Theorem lemulsd 28309

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 28306	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐷) ∈ No )
4		lemulsd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ No )
5	1, 4	mulscld 28306	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No )
6	3, 5	subscld 28234	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ∈ No )
7	6	adantr 485	. . . . 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 28306	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐷) ∈ No )
10	8, 4	mulscld 28306	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐶) ∈ No )
11	9, 10	subscld 28234	. . . . . 6 ⊢ (𝜑 → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) ∈ No )
12	11	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) ∈ No )
13	1	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐴 ∈ No )
14	8	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐵 ∈ No )
15	4	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐶 ∈ No )
16	2	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐷 ∈ No )
17		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐴 <s 𝐵)
18		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐶 <s 𝐷)
19	13, 14, 15, 16, 17, 18	ltmulsd 28308	. . . . 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 27915	. . . 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 472	. . 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 27980	. . . . . . . 8 ⊢ 0s ∈ No
23		lesid 27909	. . . . . . . 8 ⊢ ( 0s ∈ No → 0s ≤s 0s )
24	22, 23	mp1i 14	. . . . . . 7 ⊢ (𝜑 → 0s ≤s 0s )
25		subsid 28240	. . . . . . . 8 ⊢ ((𝐴 ·s 𝐷) ∈ No → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐷)) = 0s )
26	3, 25	syl 18	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐷)) = 0s )
27		subsid 28240	. . . . . . . 8 ⊢ ((𝐵 ·s 𝐷) ∈ No → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐷)) = 0s )
28	9, 27	syl 18	. . . . . . 7 ⊢ (𝜑 → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐷)) = 0s )
29	24, 26, 28	3brtr4d 5144	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐷)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐷)))
30		oveq2 7420	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (𝐴 ·s 𝐶) = (𝐴 ·s 𝐷))
31	30	oveq2d 7428	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) = ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐷)))
32		oveq2 7420	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (𝐵 ·s 𝐶) = (𝐵 ·s 𝐷))
33	32	oveq2d 7428	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) = ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐷)))
34	31, 33	breq12d 5123	. . . . . 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 250	. . . . 5 ⊢ (𝜑 → (𝐶 = 𝐷 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶))))
36	35	imp 411	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
37	36	adantlr 727	. . 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 27896	. . . . . 6 ⊢ ((𝐶 ∈ No ∧ 𝐷 ∈ No ) → (𝐶 ≤s 𝐷 ↔ (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷)))
40	4, 2, 39	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝐶 ≤s 𝐷 ↔ (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷)))
41	38, 40	mpbid 235	. . . 4 ⊢ (𝜑 → (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷))
42	41	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐴 <s 𝐵) → (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷))
43	21, 37, 42	mpjaodan 973	. 2 ⊢ ((𝜑 ∧ 𝐴 <s 𝐵) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
44		lesid 27909	. . . . 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 18	. . . 4 ⊢ (𝜑 → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
46		oveq1 7419	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ·s 𝐷) = (𝐵 ·s 𝐷))
47		oveq1 7419	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶))
48	46, 47	oveq12d 7430	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) = ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
49	48	breq1d 5120	. . . 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 250	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶))))
51	50	imp 411	. 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 27896	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
54	1, 8, 53	syl2anc 595	. . 3 ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
55	52, 54	mpbid 235	. 2 ⊢ (𝜑 → (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵))
56	43, 51, 55	mpjaodan 973	1 ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   class class class wbr 5110  (class class class)co 7412   No csur 27782   <s clts 27783   ≤s cles 27886   0_s c0s 27976   -_s csubs 28191   ·_s cmuls 28277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-1o 8454  df-2o 8455  df-nadd 8653  df-no 27785  df-lts 27786  df-bday 27787  df-les 27887  df-slts 27929  df-cuts 27931  df-0s 27978  df-made 27998  df-old 27999  df-left 28001  df-right 28002  df-norec 28109  df-norec2 28120  df-adds 28131  df-negs 28192  df-subs 28193  df-muls 28278

This theorem is referenced by:  mulsuniflem  28320