Theorem mulsge0d 28370

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		0no 28033	. . . . 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 28359	. . . . 5 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No )
6	5	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → (𝐴 ·s 𝐵) ∈ No )
7	3	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 𝐴 ∈ No )
8	4	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 𝐵 ∈ No )
9		simplr 781	. . . . 5 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s <s 𝐴)
10		simpr 490	. . . . 5 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s <s 𝐵)
11	7, 8, 9, 10	mulsgt0d 28369	. . . 4 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s <s (𝐴 ·s 𝐵))
12	2, 6, 11	ltlesd 27968	. . 3 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
13		lesid 27962	. . . . . 6 ⊢ ( 0s ∈ No → 0s ≤s 0s )
14	1, 13	ax-mp 5	. . . . 5 ⊢ 0s ≤s 0s
15		oveq2 7424	. . . . . . 7 ⊢ ( 0s = 𝐵 → (𝐴 ·s 0s ) = (𝐴 ·s 𝐵))
16	15	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 0s = 𝐵) → (𝐴 ·s 0s ) = (𝐴 ·s 𝐵))
17		muls01 28336	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
18	3, 17	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s )
19	18	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 0s = 𝐵) → (𝐴 ·s 0s ) = 0s )
20	16, 19	eqtr3d 2802	. . . . 5 ⊢ ((𝜑 ∧ 0s = 𝐵) → (𝐴 ·s 𝐵) = 0s )
21	14, 20	breqtrrid 5151	. . . 4 ⊢ ((𝜑 ∧ 0s = 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
22	21	adantlr 728	. . 3 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s = 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
23		mulsge0d.4	. . . . 5 ⊢ (𝜑 → 0s ≤s 𝐵)
24		lesloe 27949	. . . . . 6 ⊢ (( 0s ∈ No ∧ 𝐵 ∈ No ) → ( 0s ≤s 𝐵 ↔ ( 0s <s 𝐵 ∨ 0s = 𝐵)))
25	1, 4, 24	sylancr 599	. . . . 5 ⊢ (𝜑 → ( 0s ≤s 𝐵 ↔ ( 0s <s 𝐵 ∨ 0s = 𝐵)))
26	23, 25	mpbid 235	. . . 4 ⊢ (𝜑 → ( 0s <s 𝐵 ∨ 0s = 𝐵))
27	26	adantr 486	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐴) → ( 0s <s 𝐵 ∨ 0s = 𝐵))
28	12, 22, 27	mpjaodan 973	. 2 ⊢ ((𝜑 ∧ 0s <s 𝐴) → 0s ≤s (𝐴 ·s 𝐵))
29		oveq1 7423	. . . . 5 ⊢ ( 0s = 𝐴 → ( 0s ·s 𝐵) = (𝐴 ·s 𝐵))
30	29	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 0s = 𝐴) → ( 0s ·s 𝐵) = (𝐴 ·s 𝐵))
31		muls02 28365	. . . . . 6 ⊢ (𝐵 ∈ No → ( 0s ·s 𝐵) = 0s )
32	4, 31	syl 18	. . . . 5 ⊢ (𝜑 → ( 0s ·s 𝐵) = 0s )
33	32	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 0s = 𝐴) → ( 0s ·s 𝐵) = 0s )
34	30, 33	eqtr3d 2802	. . 3 ⊢ ((𝜑 ∧ 0s = 𝐴) → (𝐴 ·s 𝐵) = 0s )
35	14, 34	breqtrrid 5151	. 2 ⊢ ((𝜑 ∧ 0s = 𝐴) → 0s ≤s (𝐴 ·s 𝐵))
36		mulsge0d.3	. . 3 ⊢ (𝜑 → 0s ≤s 𝐴)
37		lesloe 27949	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ↔ ( 0s <s 𝐴 ∨ 0s = 𝐴)))
38	1, 3, 37	sylancr 599	. . 3 ⊢ (𝜑 → ( 0s ≤s 𝐴 ↔ ( 0s <s 𝐴 ∨ 0s = 𝐴)))
39	36, 38	mpbid 235	. 2 ⊢ (𝜑 → ( 0s <s 𝐴 ∨ 0s = 𝐴))
40	28, 35, 39	mpjaodan 973	1 ⊢ (𝜑 → 0s ≤s (𝐴 ·s 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   class class class wbr 5111  (class class class)co 7416   No csur 27835   <s clts 27836   ≤s cles 27939   0_s c0s 28029   ·_s cmuls 28330

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-1o 8455  df-2o 8456  df-nadd 8654  df-no 27838  df-lts 27839  df-bday 27840  df-les 27940  df-slts 27982  df-cuts 27984  df-0s 28031  df-made 28051  df-old 28052  df-left 28054  df-right 28055  df-norec 28162  df-norec2 28173  df-adds 28184  df-negs 28245  df-subs 28246  df-muls 28331

This theorem is used by:  absmuls  28468  zsoring  28633