Metamath Proof Explorer


Theorem mulsge0d

Description: The product of two non-negative surreals is non-negative. (Contributed by Scott Fenton, 6-Mar-2025)

Ref Expression
Hypotheses mulsge0d.1 φ A No
mulsge0d.2 φ B No
mulsge0d.3 φ 0 s s A
mulsge0d.4 φ 0 s s B
Assertion mulsge0d φ 0 s s A s B

Proof

Step Hyp Ref Expression
1 mulsge0d.1 φ A No
2 mulsge0d.2 φ B No
3 mulsge0d.3 φ 0 s s A
4 mulsge0d.4 φ 0 s s B
5 0sno 0 s No
6 5 a1i φ 0 s < s A 0 s < s B 0 s No
7 1 2 mulscld φ A s B No
8 7 ad2antrr φ 0 s < s A 0 s < s B A s B No
9 1 ad2antrr φ 0 s < s A 0 s < s B A No
10 2 ad2antrr φ 0 s < s A 0 s < s B B No
11 simplr φ 0 s < s A 0 s < s B 0 s < s A
12 simpr φ 0 s < s A 0 s < s B 0 s < s B
13 9 10 11 12 mulsgt0d φ 0 s < s A 0 s < s B 0 s < s A s B
14 6 8 13 sltled φ 0 s < s A 0 s < s B 0 s s A s B
15 slerflex 0 s No 0 s s 0 s
16 5 15 ax-mp 0 s s 0 s
17 oveq2 0 s = B A s 0 s = A s B
18 17 adantl φ 0 s = B A s 0 s = A s B
19 muls01 A No A s 0 s = 0 s
20 1 19 syl φ A s 0 s = 0 s
21 20 adantr φ 0 s = B A s 0 s = 0 s
22 18 21 eqtr3d φ 0 s = B A s B = 0 s
23 16 22 breqtrrid φ 0 s = B 0 s s A s B
24 23 adantlr φ 0 s < s A 0 s = B 0 s s A s B
25 sleloe 0 s No B No 0 s s B 0 s < s B 0 s = B
26 5 2 25 sylancr φ 0 s s B 0 s < s B 0 s = B
27 4 26 mpbid φ 0 s < s B 0 s = B
28 27 adantr φ 0 s < s A 0 s < s B 0 s = B
29 14 24 28 mpjaodan φ 0 s < s A 0 s s A s B
30 oveq1 0 s = A 0 s s B = A s B
31 30 adantl φ 0 s = A 0 s s B = A s B
32 muls02 B No 0 s s B = 0 s
33 2 32 syl φ 0 s s B = 0 s
34 33 adantr φ 0 s = A 0 s s B = 0 s
35 31 34 eqtr3d φ 0 s = A A s B = 0 s
36 16 35 breqtrrid φ 0 s = A 0 s s A s B
37 sleloe 0 s No A No 0 s s A 0 s < s A 0 s = A
38 5 1 37 sylancr φ 0 s s A 0 s < s A 0 s = A
39 3 38 mpbid φ 0 s < s A 0 s = A
40 29 36 39 mpjaodan φ 0 s s A s B