Theorem onmulscl 28500

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 6898	. . . 4 ⊢ ( L ‘𝐴) ∈ V
2		fvex 6898	. . . 4 ⊢ ( L ‘𝐵) ∈ V
3	1, 2	ab2rexex 7978	. . 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		leftno 28099	. . . . . . . . . 10 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ No )
6	5	adantr 486	. . . . . . . . 9 ⊢ ((𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵)) → 𝑦 ∈ No )
7	6	adantl 487	. . . . . . . 8 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → 𝑦 ∈ No )
8		onno 28477	. . . . . . . . . 10 ⊢ (𝐵 ∈ Ons → 𝐵 ∈ No )
9	8	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐵 ∈ No )
10	9	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → 𝐵 ∈ No )
11	7, 10	mulscld 28357	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝑦 ·s 𝐵) ∈ No )
12		onno 28477	. . . . . . . . . 10 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
13	12	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 ∈ No )
14	13	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → 𝐴 ∈ No )
15		leftno 28099	. . . . . . . . . 10 ⊢ (𝑧 ∈ ( L ‘𝐵) → 𝑧 ∈ No )
16	15	adantl 487	. . . . . . . . 9 ⊢ ((𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵)) → 𝑧 ∈ No )
17	16	adantl 487	. . . . . . . 8 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → 𝑧 ∈ No )
18	14, 17	mulscld 28357	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝐴 ·s 𝑧) ∈ No )
19	11, 18	addscld 28202	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → ((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) ∈ No )
20	7, 17	mulscld 28357	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝑦 ·s 𝑧) ∈ No )
21	19, 20	subscld 28285	. . . . 5 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧)) ∈ No )
22		eleq1 2853	. . . . 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 ))
23	21, 22	syl5ibrcom 250	. . . 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 ))
24	23	rexlimdvva 3224	. . 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 ))
25	24	abssdv 4022	. 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 )
26		leftssno 28095	. . . . . 6 ⊢ ( L ‘𝐴) ⊆ No
27	1	elpw 4568	. . . . . 6 ⊢ (( L ‘𝐴) ∈ 𝒫 No ↔ ( L ‘𝐴) ⊆ No )
28	26, 27	mpbir 234	. . . . 5 ⊢ ( L ‘𝐴) ∈ 𝒫 No
29		nulsgts 27998	. . . . 5 ⊢ (( L ‘𝐴) ∈ 𝒫 No → ( L ‘𝐴) <<s ∅)
30	28, 29	mp1i 14	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐴) <<s ∅)
31		leftssno 28095	. . . . . 6 ⊢ ( L ‘𝐵) ⊆ No
32	2	elpw 4568	. . . . . 6 ⊢ (( L ‘𝐵) ∈ 𝒫 No ↔ ( L ‘𝐵) ⊆ No )
33	31, 32	mpbir 234	. . . . 5 ⊢ ( L ‘𝐵) ∈ 𝒫 No
34		nulsgts 27998	. . . . 5 ⊢ (( L ‘𝐵) ∈ 𝒫 No → ( L ‘𝐵) <<s ∅)
35	33, 34	mp1i 14	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐵) <<s ∅)
36		oncutleft 28485	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
37	36	adantr 486	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 = (( L ‘𝐴) |s ∅))
38		oncutleft 28485	. . . . 5 ⊢ (𝐵 ∈ Ons → 𝐵 = (( L ‘𝐵) |s ∅))
39	38	adantl 487	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐵 = (( L ‘𝐵) |s ∅))
40	30, 35, 37, 39	mulsunif 28372	. . 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 4315	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
42	41	abf 4371	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} = ∅
43	42	uneq2i 4119	. . . . 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 4351	. . . . 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 2788	. . . 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 4315	. . . . . . . . 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 3095	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
49	48	abf 4371	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} = ∅
50		rex0 4315	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
51	50	abf 4371	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} = ∅
52	49, 51	uneq12i 4120	. . . . 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 4351	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
54	52, 53	eqtri 2788	. . . 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 7428	. . 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 2816	. 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, 25, 56	elons2d 28481	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ·s 𝐵) ∈ Ons)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2743  ∃wrex 3091  Vcvv 3457   ∪ cun 3904   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564   class class class wbr 5111  ‘cfv 6540  (class class class)co 7416   No csur 27833   <<s cslts 27979   |s ccuts 27981   L cleft 28047   +_s cadds 28181   -_s csubs 28242   ·_s cmuls 28328  On_scons 28473

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-1o 8455  df-2o 8456  df-nadd 8654  df-no 27836  df-lts 27837  df-bday 27838  df-les 27938  df-slts 27980  df-cuts 27982  df-0s 28029  df-made 28049  df-old 28050  df-left 28052  df-right 28053  df-norec 28160  df-norec2 28171  df-adds 28182  df-negs 28243  df-subs 28244  df-muls 28329  df-ons 28474

This theorem is used by: (None)