Theorem lemulsd 28130

Description: An ordering relationship for surreal multiplication. Compare theorem 8(iii) of [Conway] p. 19. (Contributed by Scott Fenton, 7-Mar-2025.)

Hypotheses

Ref	Expression
lemulsd.1	⊢ (𝜑 → 𝐴 ∈ No )
lemulsd.2	⊢ (𝜑 → 𝐵 ∈ No )
lemulsd.3	⊢ (𝜑 → 𝐶 ∈ No )
lemulsd.4	⊢ (𝜑 → 𝐷 ∈ No )
lemulsd.5	⊢ (𝜑 → 𝐴 ≤s 𝐵)
lemulsd.6	⊢ (𝜑 → 𝐶 ≤s 𝐷)

Assertion

Ref	Expression
lemulsd	⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))

Proof of Theorem lemulsd

Step	Hyp	Ref	Expression
1		lemulsd.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ No )
2		lemulsd.4	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ No )
3	1, 2	mulscld 28127	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐷) ∈ No )
4		lemulsd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ No )
5	1, 4	mulscld 28127	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No )
6	3, 5	subscld 28055	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ∈ No )
7	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ∈ No )
8		lemulsd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ No )
9	8, 2	mulscld 28127	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐷) ∈ No )
10	8, 4	mulscld 28127	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐶) ∈ No )
11	9, 10	subscld 28055	. . . . . 6 ⊢ (𝜑 → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) ∈ No )
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) ∈ No )
13	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐴 ∈ No )
14	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐵 ∈ No )
15	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐶 ∈ No )
16	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐷 ∈ No )
17		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐴 <s 𝐵)
18		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐶 <s 𝐷)
19	13, 14, 15, 16, 17, 18	ltmulsd 28129	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) <s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
20	7, 12, 19	ltlesd 27737	. . . 4 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
21	20	anassrs 467	. . 3 ⊢ (((𝜑 ∧ 𝐴 <s 𝐵) ∧ 𝐶 <s 𝐷) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
22		0no 27801	. . . . . . . 8 ⊢ 0s ∈ No
23		lesid 27731	. . . . . . . 8 ⊢ ( 0s ∈ No → 0s ≤s 0s )
24	22, 23	mp1i 13	. . . . . . 7 ⊢ (𝜑 → 0s ≤s 0s )
25		subsid 28061	. . . . . . . 8 ⊢ ((𝐴 ·s 𝐷) ∈ No → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐷)) = 0s )
26	3, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐷)) = 0s )
27		subsid 28061	. . . . . . . 8 ⊢ ((𝐵 ·s 𝐷) ∈ No → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐷)) = 0s )
28	9, 27	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐷)) = 0s )
29	24, 26, 28	3brtr4d 5117	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐷)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐷)))
30		oveq2 7375	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (𝐴 ·s 𝐶) = (𝐴 ·s 𝐷))
31	30	oveq2d 7383	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) = ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐷)))
32		oveq2 7375	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (𝐵 ·s 𝐶) = (𝐵 ·s 𝐷))
33	32	oveq2d 7383	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) = ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐷)))
34	31, 33	breq12d 5098	. . . . . 6 ⊢ (𝐶 = 𝐷 → (((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) ↔ ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐷)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐷))))
35	29, 34	syl5ibrcom 247	. . . . 5 ⊢ (𝜑 → (𝐶 = 𝐷 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶))))
36	35	imp 406	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
37	36	adantlr 716	. . 3 ⊢ (((𝜑 ∧ 𝐴 <s 𝐵) ∧ 𝐶 = 𝐷) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
38		lemulsd.6	. . . . 5 ⊢ (𝜑 → 𝐶 ≤s 𝐷)
39		lesloe 27718	. . . . . 6 ⊢ ((𝐶 ∈ No ∧ 𝐷 ∈ No ) → (𝐶 ≤s 𝐷 ↔ (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷)))
40	4, 2, 39	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝐶 ≤s 𝐷 ↔ (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷)))
41	38, 40	mpbid 232	. . . 4 ⊢ (𝜑 → (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷))
42	41	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 <s 𝐵) → (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷))
43	21, 37, 42	mpjaodan 961	. 2 ⊢ ((𝜑 ∧ 𝐴 <s 𝐵) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
44		lesid 27731	. . . . 5 ⊢ (((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) ∈ No → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
45	11, 44	syl 17	. . . 4 ⊢ (𝜑 → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
46		oveq1 7374	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ·s 𝐷) = (𝐵 ·s 𝐷))
47		oveq1 7374	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶))
48	46, 47	oveq12d 7385	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) = ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
49	48	breq1d 5095	. . . 4 ⊢ (𝐴 = 𝐵 → (((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) ↔ ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶))))
50	45, 49	syl5ibrcom 247	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶))))
51	50	imp 406	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
52		lemulsd.5	. . 3 ⊢ (𝜑 → 𝐴 ≤s 𝐵)
53		lesloe 27718	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
54	1, 8, 53	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
55	52, 54	mpbid 232	. 2 ⊢ (𝜑 → (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵))
56	43, 51, 55	mpjaodan 961	1 ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   class class class wbr 5085  (class class class)co 7367   No csur 27603   <s clts 27604   ≤s cles 27708   0_s c0s 27797   -_s csubs 28012   ·_s cmuls 28098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-nadd 8602  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-norec2 27941  df-adds 27952  df-negs 28013  df-subs 28014  df-muls 28099

This theorem is referenced by:  mulsuniflem  28141