Theorem lemulsd 28557

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 28554	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐷) ∈ No)
4		lemulsd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ No)
5	1, 4	mulscld 28554	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No)
6	3, 5	subscld 28482	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s 𝐷) −s (𝐴 ·s 𝐶)) ∈ No)
7	6	adantr 486	. . . . 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 28554	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐷) ∈ No)
10	8, 4	mulscld 28554	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐶) ∈ No)
11	9, 10	subscld 28482	. . . . . 6 ⊢ (𝜑 → ((𝐵 ·s 𝐷) −s (𝐵 ·s 𝐶)) ∈ No)
12	11	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → ((𝐵 ·s 𝐷) −s (𝐵 ·s 𝐶)) ∈ No)
13	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐴 ∈ No)
14	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐵 ∈ No)
15	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐶 ∈ No)
16	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐷 ∈ No)
17		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐴 <s 𝐵)
18		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐶 <s 𝐷)
19	13, 14, 15, 16, 17, 18	ltmulsd 28556	. . . . 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 28163	. . . 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 473	. . 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 28228	. . . . . . . 8 ⊢ 0s ∈ No
23		lesid 28157	. . . . . . . 8 ⊢ ( 0s ∈ No → 0s ≤s 0s )
24	22, 23	mp1i 14	. . . . . . 7 ⊢ (𝜑 → 0s ≤s 0s )
25		subsid 28488	. . . . . . . 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 28488	. . . . . . . 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 5141	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s 𝐷) −s (𝐴 ·s 𝐷)) ≤s ((𝐵 ·s 𝐷) −s (𝐵 ·s 𝐷)))
30		oveq2 7454	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (𝐴 ·s 𝐶) = (𝐴 ·s 𝐷))
31	30	oveq2d 7462	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝐴 ·s 𝐷) −s (𝐴 ·s 𝐶)) = ((𝐴 ·s 𝐷) −s (𝐴 ·s 𝐷)))
32		oveq2 7454	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (𝐵 ·s 𝐶) = (𝐵 ·s 𝐷))
33	32	oveq2d 7462	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝐵 ·s 𝐷) −s (𝐵 ·s 𝐶)) = ((𝐵 ·s 𝐷) −s (𝐵 ·s 𝐷)))
34	31, 33	breq12d 5119	. . . . . 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 412	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → ((𝐴 ·s 𝐷) −s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) −s (𝐵 ·s 𝐶)))
37	36	adantlr 728	. . 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 28144	. . . . . 6 ⊢ ((𝐶 ∈ No ∧ 𝐷 ∈ No) → (𝐶 ≤s 𝐷 ↔ (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷)))
40	4, 2, 39	syl2anc 596	. . . . 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 486	. . 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 28157	. . . . 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 7453	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ·s 𝐷) = (𝐵 ·s 𝐷))
47		oveq1 7453	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶))
48	46, 47	oveq12d 7464	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐴 ·s 𝐷) −s (𝐴 ·s 𝐶)) = ((𝐵 ·s 𝐷) −s (𝐵 ·s 𝐶)))
49	48	breq1d 5116	. . . 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 412	. 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 28144	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
54	1, 8, 53	syl2anc 596	. . 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   class class class wbr 5106  (class class class)co 7446  Nocsur 28030   <s clts 28031   ≤s cles 28134   0_s c0s 28224   −_s csubs 28439   ·_s cmuls 28525

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-nadd 8706  df-no 28033  df-lts 28034  df-bday 28035  df-les 28135  df-slts 28177  df-cuts 28179  df-0s 28226  df-made 28246  df-old 28247  df-left 28249  df-right 28250  df-norec 28357  df-norec2 28368  df-adds 28379  df-negs 28440  df-subs 28441  df-muls 28526

This theorem is used by:  mulsuniflem  28568