Theorem mulsge0d 28190

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 27889	. . . . 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 28179	. . . . 5 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No )
6	5	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → (𝐴 ·s 𝐵) ∈ No )
7	3	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 𝐴 ∈ No )
8	4	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 𝐵 ∈ No )
9		simplr 768	. . . . 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 28189	. . . 4 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s <s (𝐴 ·s 𝐵))
12	2, 6, 11	sltled 27832	. . 3 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
13		slerflex 27826	. . . . . 6 ⊢ ( 0s ∈ No → 0s ≤s 0s )
14	1, 13	ax-mp 5	. . . . 5 ⊢ 0s ≤s 0s
15		oveq2 7456	. . . . . . 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 28156	. . . . . . . 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 2782	. . . . 5 ⊢ ((𝜑 ∧ 0s = 𝐵) → (𝐴 ·s 𝐵) = 0s )
21	14, 20	breqtrrid 5204	. . . 4 ⊢ ((𝜑 ∧ 0s = 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
22	21	adantlr 714	. . 3 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s = 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
23		mulsge0d.4	. . . . 5 ⊢ (𝜑 → 0s ≤s 𝐵)
24		sleloe 27817	. . . . . 6 ⊢ (( 0s ∈ No ∧ 𝐵 ∈ No ) → ( 0s ≤s 𝐵 ↔ ( 0s <s 𝐵 ∨ 0s = 𝐵)))
25	1, 4, 24	sylancr 586	. . . . 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 959	. 2 ⊢ ((𝜑 ∧ 0s <s 𝐴) → 0s ≤s (𝐴 ·s 𝐵))
29		oveq1 7455	. . . . 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 28185	. . . . . 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 2782	. . 3 ⊢ ((𝜑 ∧ 0s = 𝐴) → (𝐴 ·s 𝐵) = 0s )
35	14, 34	breqtrrid 5204	. 2 ⊢ ((𝜑 ∧ 0s = 𝐴) → 0s ≤s (𝐴 ·s 𝐵))
36		mulsge0d.3	. . 3 ⊢ (𝜑 → 0s ≤s 𝐴)
37		sleloe 27817	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ↔ ( 0s <s 𝐴 ∨ 0s = 𝐴)))
38	1, 3, 37	sylancr 586	. . 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 959	1 ⊢ (𝜑 → 0s ≤s (𝐴 ·s 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108   class class class wbr 5166  (class class class)co 7448   No csur 27702   <s cslt 27703   ≤s csle 27807   0_s c0s 27885   ·_s cmuls 28150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-muls 28151

This theorem is referenced by:  absmuls  28286