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Theorem absmuls 28268
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 28160 . . . . . . 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 28172 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
8 abssid 28265 . . . . . 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 28265 . . . . . . 7 ((𝐵 No ∧ 0s ≤s 𝐵) → (abss𝐵) = 𝐵)
1110ad4ant24 754 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss𝐵) = 𝐵)
1211oveq2d 7447 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (𝐴 ·s (abss𝐵)) = (𝐴 ·s 𝐵))
139, 12eqtr4d 2780 . . . 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 28187 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s ( -us𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
17 abssnid 28267 . . . . . . 7 ((𝐵 No 𝐵 ≤s 0s ) → (abss𝐵) = ( -us𝐵))
1817ad4ant24 754 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss𝐵) = ( -us𝐵))
1918oveq2d 7447 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s (abss𝐵)) = (𝐴 ·s ( -us𝐵)))
20 negs0s 28058 . . . . . . . 8 ( -us ‘ 0s ) = 0s
2115negscld 28069 . . . . . . . . . 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 27871 . . . . . . . . . . . . . 14 0s No
2524a1i 11 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s No )
2615, 25slenegd 28080 . . . . . . . . . . . 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 5179 . . . . . . . . . 10 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us𝐵))
2914, 21, 22, 28mulsge0d 28172 . . . . . . . . 9 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s ( -us𝐵)))
3029, 16breqtrd 5169 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘(𝐴 ·s 𝐵)))
3120, 30eqbrtrid 5178 . . . . . . 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 28080 . . . . . . 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 28267 . . . . . 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 2788 . . . 4 ((((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss𝐵)))
38 sletric 27809 . . . . . 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 961 . . 3 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss𝐵)))
42 abssid 28265 . . . . 5 ((𝐴 No ∧ 0s ≤s 𝐴) → (abss𝐴) = 𝐴)
4342adantlr 715 . . . 4 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → (abss𝐴) = 𝐴)
4443oveq1d 7446 . . 3 (((𝐴 No 𝐵 No ) ∧ 0s ≤s 𝐴) → ((abss𝐴) ·s (abss𝐵)) = (𝐴 ·s (abss𝐵)))
4541, 44eqtr4d 2780 . 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 28186 . . . . 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 7447 . . . . 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 28069 . . . . . . . . . 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 28080 . . . . . . . . . . . 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 5179 . . . . . . . . . 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 28172 . . . . . . . . 9 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s (( -us𝐴) ·s 𝐵))
6059, 48breqtrd 5169 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s ( -us ‘(𝐴 ·s 𝐵)))
6120, 60eqbrtrid 5178 . . . . . . 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 28080 . . . . . . 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 2788 . . . 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 28188 . . . . 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 7447 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -us𝐴) ·s (abss𝐵)) = (( -us𝐴) ·s ( -us𝐵)))
7267negscld 28069 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us𝐴) ∈ No )
7368negscld 28069 . . . . . . . 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 28080 . . . . . . . . . 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 5179 . . . . . . . 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 28080 . . . . . . . . . 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 5179 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us𝐵))
8372, 73, 78, 82mulsge0d 28172 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (( -us𝐴) ·s ( -us𝐵)))
8483, 69breqtrd 5169 . . . . . 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 2788 . . . 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 961 . . 3 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (( -us𝐴) ·s (abss𝐵)))
89 abssnid 28267 . . . . 5 ((𝐴 No 𝐴 ≤s 0s ) → (abss𝐴) = ( -us𝐴))
9089oveq1d 7446 . . . 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 2780 . 2 (((𝐴 No 𝐵 No ) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ((abss𝐴) ·s (abss𝐵)))
93 sletric 27809 . . . 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 961 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 848   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  (class class class)co 7431   No csur 27684   ≤s csle 27789   0s c0s 27867   -us cnegs 28051   ·s cmuls 28132  absscabss 28261
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  df-subs 28054  df-muls 28133  df-abss 28262
This theorem is referenced by:  remulscllem2  28433
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