Theorem slemul1ad 28226

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 28219	. . 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 27889	. . . . . 6 ⊢ 0s ∈ No
13		slerflex 27826	. . . . . 6 ⊢ ( 0s ∈ No → 0s ≤s 0s )
14	12, 13	mp1i 13	. . . . 5 ⊢ (𝜑 → 0s ≤s 0s )
15		muls01 28156	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
16	3, 15	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s )
17		muls01 28156	. . . . . 6 ⊢ (𝐵 ∈ No → (𝐵 ·s 0s ) = 0s )
18	5, 17	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 ·s 0s ) = 0s )
19	14, 16, 18	3brtr4d 5198	. . . 4 ⊢ (𝜑 → (𝐴 ·s 0s ) ≤s (𝐵 ·s 0s ))
20		oveq2 7456	. . . . 5 ⊢ ( 0s = 𝐶 → (𝐴 ·s 0s ) = (𝐴 ·s 𝐶))
21		oveq2 7456	. . . . 5 ⊢ ( 0s = 𝐶 → (𝐵 ·s 0s ) = (𝐵 ·s 𝐶))
22	20, 21	breq12d 5179	. . . 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 27817	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐶 ∈ No ) → ( 0s ≤s 𝐶 ↔ ( 0s <s 𝐶 ∨ 0s = 𝐶)))
27	12, 7, 26	sylancr 586	. . 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 959	1 ⊢ (𝜑 → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108   class class class wbr 5166  (class class class)co 7448   No csur 27702   <s cslt 27703   ≤s csle 27807   0_s c0s 27885   ·_s cmuls 28150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-muls 28151

This theorem is referenced by:  sltmul12ad  28227