Theorem lemulsd 28208

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 28205	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐷) ∈ No )
4		lemulsd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ No )
5	1, 4	mulscld 28205	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No )
6	3, 5	subscld 28133	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ∈ No )
7	6	adantr 484	. . . . 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 28205	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐷) ∈ No )
10	8, 4	mulscld 28205	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐶) ∈ No )
11	9, 10	subscld 28133	. . . . . 6 ⊢ (𝜑 → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) ∈ No )
12	11	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) ∈ No )
13	1	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐴 ∈ No )
14	8	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐵 ∈ No )
15	4	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐶 ∈ No )
16	2	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐷 ∈ No )
17		simprl 780	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐴 <s 𝐵)
18		simprr 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐶 <s 𝐷)
19	13, 14, 15, 16, 17, 18	ltmulsd 28207	. . . . 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 27814	. . . 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 471	. . 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 27879	. . . . . . . 8 ⊢ 0s ∈ No
23		lesid 27808	. . . . . . . 8 ⊢ ( 0s ∈ No → 0s ≤s 0s )
24	22, 23	mp1i 13	. . . . . . 7 ⊢ (𝜑 → 0s ≤s 0s )
25		subsid 28139	. . . . . . . 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 28139	. . . . . . . 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 5131	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐷)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐷)))
30		oveq2 7400	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (𝐴 ·s 𝐶) = (𝐴 ·s 𝐷))
31	30	oveq2d 7408	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) = ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐷)))
32		oveq2 7400	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (𝐵 ·s 𝐶) = (𝐵 ·s 𝐷))
33	32	oveq2d 7408	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)) = ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐷)))
34	31, 33	breq12d 5112	. . . . . 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 249	. . . . 5 ⊢ (𝜑 → (𝐶 = 𝐷 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶))))
36	35	imp 410	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
37	36	adantlr 725	. . 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 27795	. . . . . 6 ⊢ ((𝐶 ∈ No ∧ 𝐷 ∈ No ) → (𝐶 ≤s 𝐷 ↔ (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷)))
40	4, 2, 39	syl2anc 593	. . . . 5 ⊢ (𝜑 → (𝐶 ≤s 𝐷 ↔ (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷)))
41	38, 40	mpbid 234	. . . 4 ⊢ (𝜑 → (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷))
42	41	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 <s 𝐵) → (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷))
43	21, 37, 42	mpjaodan 971	. 2 ⊢ ((𝜑 ∧ 𝐴 <s 𝐵) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
44		lesid 27808	. . . . 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 7399	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ·s 𝐷) = (𝐵 ·s 𝐷))
47		oveq1 7399	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶))
48	46, 47	oveq12d 7410	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) = ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
49	48	breq1d 5109	. . . 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 249	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶))))
51	50	imp 410	. 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 27795	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
54	1, 8, 53	syl2anc 593	. . 3 ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
55	52, 54	mpbid 234	. 2 ⊢ (𝜑 → (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵))
56	43, 51, 55	mpjaodan 971	1 ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1559   ∈ wcel 2141   class class class wbr 5099  (class class class)co 7392   No csur 27681   <s clts 27682   ≤s cles 27785   0_s c0s 27875   -_s csubs 28090   ·_s cmuls 28176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-1o 8432  df-2o 8433  df-nadd 8631  df-no 27684  df-lts 27685  df-bday 27686  df-les 27786  df-slts 27828  df-cuts 27830  df-0s 27877  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-norec 28008  df-norec2 28019  df-adds 28030  df-negs 28091  df-subs 28092  df-muls 28177

This theorem is referenced by:  mulsuniflem  28219