Theorem lemulsd 28529

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 28526	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐷) ∈ No)
4		lemulsd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ No)
5	1, 4	mulscld 28526	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No)
6	3, 5	subscld 28454	. . . . . 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 28526	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐷) ∈ No)
10	8, 4	mulscld 28526	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐶) ∈ No)
11	9, 10	subscld 28454	. . . . . 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 28528	. . . . 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 28135	. . . 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 28200	. . . . . . . 8 ⊢ 0s ∈ No
23		lesid 28129	. . . . . . . 8 ⊢ ( 0s ∈ No → 0s ≤s 0s )
24	22, 23	mp1i 14	. . . . . . 7 ⊢ (𝜑 → 0s ≤s 0s )
25		subsid 28460	. . . . . . . 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 28460	. . . . . . . 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 5136	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s 𝐷) −s (𝐴 ·s 𝐷)) ≤s ((𝐵 ·s 𝐷) −s (𝐵 ·s 𝐷)))
30		oveq2 7428	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (𝐴 ·s 𝐶) = (𝐴 ·s 𝐷))
31	30	oveq2d 7436	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝐴 ·s 𝐷) −s (𝐴 ·s 𝐶)) = ((𝐴 ·s 𝐷) −s (𝐴 ·s 𝐷)))
32		oveq2 7428	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (𝐵 ·s 𝐶) = (𝐵 ·s 𝐷))
33	32	oveq2d 7436	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝐵 ·s 𝐷) −s (𝐵 ·s 𝐶)) = ((𝐵 ·s 𝐷) −s (𝐵 ·s 𝐷)))
34	31, 33	breq12d 5115	. . . . . 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 28116	. . . . . 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 28129	. . . . 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 7427	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ·s 𝐷) = (𝐵 ·s 𝐷))
47		oveq1 7427	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶))
48	46, 47	oveq12d 7438	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐴 ·s 𝐷) −s (𝐴 ·s 𝐶)) = ((𝐵 ·s 𝐷) −s (𝐵 ·s 𝐶)))
49	48	breq1d 5112	. . . 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 28116	. . . 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 1569   ∈ wcel 2144   class class class wbr 5102  (class class class)co 7420  Nocsur 28002   <s clts 28003   ≤s cles 28106   0_s c0s 28196   −_s csubs 28411   ·_s cmuls 28497

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-nadd 8678  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-0s 28198  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec 28329  df-norec2 28340  df-adds 28351  df-negs 28412  df-subs 28413  df-muls 28498

This theorem is used by:  mulsuniflem  28540