Theorem slemul1ad 28108

Description: Multiplication of both sides of surreal less-than or equal by a non-negative number. (Contributed by Scott Fenton, 17-Apr-2025.)

Hypotheses

Ref	Expression
slemul1ad.1	⊢ (𝜑 → 𝐴 ∈ No )
slemul1ad.2	⊢ (𝜑 → 𝐵 ∈ No )
slemul1ad.3	⊢ (𝜑 → 𝐶 ∈ No )
slemul1ad.4	⊢ (𝜑 → 0s ≤s 𝐶)
slemul1ad.5	⊢ (𝜑 → 𝐴 ≤s 𝐵)

Assertion

Ref	Expression
slemul1ad	⊢ (𝜑 → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))

Proof of Theorem slemul1ad

Step	Hyp	Ref	Expression
1		slemul1ad.5	. . . 4 ⊢ (𝜑 → 𝐴 ≤s 𝐵)
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐴 ≤s 𝐵)
3		slemul1ad.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ No )
4	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐴 ∈ No )
5		slemul1ad.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ No )
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐵 ∈ No )
7		slemul1ad.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ No )
8	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐶 ∈ No )
9		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 0s <s 𝐶)
10	4, 6, 8, 9	slemul1d 28101	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → (𝐴 ≤s 𝐵 ↔ (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)))
11	2, 10	mpbid 232	. 2 ⊢ ((𝜑 ∧ 0s <s 𝐶) → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))
12		0sno 27758	. . . . . 6 ⊢ 0s ∈ No
13		slerflex 27691	. . . . . 6 ⊢ ( 0s ∈ No → 0s ≤s 0s )
14	12, 13	mp1i 13	. . . . 5 ⊢ (𝜑 → 0s ≤s 0s )
15		muls01 28038	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
16	3, 15	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s )
17		muls01 28038	. . . . . 6 ⊢ (𝐵 ∈ No → (𝐵 ·s 0s ) = 0s )
18	5, 17	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 ·s 0s ) = 0s )
19	14, 16, 18	3brtr4d 5127	. . . 4 ⊢ (𝜑 → (𝐴 ·s 0s ) ≤s (𝐵 ·s 0s ))
20		oveq2 7361	. . . . 5 ⊢ ( 0s = 𝐶 → (𝐴 ·s 0s ) = (𝐴 ·s 𝐶))
21		oveq2 7361	. . . . 5 ⊢ ( 0s = 𝐶 → (𝐵 ·s 0s ) = (𝐵 ·s 𝐶))
22	20, 21	breq12d 5108	. . . 4 ⊢ ( 0s = 𝐶 → ((𝐴 ·s 0s ) ≤s (𝐵 ·s 0s ) ↔ (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)))
23	19, 22	syl5ibcom 245	. . 3 ⊢ (𝜑 → ( 0s = 𝐶 → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)))
24	23	imp 406	. 2 ⊢ ((𝜑 ∧ 0s = 𝐶) → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))
25		slemul1ad.4	. . 3 ⊢ (𝜑 → 0s ≤s 𝐶)
26		sleloe 27682	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐶 ∈ No ) → ( 0s ≤s 𝐶 ↔ ( 0s <s 𝐶 ∨ 0s = 𝐶)))
27	12, 7, 26	sylancr 587	. . 3 ⊢ (𝜑 → ( 0s ≤s 𝐶 ↔ ( 0s <s 𝐶 ∨ 0s = 𝐶)))
28	25, 27	mpbid 232	. 2 ⊢ (𝜑 → ( 0s <s 𝐶 ∨ 0s = 𝐶))
29	11, 24, 28	mpjaodan 960	1 ⊢ (𝜑 → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   class class class wbr 5095  (class class class)co 7353   No csur 27567   <s cslt 27568   ≤s csle 27672   0_s c0s 27754   ·_s cmuls 28032

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 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-2o 8396  df-nadd 8591  df-no 27570  df-slt 27571  df-bday 27572  df-sle 27673  df-sslt 27710  df-scut 27712  df-0s 27756  df-made 27775  df-old 27776  df-left 27778  df-right 27779  df-norec 27868  df-norec2 27879  df-adds 27890  df-negs 27950  df-subs 27951  df-muls 28033

This theorem is referenced by:  sltmul12ad  28109