Theorem mulsge0d 28172

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

Hypotheses

Ref	Expression
mulsge0d.1	⊢ (𝜑 → 𝐴 ∈ No )
mulsge0d.2	⊢ (𝜑 → 𝐵 ∈ No )
mulsge0d.3	⊢ (𝜑 → 0s ≤s 𝐴)
mulsge0d.4	⊢ (𝜑 → 0s ≤s 𝐵)

Assertion

Ref	Expression
mulsge0d	⊢ (𝜑 → 0s ≤s (𝐴 ·s 𝐵))

Proof of Theorem mulsge0d

Step	Hyp	Ref	Expression
1		0sno 27871	. . . . 5 ⊢ 0s ∈ No
2	1	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s ∈ No )
3		mulsge0d.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
4		mulsge0d.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No )
5	3, 4	mulscld 28161	. . . . 5 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No )
6	5	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → (𝐴 ·s 𝐵) ∈ No )
7	3	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 𝐴 ∈ No )
8	4	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 𝐵 ∈ No )
9		simplr 769	. . . . 5 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s <s 𝐴)
10		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s <s 𝐵)
11	7, 8, 9, 10	mulsgt0d 28171	. . . 4 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s <s (𝐴 ·s 𝐵))
12	2, 6, 11	sltled 27814	. . 3 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
13		slerflex 27808	. . . . . 6 ⊢ ( 0s ∈ No → 0s ≤s 0s )
14	1, 13	ax-mp 5	. . . . 5 ⊢ 0s ≤s 0s
15		oveq2 7439	. . . . . . 7 ⊢ ( 0s = 𝐵 → (𝐴 ·s 0s ) = (𝐴 ·s 𝐵))
16	15	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 0s = 𝐵) → (𝐴 ·s 0s ) = (𝐴 ·s 𝐵))
17		muls01 28138	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
18	3, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s )
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 0s = 𝐵) → (𝐴 ·s 0s ) = 0s )
20	16, 19	eqtr3d 2779	. . . . 5 ⊢ ((𝜑 ∧ 0s = 𝐵) → (𝐴 ·s 𝐵) = 0s )
21	14, 20	breqtrrid 5181	. . . 4 ⊢ ((𝜑 ∧ 0s = 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
22	21	adantlr 715	. . 3 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s = 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
23		mulsge0d.4	. . . . 5 ⊢ (𝜑 → 0s ≤s 𝐵)
24		sleloe 27799	. . . . . 6 ⊢ (( 0s ∈ No ∧ 𝐵 ∈ No ) → ( 0s ≤s 𝐵 ↔ ( 0s <s 𝐵 ∨ 0s = 𝐵)))
25	1, 4, 24	sylancr 587	. . . . 5 ⊢ (𝜑 → ( 0s ≤s 𝐵 ↔ ( 0s <s 𝐵 ∨ 0s = 𝐵)))
26	23, 25	mpbid 232	. . . 4 ⊢ (𝜑 → ( 0s <s 𝐵 ∨ 0s = 𝐵))
27	26	adantr 480	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐴) → ( 0s <s 𝐵 ∨ 0s = 𝐵))
28	12, 22, 27	mpjaodan 961	. 2 ⊢ ((𝜑 ∧ 0s <s 𝐴) → 0s ≤s (𝐴 ·s 𝐵))
29		oveq1 7438	. . . . 5 ⊢ ( 0s = 𝐴 → ( 0s ·s 𝐵) = (𝐴 ·s 𝐵))
30	29	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 0s = 𝐴) → ( 0s ·s 𝐵) = (𝐴 ·s 𝐵))
31		muls02 28167	. . . . . 6 ⊢ (𝐵 ∈ No → ( 0s ·s 𝐵) = 0s )
32	4, 31	syl 17	. . . . 5 ⊢ (𝜑 → ( 0s ·s 𝐵) = 0s )
33	32	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0s = 𝐴) → ( 0s ·s 𝐵) = 0s )
34	30, 33	eqtr3d 2779	. . 3 ⊢ ((𝜑 ∧ 0s = 𝐴) → (𝐴 ·s 𝐵) = 0s )
35	14, 34	breqtrrid 5181	. 2 ⊢ ((𝜑 ∧ 0s = 𝐴) → 0s ≤s (𝐴 ·s 𝐵))
36		mulsge0d.3	. . 3 ⊢ (𝜑 → 0s ≤s 𝐴)
37		sleloe 27799	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ↔ ( 0s <s 𝐴 ∨ 0s = 𝐴)))
38	1, 3, 37	sylancr 587	. . 3 ⊢ (𝜑 → ( 0s ≤s 𝐴 ↔ ( 0s <s 𝐴 ∨ 0s = 𝐴)))
39	36, 38	mpbid 232	. 2 ⊢ (𝜑 → ( 0s <s 𝐴 ∨ 0s = 𝐴))
40	28, 35, 39	mpjaodan 961	1 ⊢ (𝜑 → 0s ≤s (𝐴 ·s 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108   class class class wbr 5143  (class class class)co 7431   No csur 27684   <s cslt 27685   ≤s csle 27789   0_s c0s 27867   ·_s cmuls 28132

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

This theorem is referenced by:  absmuls  28268