Theorem lemuls1ad 28199

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 481	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐴 ≤s 𝐵)
3		lemuls1ad.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ No )
4	3	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐴 ∈ No )
5		lemuls1ad.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ No )
6	5	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐵 ∈ No )
7		lemuls1ad.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ No )
8	7	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐶 ∈ No )
9		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 0s <s 𝐶)
10	4, 6, 8, 9	lemuls1d 28192	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → (𝐴 ≤s 𝐵 ↔ (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)))
11	2, 10	mpbid 233	. 2 ⊢ ((𝜑 ∧ 0s <s 𝐶) → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))
12		0no 27826	. . . . . 6 ⊢ 0s ∈ No
13		lesid 27756	. . . . . 6 ⊢ ( 0s ∈ No → 0s ≤s 0s )
14	12, 13	mp1i 13	. . . . 5 ⊢ (𝜑 → 0s ≤s 0s )
15		muls01 28129	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
16	3, 15	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s )
17		muls01 28129	. . . . . 6 ⊢ (𝐵 ∈ No → (𝐵 ·s 0s ) = 0s )
18	5, 17	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 ·s 0s ) = 0s )
19	14, 16, 18	3brtr4d 5111	. . . 4 ⊢ (𝜑 → (𝐴 ·s 0s ) ≤s (𝐵 ·s 0s ))
20		oveq2 7371	. . . . 5 ⊢ ( 0s = 𝐶 → (𝐴 ·s 0s ) = (𝐴 ·s 𝐶))
21		oveq2 7371	. . . . 5 ⊢ ( 0s = 𝐶 → (𝐵 ·s 0s ) = (𝐵 ·s 𝐶))
22	20, 21	breq12d 5092	. . . 4 ⊢ ( 0s = 𝐶 → ((𝐴 ·s 0s ) ≤s (𝐵 ·s 0s ) ↔ (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)))
23	19, 22	syl5ibcom 246	. . 3 ⊢ (𝜑 → ( 0s = 𝐶 → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)))
24	23	imp 407	. 2 ⊢ ((𝜑 ∧ 0s = 𝐶) → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))
25		lemuls1ad.4	. . 3 ⊢ (𝜑 → 0s ≤s 𝐶)
26		lesloe 27743	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐶 ∈ No ) → ( 0s ≤s 𝐶 ↔ ( 0s <s 𝐶 ∨ 0s = 𝐶)))
27	12, 7, 26	sylancr 593	. . 3 ⊢ (𝜑 → ( 0s ≤s 𝐶 ↔ ( 0s <s 𝐶 ∨ 0s = 𝐶)))
28	25, 27	mpbid 233	. 2 ⊢ (𝜑 → ( 0s <s 𝐶 ∨ 0s = 𝐶))
29	11, 24, 28	mpjaodan 966	1 ⊢ (𝜑 → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   class class class wbr 5079  (class class class)co 7363   No csur 27628   <s clts 27629   ≤s cles 27733   0_s c0s 27822   ·_s cmuls 28123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-1o 8402  df-2o 8403  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec 27955  df-norec2 27966  df-adds 27977  df-negs 28038  df-subs 28039  df-muls 28124

This theorem is referenced by:  ltmuls12ad  28200