Theorem lemuls1ad 28573

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
lemuls1ad.1	⊢ (𝜑 → 𝐴 ∈ No)
lemuls1ad.2	⊢ (𝜑 → 𝐵 ∈ No)
lemuls1ad.3	⊢ (𝜑 → 𝐶 ∈ No)
lemuls1ad.4	⊢ (𝜑 → 0s ≤s 𝐶)
lemuls1ad.5	⊢ (𝜑 → 𝐴 ≤s 𝐵)

Assertion

Ref	Expression
lemuls1ad	⊢ (𝜑 → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))

Proof of Theorem lemuls1ad

Step	Hyp	Ref	Expression
1		lemuls1ad.5	. . . 4 ⊢ (𝜑 → 𝐴 ≤s 𝐵)
2	1	adantr 486	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐴 ≤s 𝐵)
3		lemuls1ad.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ No)
4	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐴 ∈ No)
5		lemuls1ad.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ No)
6	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐵 ∈ No)
7		lemuls1ad.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ No)
8	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐶 ∈ No)
9		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 0s <s 𝐶)
10	4, 6, 8, 9	lemuls1d 28566	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → (𝐴 ≤s 𝐵 ↔ (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)))
11	2, 10	mpbid 235	. 2 ⊢ ((𝜑 ∧ 0s <s 𝐶) → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))
12		0no 28200	. . . . . 6 ⊢ 0s ∈ No
13		lesid 28129	. . . . . 6 ⊢ ( 0s ∈ No → 0s ≤s 0s )
14	12, 13	mp1i 14	. . . . 5 ⊢ (𝜑 → 0s ≤s 0s )
15		muls01 28503	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
16	3, 15	syl 18	. . . . 5 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s )
17		muls01 28503	. . . . . 6 ⊢ (𝐵 ∈ No → (𝐵 ·s 0s ) = 0s )
18	5, 17	syl 18	. . . . 5 ⊢ (𝜑 → (𝐵 ·s 0s ) = 0s )
19	14, 16, 18	3brtr4d 5136	. . . 4 ⊢ (𝜑 → (𝐴 ·s 0s ) ≤s (𝐵 ·s 0s ))
20		oveq2 7428	. . . . 5 ⊢ ( 0s = 𝐶 → (𝐴 ·s 0s ) = (𝐴 ·s 𝐶))
21		oveq2 7428	. . . . 5 ⊢ ( 0s = 𝐶 → (𝐵 ·s 0s ) = (𝐵 ·s 𝐶))
22	20, 21	breq12d 5115	. . . 4 ⊢ ( 0s = 𝐶 → ((𝐴 ·s 0s ) ≤s (𝐵 ·s 0s ) ↔ (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)))
23	19, 22	syl5ibcom 248	. . 3 ⊢ (𝜑 → ( 0s = 𝐶 → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)))
24	23	imp 412	. 2 ⊢ ((𝜑 ∧ 0s = 𝐶) → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))
25		lemuls1ad.4	. . 3 ⊢ (𝜑 → 0s ≤s 𝐶)
26		lesloe 28116	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐶 ∈ No) → ( 0s ≤s 𝐶 ↔ ( 0s <s 𝐶 ∨ 0s = 𝐶)))
27	12, 7, 26	sylancr 599	. . 3 ⊢ (𝜑 → ( 0s ≤s 𝐶 ↔ ( 0s <s 𝐶 ∨ 0s = 𝐶)))
28	25, 27	mpbid 235	. 2 ⊢ (𝜑 → ( 0s <s 𝐶 ∨ 0s = 𝐶))
29	11, 24, 28	mpjaodan 973	1 ⊢ (𝜑 → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   class class class wbr 5102  (class class class)co 7420  Nocsur 28002   <s clts 28003   ≤s cles 28106   0_s c0s 28196   ·_s cmuls 28497

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-nadd 8678  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-0s 28198  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec 28329  df-norec2 28340  df-adds 28351  df-negs 28412  df-subs 28413  df-muls 28498

This theorem is used by:  ltmuls12ad  28574