Theorem onmulscl 28182

Description: The surreal ordinals are closed under multiplication. (Contributed by Scott Fenton, 22-Aug-2025.)

Assertion

Ref	Expression
onmulscl	⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ·s 𝐵) ∈ Ons)

Proof of Theorem onmulscl

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6878	. . . 4 ⊢ ( L ‘𝐴) ∈ V
2		fvex 6878	. . . 4 ⊢ ( L ‘𝐵) ∈ V
3	1, 2	ab2rexex 7967	. . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} ∈ V
4	3	a1i 11	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} ∈ V)
5		leftssno 27799	. . . . . . . . . . 11 ⊢ ( L ‘𝐴) ⊆ No
6	5	sseli 3950	. . . . . . . . . 10 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ No )
7	6	adantr 480	. . . . . . . . 9 ⊢ ((𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵)) → 𝑦 ∈ No )
8	7	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → 𝑦 ∈ No )
9		onsno 28163	. . . . . . . . . 10 ⊢ (𝐵 ∈ Ons → 𝐵 ∈ No )
10	9	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐵 ∈ No )
11	10	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → 𝐵 ∈ No )
12	8, 11	mulscld 28045	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝑦 ·s 𝐵) ∈ No )
13		onsno 28163	. . . . . . . . . 10 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
14	13	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 ∈ No )
15	14	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → 𝐴 ∈ No )
16		leftssno 27799	. . . . . . . . . . 11 ⊢ ( L ‘𝐵) ⊆ No
17	16	sseli 3950	. . . . . . . . . 10 ⊢ (𝑧 ∈ ( L ‘𝐵) → 𝑧 ∈ No )
18	17	adantl 481	. . . . . . . . 9 ⊢ ((𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵)) → 𝑧 ∈ No )
19	18	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → 𝑧 ∈ No )
20	15, 19	mulscld 28045	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝐴 ·s 𝑧) ∈ No )
21	12, 20	addscld 27894	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → ((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) ∈ No )
22	8, 19	mulscld 28045	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝑦 ·s 𝑧) ∈ No )
23	21, 22	subscld 27974	. . . . 5 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧)) ∈ No )
24		eleq1 2817	. . . . 5 ⊢ (𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧)) → (𝑥 ∈ No ↔ (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧)) ∈ No ))
25	23, 24	syl5ibrcom 247	. . . 4 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧)) → 𝑥 ∈ No ))
26	25	rexlimdvva 3196	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧)) → 𝑥 ∈ No ))
27	26	abssdv 4039	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} ⊆ No )
28	1	elpw 4575	. . . . . 6 ⊢ (( L ‘𝐴) ∈ 𝒫 No ↔ ( L ‘𝐴) ⊆ No )
29	5, 28	mpbir 231	. . . . 5 ⊢ ( L ‘𝐴) ∈ 𝒫 No
30		nulssgt 27717	. . . . 5 ⊢ (( L ‘𝐴) ∈ 𝒫 No → ( L ‘𝐴) <<s ∅)
31	29, 30	mp1i 13	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐴) <<s ∅)
32	2	elpw 4575	. . . . . 6 ⊢ (( L ‘𝐵) ∈ 𝒫 No ↔ ( L ‘𝐵) ⊆ No )
33	16, 32	mpbir 231	. . . . 5 ⊢ ( L ‘𝐵) ∈ 𝒫 No
34		nulssgt 27717	. . . . 5 ⊢ (( L ‘𝐵) ∈ 𝒫 No → ( L ‘𝐵) <<s ∅)
35	33, 34	mp1i 13	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐵) <<s ∅)
36		onscutleft 28171	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
37	36	adantr 480	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 = (( L ‘𝐴) |s ∅))
38		onscutleft 28171	. . . . 5 ⊢ (𝐵 ∈ Ons → 𝐵 = (( L ‘𝐵) |s ∅))
39	38	adantl 481	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐵 = (( L ‘𝐵) |s ∅))
40	31, 35, 37, 39	mulsunif 28060	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ·s 𝐵) = (({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))}) |s ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))})))
41		rex0 4331	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
42	41	abf 4377	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} = ∅
43	42	uneq2i 4136	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))}) = ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} ∪ ∅)
44		un0 4365	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} ∪ ∅) = {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))}
45	43, 44	eqtri 2753	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))}) = {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))}
46		rex0 4331	. . . . . . . . 9 ⊢ ¬ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
47	46	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ ( L ‘𝐴) → ¬ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧)))
48	47	nrex 3059	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
49	48	abf 4377	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} = ∅
50		rex0 4331	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
51	50	abf 4377	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} = ∅
52	49, 51	uneq12i 4137	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))}) = (∅ ∪ ∅)
53		un0 4365	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
54	52, 53	eqtri 2753	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))}) = ∅
55	45, 54	oveq12i 7406	. . 3 ⊢ (({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))}) |s ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))})) = ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} |s ∅)
56	40, 55	eqtrdi 2781	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ·s 𝐵) = ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} |s ∅))
57	4, 27, 56	elons2d 28167	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ·s 𝐵) ∈ Ons)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2708  ∃wrex 3055  Vcvv 3455   ∪ cun 3920   ⊆ wss 3922  ∅c0 4304  𝒫 cpw 4571   class class class wbr 5115  ‘cfv 6519  (class class class)co 7394   No csur 27558   <<s csslt 27699   |s cscut 27701   L cleft 27760   +_s cadds 27873   -_s csubs 27933   ·_s cmuls 28016  On_scons 28159

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 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

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 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rmo 3357  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-pss 3942  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-tp 4602  df-op 4604  df-ot 4606  df-uni 4880  df-int 4919  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5541  df-eprel 5546  df-po 5554  df-so 5555  df-fr 5599  df-se 5600  df-we 5601  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-pred 6282  df-ord 6343  df-on 6344  df-suc 6346  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-riota 7351  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-frecs 8269  df-wrecs 8300  df-recs 8349  df-1o 8443  df-2o 8444  df-nadd 8641  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  df-ons 28160

This theorem is referenced by: (None)