Theorem mulsge0d 28186

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 27885	. . . . 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 28175	. . . . 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 28185	. . . 4 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s <s (𝐴 ·s 𝐵))
12	2, 6, 11	sltled 27828	. . 3 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
13		slerflex 27822	. . . . . 6 ⊢ ( 0s ∈ No → 0s ≤s 0s )
14	1, 13	ax-mp 5	. . . . 5 ⊢ 0s ≤s 0s
15		oveq2 7438	. . . . . . 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 28152	. . . . . . . 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 2776	. . . . 5 ⊢ ((𝜑 ∧ 0s = 𝐵) → (𝐴 ·s 𝐵) = 0s )
21	14, 20	breqtrrid 5185	. . . 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 27813	. . . . . 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 960	. 2 ⊢ ((𝜑 ∧ 0s <s 𝐴) → 0s ≤s (𝐴 ·s 𝐵))
29		oveq1 7437	. . . . 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 28181	. . . . . 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 2776	. . 3 ⊢ ((𝜑 ∧ 0s = 𝐴) → (𝐴 ·s 𝐵) = 0s )
35	14, 34	breqtrrid 5185	. 2 ⊢ ((𝜑 ∧ 0s = 𝐴) → 0s ≤s (𝐴 ·s 𝐵))
36		mulsge0d.3	. . 3 ⊢ (𝜑 → 0s ≤s 𝐴)
37		sleloe 27813	. . . 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 960	1 ⊢ (𝜑 → 0s ≤s (𝐴 ·s 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1536   ∈ wcel 2105   class class class wbr 5147  (class class class)co 7430   No csur 27698   <s cslt 27699   ≤s csle 27803   0_s c0s 27881   ·_s cmuls 28146

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

This theorem is referenced by:  absmuls  28282