Theorem slemuld 28048

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

Assertion

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

Proof of Theorem slemuld

Step	Hyp	Ref	Expression
1		slemuld.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ No )
2		slemuld.4	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ No )
3	1, 2	mulscld 28045	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐷) ∈ No )
4		slemuld.3	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ No )
5	1, 4	mulscld 28045	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No )
6	3, 5	subscld 27974	. . . . . 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		slemuld.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ No )
9	8, 2	mulscld 28045	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐷) ∈ No )
10	8, 4	mulscld 28045	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐶) ∈ No )
11	9, 10	subscld 27974	. . . . . 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 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐴 <s 𝐵)
18		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷)) → 𝐶 <s 𝐷)
19	13, 14, 15, 16, 17, 18	sltmuld 28047	. . . . 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	sltled 27688	. . . 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		0sno 27745	. . . . . . . 8 ⊢ 0s ∈ No
23		slerflex 27682	. . . . . . . 8 ⊢ ( 0s ∈ No → 0s ≤s 0s )
24	22, 23	mp1i 13	. . . . . . 7 ⊢ (𝜑 → 0s ≤s 0s )
25		subsid 27980	. . . . . . . 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 27980	. . . . . . . 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 5142	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐷)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐷)))
30		oveq2 7398	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (𝐴 ·s 𝐶) = (𝐴 ·s 𝐷))
31	30	oveq2d 7406	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) = ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐷)))
32		oveq2 7398	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (𝐵 ·s 𝐶) = (𝐵 ·s 𝐷))
33	32	oveq2d 7406	. . . . . . 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 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 715	. . 3 ⊢ (((𝜑 ∧ 𝐴 <s 𝐵) ∧ 𝐶 = 𝐷) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
38		slemuld.6	. . . . 5 ⊢ (𝜑 → 𝐶 ≤s 𝐷)
39		sleloe 27673	. . . . . 6 ⊢ ((𝐶 ∈ No ∧ 𝐷 ∈ No ) → (𝐶 ≤s 𝐷 ↔ (𝐶 <s 𝐷 ∨ 𝐶 = 𝐷)))
40	4, 2, 39	syl2anc 584	. . . . 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 960	. 2 ⊢ ((𝜑 ∧ 𝐴 <s 𝐵) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
44		slerflex 27682	. . . . 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 7397	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ·s 𝐷) = (𝐵 ·s 𝐷))
47		oveq1 7397	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶))
48	46, 47	oveq12d 7408	. . . . 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 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		slemuld.5	. . 3 ⊢ (𝜑 → 𝐴 ≤s 𝐵)
53		sleloe 27673	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
54	1, 8, 53	syl2anc 584	. . 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 960	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 847   = wceq 1540   ∈ wcel 2109   class class class wbr 5110  (class class class)co 7390   No csur 27558   <s cslt 27559   ≤s csle 27663   0_s c0s 27741   -_s csubs 27933   ·_s cmuls 28016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-1o 8437  df-2o 8438  df-nadd 8633  df-no 27561  df-slt 27562  df-bday 27563  df-sle 27664  df-sslt 27700  df-scut 27702  df-0s 27743  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec 27852  df-norec2 27863  df-adds 27874  df-negs 27934  df-subs 27935  df-muls 28017

This theorem is referenced by:  mulsuniflem  28059