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Theorem absmuls 28151
Description: Surreal absolute value distributes over multiplication. (Contributed by Scott Fenton, 16-Apr-2025.)
Assertion
Ref Expression
absmuls ((𝐴 No 𝐵 No ) → (abss‘(𝐴 ·s 𝐵)) = ((abss𝐴) ·s (abss𝐵)))

Proof of Theorem absmuls
StepHypRef Expression
1 mulscl 28047 . . . . . . 7 ((𝐴 No 𝐵 No ) → (𝐴 ·s 𝐵) ∈ No )
21adantr 480 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → (𝐴 ·s 𝐵) ∈ No )
3 simplll 774 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 𝐴 No )
4 simpllr 775 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 𝐵 No )
5 simplr 768 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s 𝐴)
6 simpr 484 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s 𝐵)
73, 4, 5, 6mulsge0d 28059 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
8 abssid 28148 . . . . . 6 (((𝐴 ·s 𝐵) ∈ No ∧ 0s ≤s (𝐴 ·s 𝐵)) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
92, 7, 8syl2an2r 684 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
10 abssid 28148 . . . . . . 7 ((𝐵 No ∧ 0s ≤s 𝐵) → (abss𝐵) = 𝐵)
1110ad4ant24 753 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss𝐵) = 𝐵)
1211oveq2d 7436 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (𝐴 ·s (abss𝐵)) = (𝐴 ·s 𝐵))
139, 12eqtr4d 2771 . . . 4 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss𝐵)))
14 simplll 774 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐴 No )
15 simpllr 775 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐵 No )
1614, 15mulnegs2d 28074 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s ( -us𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
17 abssnid 28150 . . . . . . 7 ((𝐵 No 𝐵 ≤s 0s ) → (abss𝐵) = ( -us𝐵))
1817ad4ant24 753 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss𝐵) = ( -us𝐵))
1918oveq2d 7436 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s (abss𝐵)) = (𝐴 ·s ( -us𝐵)))
20 negs0s 27952 . . . . . . . 8 ( -us ‘ 0s ) = 0s
2115negscld 27962 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -us𝐵) ∈ No )
22 simplr 768 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s 𝐴)
23 simpr 484 . . . . . . . . . . . 12 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐵 ≤s 0s )
24 0sno 27772 . . . . . . . . . . . . . 14 0s No
2524a1i 11 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s No )
2615, 25slenegd 27973 . . . . . . . . . . . 12 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐵 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us𝐵)))
2723, 26mpbid 231 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us𝐵))
2820, 27eqbrtrrid 5184 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us𝐵))
2914, 21, 22, 28mulsge0d 28059 . . . . . . . . 9 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s ( -us𝐵)))
3029, 16breqtrd 5174 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘(𝐴 ·s 𝐵)))
3120, 30eqbrtrid 5183 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us ‘(𝐴 ·s 𝐵)))
322adantr 480 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s 𝐵) ∈ No )
3332, 25slenegd 27973 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ((𝐴 ·s 𝐵) ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘(𝐴 ·s 𝐵))))
3431, 33mpbird 257 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s 𝐵) ≤s 0s )
35 abssnid 28150 . . . . . 6 (((𝐴 ·s 𝐵) ∈ No ∧ (𝐴 ·s 𝐵) ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
362, 34, 35syl2an2r 684 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
3716, 19, 363eqtr4rd 2779 . . . 4 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss𝐵)))
38 sletric 27710 . . . . . 6 (( 0s No 𝐵 No ) → ( 0s ≤s 𝐵𝐵 ≤s 0s ))
3924, 38mpan 689 . . . . 5 (𝐵 No → ( 0s ≤s 𝐵𝐵 ≤s 0s ))
4039ad2antlr 726 . . . 4 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → ( 0s ≤s 𝐵𝐵 ≤s 0s ))
4113, 37, 40mpjaodan 957 . . 3 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss𝐵)))
42 abssid 28148 . . . . 5 ((𝐴 No ∧ 0s ≤s 𝐴) → (abss𝐴) = 𝐴)
4342adantlr 714 . . . 4 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → (abss𝐴) = 𝐴)
4443oveq1d 7435 . . 3 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → ((abss𝐴) ·s (abss𝐵)) = (𝐴 ·s (abss𝐵)))
4541, 44eqtr4d 2771 . 2 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → (abss‘(𝐴 ·s 𝐵)) = ((abss𝐴) ·s (abss𝐵)))
46 simplll 774 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 𝐴 No )
47 simpllr 775 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 𝐵 No )
4846, 47mulnegs1d 28073 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (( -us𝐴) ·s 𝐵) = ( -us ‘(𝐴 ·s 𝐵)))
4910ad4ant24 753 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss𝐵) = 𝐵)
5049oveq2d 7436 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (( -us𝐴) ·s (abss𝐵)) = (( -us𝐴) ·s 𝐵))
511adantr 480 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) → (𝐴 ·s 𝐵) ∈ No )
5246negscld 27962 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -us𝐴) ∈ No )
53 simplr 768 . . . . . . . . . . . 12 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 𝐴 ≤s 0s )
5424a1i 11 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s No )
5546, 54slenegd 27973 . . . . . . . . . . . 12 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us𝐴)))
5653, 55mpbid 231 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -us ‘ 0s ) ≤s ( -us𝐴))
5720, 56eqbrtrrid 5184 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s ( -us𝐴))
58 simpr 484 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s 𝐵)
5952, 47, 57, 58mulsge0d 28059 . . . . . . . . 9 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s (( -us𝐴) ·s 𝐵))
6059, 48breqtrd 5174 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s ( -us ‘(𝐴 ·s 𝐵)))
6120, 60eqbrtrid 5183 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -us ‘ 0s ) ≤s ( -us ‘(𝐴 ·s 𝐵)))
6251adantr 480 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (𝐴 ·s 𝐵) ∈ No )
6362, 54slenegd 27973 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ((𝐴 ·s 𝐵) ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘(𝐴 ·s 𝐵))))
6461, 63mpbird 257 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (𝐴 ·s 𝐵) ≤s 0s )
6551, 64, 35syl2an2r 684 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
6648, 50, 653eqtr4rd 2779 . . . 4 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (( -us𝐴) ·s (abss𝐵)))
67 simplll 774 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐴 No )
68 simpllr 775 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐵 No )
6967, 68mul2negsd 28075 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -us𝐴) ·s ( -us𝐵)) = (𝐴 ·s 𝐵))
7017ad4ant24 753 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss𝐵) = ( -us𝐵))
7170oveq2d 7436 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -us𝐴) ·s (abss𝐵)) = (( -us𝐴) ·s ( -us𝐵)))
7267negscld 27962 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us𝐴) ∈ No )
7368negscld 27962 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us𝐵) ∈ No )
74 simplr 768 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐴 ≤s 0s )
7524a1i 11 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s No )
7667, 75slenegd 27973 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us𝐴)))
7774, 76mpbid 231 . . . . . . . . 9 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us𝐴))
7820, 77eqbrtrrid 5184 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us𝐴))
79 simpr 484 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐵 ≤s 0s )
8068, 75slenegd 27973 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (𝐵 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us𝐵)))
8179, 80mpbid 231 . . . . . . . . 9 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us𝐵))
8220, 81eqbrtrrid 5184 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us𝐵))
8372, 73, 78, 82mulsge0d 28059 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (( -us𝐴) ·s ( -us𝐵)))
8483, 69breqtrd 5174 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s 𝐵))
8551, 84, 8syl2an2r 684 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
8669, 71, 853eqtr4rd 2779 . . . 4 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (( -us𝐴) ·s (abss𝐵)))
8739ad2antlr 726 . . . 4 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) → ( 0s ≤s 𝐵𝐵 ≤s 0s ))
8866, 86, 87mpjaodan 957 . . 3 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (( -us𝐴) ·s (abss𝐵)))
89 abssnid 28150 . . . . 5 ((𝐴 No 𝐴 ≤s 0s ) → (abss𝐴) = ( -us𝐴))
9089oveq1d 7435 . . . 4 ((𝐴 No 𝐴 ≤s 0s ) → ((abss𝐴) ·s (abss𝐵)) = (( -us𝐴) ·s (abss𝐵)))
9190adantlr 714 . . 3 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) → ((abss𝐴) ·s (abss𝐵)) = (( -us𝐴) ·s (abss𝐵)))
9288, 91eqtr4d 2771 . 2 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ((abss𝐴) ·s (abss𝐵)))
93 sletric 27710 . . . 4 (( 0s No 𝐴 No ) → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
9424, 93mpan 689 . . 3 (𝐴 No → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
9594adantr 480 . 2 ((𝐴 No 𝐵 No ) → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
9645, 92, 95mpjaodan 957 1 ((𝐴 No 𝐵 No ) → (abss‘(𝐴 ·s 𝐵)) = ((abss𝐴) ·s (abss𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846   = wceq 1534  wcel 2099   class class class wbr 5148  cfv 6548  (class class class)co 7420   No csur 27586   ≤s csle 27690   0s c0s 27768   -us cnegs 27945   ·s cmuls 28019  absscabss 28144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-ot 4638  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-1o 8487  df-2o 8488  df-nadd 8687  df-no 27589  df-slt 27590  df-bday 27591  df-sle 27691  df-sslt 27727  df-scut 27729  df-0s 27770  df-made 27787  df-old 27788  df-left 27790  df-right 27791  df-norec 27868  df-norec2 27879  df-adds 27890  df-negs 27947  df-subs 27948  df-muls 28020  df-abss 28145
This theorem is referenced by:  remulscllem2  28242
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