Theorem slemul1ad 28119

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 28112	. . 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 27768	. . . . . 6 ⊢ 0s ∈ No
13		slerflex 27700	. . . . . 6 ⊢ ( 0s ∈ No → 0s ≤s 0s )
14	12, 13	mp1i 13	. . . . 5 ⊢ (𝜑 → 0s ≤s 0s )
15		muls01 28049	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
16	3, 15	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s )
17		muls01 28049	. . . . . 6 ⊢ (𝐵 ∈ No → (𝐵 ·s 0s ) = 0s )
18	5, 17	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 ·s 0s ) = 0s )
19	14, 16, 18	3brtr4d 5123	. . . 4 ⊢ (𝜑 → (𝐴 ·s 0s ) ≤s (𝐵 ·s 0s ))
20		oveq2 7354	. . . . 5 ⊢ ( 0s = 𝐶 → (𝐴 ·s 0s ) = (𝐴 ·s 𝐶))
21		oveq2 7354	. . . . 5 ⊢ ( 0s = 𝐶 → (𝐵 ·s 0s ) = (𝐵 ·s 𝐶))
22	20, 21	breq12d 5104	. . . 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 27691	. . . 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 1541   ∈ wcel 2111   class class class wbr 5091  (class class class)co 7346   No csur 27576   <s cslt 27577   ≤s csle 27681   0_s c0s 27764   ·_s cmuls 28043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-ot 4585  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-2o 8386  df-nadd 8581  df-no 27579  df-slt 27580  df-bday 27581  df-sle 27682  df-sslt 27719  df-scut 27721  df-0s 27766  df-made 27786  df-old 27787  df-left 27789  df-right 27790  df-norec 27879  df-norec2 27890  df-adds 27901  df-negs 27961  df-subs 27962  df-muls 28044

This theorem is referenced by:  sltmul12ad  28120