s ( -us𝑐)))
125122, 123, 124syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑐𝐿) → (𝑐 <s 𝐴 ↔ ( -us𝐴) <s ( -us𝑐)))
126121, 125mpbid 231 . . . . . . . . . . 11 ((𝜑𝑐𝐿) → ( -us𝐴) <s ( -us𝑐))
127 breq2 5145 . . . . . . . . . . 11 (( -us𝑐) = 𝑏 → (( -us𝐴) <s ( -us𝑐) ↔ ( -us𝐴) <s 𝑏))
128126, 127syl5ibcom 244 . . . . . . . . . 10 ((𝜑𝑐𝐿) → (( -us𝑐) = 𝑏 → ( -us𝐴) <s 𝑏))
129128rexlimdva 3154 . . . . . . . . 9 (𝜑 → (∃𝑐𝐿 ( -us𝑐) = 𝑏 → ( -us𝐴) <s 𝑏))
130113, 129sylbid 239 . . . . . . . 8 (𝜑 → (𝑏 ∈ ( -us𝐿) → ( -us𝐴) <s 𝑏))
131 breq1 5144 . . . . . . . . 9 (𝑎 = ( -us𝐴) → (𝑎 <s 𝑏 ↔ ( -us𝐴) <s 𝑏))
132131imbi2d 340 . . . . . . . 8 (𝑎 = ( -us𝐴) → ((𝑏 ∈ ( -us𝐿) → 𝑎 <s 𝑏) ↔ (𝑏 ∈ ( -us𝐿) → ( -us𝐴) <s 𝑏)))
133130, 132syl5ibrcom 246 . . . . . . 7 (𝜑 → (𝑎 = ( -us𝐴) → (𝑏 ∈ ( -us𝐿) → 𝑎 <s 𝑏)))
13472, 133biimtrid 241 . . . . . 6 (𝜑 → (𝑎 ∈ {( -us𝐴)} → (𝑏 ∈ ( -us𝐿) → 𝑎 <s 𝑏)))
1351343imp 1111 . . . . 5 ((𝜑𝑎 ∈ {( -us𝐴)} ∧ 𝑏 ∈ ( -us𝐿)) → 𝑎 <s 𝑏)
13663, 108, 71, 111, 135ssltd 27219 . . . 4 (𝜑 → {( -us𝐴)} <<s ( -us𝐿))
137103, 136eqbrtrrd 5165 . . 3 (𝜑 → {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} <<s ( -us𝐿))
1388, 32, 55, 104, 137cofcut1d 27328 . 2 (𝜑 → (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) = (( -us𝑅) |s ( -us𝐿)))
1396, 138eqtrd 2771 1 (𝜑 → ( -us𝐴) = (( -us𝑅) |s ( -us𝐿)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3060  wrex 3069  Vcvv 3473  wss 3944  {csn 4622   class class class wbr 5141  ran crn 5670  cima 5672  Fun wfun 6526   Fn wfn 6527  ontowfo 6530  cfv 6532  (class class class)co 7393   No csur 27070   <s cslt 27071   ≤s csle 27174   <<s csslt 27208   |s cscut 27210   L cleft 27263   R cright 27264   -us cnegs 27410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-ot 4631  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-1o 8448  df-2o 8449  df-nadd 8648  df-no 27073  df-slt 27074  df-bday 27075  df-sle 27175  df-sslt 27209  df-scut 27211  df-0s 27251  df-made 27265  df-old 27266  df-left 27268  df-right 27269  df-norec 27338  df-norec2 27349  df-adds 27360  df-negs 27412
This theorem is referenced by: (None)
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