Theorem mulsge0d 28056

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 27745	. . . . 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 28045	. . . . 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 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 28055	. . . 4 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s <s (𝐴 ·s 𝐵))
12	2, 6, 11	sltled 27688	. . 3 ⊢ (((𝜑 ∧ 0s <s 𝐴) ∧ 0s <s 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
13		slerflex 27682	. . . . . 6 ⊢ ( 0s ∈ No → 0s ≤s 0s )
14	1, 13	ax-mp 5	. . . . 5 ⊢ 0s ≤s 0s
15		oveq2 7398	. . . . . . 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 28022	. . . . . . . 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 2767	. . . . 5 ⊢ ((𝜑 ∧ 0s = 𝐵) → (𝐴 ·s 𝐵) = 0s )
21	14, 20	breqtrrid 5148	. . . 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 27673	. . . . . 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 7397	. . . . 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 28051	. . . . . 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 2767	. . 3 ⊢ ((𝜑 ∧ 0s = 𝐴) → (𝐴 ·s 𝐵) = 0s )
35	14, 34	breqtrrid 5148	. 2 ⊢ ((𝜑 ∧ 0s = 𝐴) → 0s ≤s (𝐴 ·s 𝐵))
36		mulsge0d.3	. . 3 ⊢ (𝜑 → 0s ≤s 𝐴)
37		sleloe 27673	. . . 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 1540   ∈ wcel 2109   class class class wbr 5110  (class class class)co 7390   No csur 27558   <s cslt 27559   ≤s csle 27663   0_s c0s 27741   ·_s cmuls 28016

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-1o 8437  df-2o 8438  df-nadd 8633  df-no 27561  df-slt 27562  df-bday 27563  df-sle 27664  df-sslt 27700  df-scut 27702  df-0s 27743  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec 27852  df-norec2 27863  df-adds 27874  df-negs 27934  df-subs 27935  df-muls 28017

This theorem is referenced by:  absmuls  28153