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Theorem absmuls 28282
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 28174 . . . . . . 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 775 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 𝐴 No )
4 simpllr 776 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 𝐵 No )
5 simplr 769 . . . . . . 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 28186 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
8 abssid 28279 . . . . . 6 (((𝐴 ·s 𝐵) ∈ No ∧ 0s ≤s (𝐴 ·s 𝐵)) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
92, 7, 8syl2an2r 685 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
10 abssid 28279 . . . . . . 7 ((𝐵 No ∧ 0s ≤s 𝐵) → (abss𝐵) = 𝐵)
1110ad4ant24 754 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss𝐵) = 𝐵)
1211oveq2d 7446 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (𝐴 ·s (abss𝐵)) = (𝐴 ·s 𝐵))
139, 12eqtr4d 2777 . . . 4 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss𝐵)))
14 simplll 775 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐴 No )
15 simpllr 776 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐵 No )
1614, 15mulnegs2d 28201 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s ( -us𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
17 abssnid 28281 . . . . . . 7 ((𝐵 No 𝐵 ≤s 0s ) → (abss𝐵) = ( -us𝐵))
1817ad4ant24 754 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss𝐵) = ( -us𝐵))
1918oveq2d 7446 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s (abss𝐵)) = (𝐴 ·s ( -us𝐵)))
20 negs0s 28072 . . . . . . . 8 ( -us ‘ 0s ) = 0s
2115negscld 28083 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -us𝐵) ∈ No )
22 simplr 769 . . . . . . . . . 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 27885 . . . . . . . . . . . . . 14 0s No
2524a1i 11 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s No )
2615, 25slenegd 28094 . . . . . . . . . . . 12 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐵 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us𝐵)))
2723, 26mpbid 232 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us𝐵))
2820, 27eqbrtrrid 5183 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us𝐵))
2914, 21, 22, 28mulsge0d 28186 . . . . . . . . 9 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s ( -us𝐵)))
3029, 16breqtrd 5173 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘(𝐴 ·s 𝐵)))
3120, 30eqbrtrid 5182 . . . . . . 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 28094 . . . . . . 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 28281 . . . . . 6 (((𝐴 ·s 𝐵) ∈ No ∧ (𝐴 ·s 𝐵) ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
362, 34, 35syl2an2r 685 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
3716, 19, 363eqtr4rd 2785 . . . 4 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss𝐵)))
38 sletric 27823 . . . . . 6 (( 0s No 𝐵 No ) → ( 0s ≤s 𝐵𝐵 ≤s 0s ))
3924, 38mpan 690 . . . . 5 (𝐵 No → ( 0s ≤s 𝐵𝐵 ≤s 0s ))
4039ad2antlr 727 . . . 4 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → ( 0s ≤s 𝐵𝐵 ≤s 0s ))
4113, 37, 40mpjaodan 960 . . 3 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss𝐵)))
42 abssid 28279 . . . . 5 ((𝐴 No ∧ 0s ≤s 𝐴) → (abss𝐴) = 𝐴)
4342adantlr 715 . . . 4 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → (abss𝐴) = 𝐴)
4443oveq1d 7445 . . 3 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → ((abss𝐴) ·s (abss𝐵)) = (𝐴 ·s (abss𝐵)))
4541, 44eqtr4d 2777 . 2 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → (abss‘(𝐴 ·s 𝐵)) = ((abss𝐴) ·s (abss𝐵)))
46 simplll 775 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 𝐴 No )
47 simpllr 776 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 𝐵 No )
4846, 47mulnegs1d 28200 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (( -us𝐴) ·s 𝐵) = ( -us ‘(𝐴 ·s 𝐵)))
4910ad4ant24 754 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss𝐵) = 𝐵)
5049oveq2d 7446 . . . . 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 28083 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -us𝐴) ∈ No )
53 simplr 769 . . . . . . . . . . . 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 28094 . . . . . . . . . . . 12 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us𝐴)))
5653, 55mpbid 232 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -us ‘ 0s ) ≤s ( -us𝐴))
5720, 56eqbrtrrid 5183 . . . . . . . . . 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 28186 . . . . . . . . 9 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s (( -us𝐴) ·s 𝐵))
6059, 48breqtrd 5173 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s ( -us ‘(𝐴 ·s 𝐵)))
6120, 60eqbrtrid 5182 . . . . . . 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 28094 . . . . . . 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 685 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
6648, 50, 653eqtr4rd 2785 . . . 4 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (( -us𝐴) ·s (abss𝐵)))
67 simplll 775 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐴 No )
68 simpllr 776 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐵 No )
6967, 68mul2negsd 28202 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -us𝐴) ·s ( -us𝐵)) = (𝐴 ·s 𝐵))
7017ad4ant24 754 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss𝐵) = ( -us𝐵))
7170oveq2d 7446 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -us𝐴) ·s (abss𝐵)) = (( -us𝐴) ·s ( -us𝐵)))
7267negscld 28083 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us𝐴) ∈ No )
7368negscld 28083 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us𝐵) ∈ No )
74 simplr 769 . . . . . . . . . 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 28094 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us𝐴)))
7774, 76mpbid 232 . . . . . . . . 9 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us𝐴))
7820, 77eqbrtrrid 5183 . . . . . . . 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 28094 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (𝐵 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us𝐵)))
8179, 80mpbid 232 . . . . . . . . 9 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us𝐵))
8220, 81eqbrtrrid 5183 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us𝐵))
8372, 73, 78, 82mulsge0d 28186 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (( -us𝐴) ·s ( -us𝐵)))
8483, 69breqtrd 5173 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s 𝐵))
8551, 84, 8syl2an2r 685 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
8669, 71, 853eqtr4rd 2785 . . . 4 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (( -us𝐴) ·s (abss𝐵)))
8739ad2antlr 727 . . . 4 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) → ( 0s ≤s 𝐵𝐵 ≤s 0s ))
8866, 86, 87mpjaodan 960 . . 3 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (( -us𝐴) ·s (abss𝐵)))
89 abssnid 28281 . . . . 5 ((𝐴 No 𝐴 ≤s 0s ) → (abss𝐴) = ( -us𝐴))
9089oveq1d 7445 . . . 4 ((𝐴 No 𝐴 ≤s 0s ) → ((abss𝐴) ·s (abss𝐵)) = (( -us𝐴) ·s (abss𝐵)))
9190adantlr 715 . . 3 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) → ((abss𝐴) ·s (abss𝐵)) = (( -us𝐴) ·s (abss𝐵)))
9288, 91eqtr4d 2777 . 2 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ((abss𝐴) ·s (abss𝐵)))
93 sletric 27823 . . . 4 (( 0s No 𝐴 No ) → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
9424, 93mpan 690 . . 3 (𝐴 No → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
9594adantr 480 . 2 ((𝐴 No 𝐵 No ) → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
9645, 92, 95mpjaodan 960 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 847   = wceq 1536  wcel 2105   class class class wbr 5147  cfv 6562  (class class class)co 7430   No csur 27698   ≤s csle 27803   0s c0s 27881   -us cnegs 28065   ·s cmuls 28146  absscabss 28275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-ot 4639  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-1o 8504  df-2o 8505  df-nadd 8702  df-no 27701  df-slt 27702  df-bday 27703  df-sle 27804  df-sslt 27840  df-scut 27842  df-0s 27883  df-made 27900  df-old 27901  df-left 27903  df-right 27904  df-norec 27985  df-norec2 27996  df-adds 28007  df-negs 28067  df-subs 28068  df-muls 28147  df-abss 28276
This theorem is referenced by:  remulscllem2  28447
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