Theorem onmulscl 28710

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 6927	. . . 4 ⊢ (L‘𝐴) ∈ V
2		fvex 6927	. . . 4 ⊢ (L‘𝐵) ∈ V
3	1, 2	ab2rexex 8023	. . 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 28309	. . . . . . . . . 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 28687	. . . . . . . . . 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 28567	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ (L‘𝐴) ∧ 𝑧 ∈ (L‘𝐵))) → (𝑦 ·s 𝐵) ∈ No)
12		onno 28687	. . . . . . . . . 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 28309	. . . . . . . . . 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 28567	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ (L‘𝐴) ∧ 𝑧 ∈ (L‘𝐵))) → (𝐴 ·s 𝑧) ∈ No)
19	11, 18	addscld 28412	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ (L‘𝐴) ∧ 𝑧 ∈ (L‘𝐵))) → ((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) ∈ No)
20	7, 17	mulscld 28567	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ (L‘𝐴) ∧ 𝑧 ∈ (L‘𝐵))) → (𝑦 ·s 𝑧) ∈ No)
21	19, 20	subscld 28495	. . . . 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 2852	. . . . 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 3223	. . 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 3995	. 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 28305	. . . . . 6 ⊢ (L‘𝐴) ⊆ No
27	1	elpw 4564	. . . . . 6 ⊢ ((L‘𝐴) ∈ 𝒫 No ↔ (L‘𝐴) ⊆ No)
28	26, 27	mpbir 234	. . . . 5 ⊢ (L‘𝐴) ∈ 𝒫 No
29		nulsgts 28208	. . . . 5 ⊢ ((L‘𝐴) ∈ 𝒫 No → (L‘𝐴) <<s ∅)
30	28, 29	mp1i 14	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (L‘𝐴) <<s ∅)
31		leftssno 28305	. . . . . 6 ⊢ (L‘𝐵) ⊆ No
32	2	elpw 4564	. . . . . 6 ⊢ ((L‘𝐵) ∈ 𝒫 No ↔ (L‘𝐵) ⊆ No)
33	31, 32	mpbir 234	. . . . 5 ⊢ (L‘𝐵) ∈ 𝒫 No
34		nulsgts 28208	. . . . 5 ⊢ ((L‘𝐵) ∈ 𝒫 No → (L‘𝐵) <<s ∅)
35	33, 34	mp1i 14	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (L‘𝐵) <<s ∅)
36		oncutleft 28695	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 = ((L‘𝐴) |s ∅))
37	36	adantr 486	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 = ((L‘𝐴) |s ∅))
38		oncutleft 28695	. . . . 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 28582	. . 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 4311	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) −s (𝑦 ·s 𝑧))
42	41	abf 4367	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) −s (𝑦 ·s 𝑧))} = ∅
43	42	uneq2i 4167	. . . . 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 4347	. . . . 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 2786	. . . 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 4311	. . . . . . . . 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 3094	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ (L‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) −s (𝑦 ·s 𝑧))
49	48	abf 4367	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ (L‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) −s (𝑦 ·s 𝑧))} = ∅
50		rex0 4311	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ (L‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) −s (𝑦 ·s 𝑧))
51	50	abf 4367	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ (L‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) −s (𝑦 ·s 𝑧))} = ∅
52	49, 51	uneq12i 4168	. . . . 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 4347	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
54	52, 53	eqtri 2786	. . . 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 7461	. . 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 2814	. 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 28691	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ·s 𝐵) ∈ Ons)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  𝒫 cpw 4560   class class class wbr 5106  ‘cfv 6564  (class class class)co 7449  Nocsur 28043   <<s cslts 28189   |s ccuts 28191  Lcleft 28257   +_s cadds 28391   −_s csubs 28452   ·_s cmuls 28538  On_scons 28683

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-1o 8512  df-2o 8513  df-nadd 8711  df-no 28046  df-lts 28047  df-bday 28048  df-les 28148  df-slts 28190  df-cuts 28192  df-0s 28239  df-made 28259  df-old 28260  df-left 28262  df-right 28263  df-norec 28370  df-norec2 28381  df-adds 28392  df-negs 28453  df-subs 28454  df-muls 28539  df-ons 28684

This theorem is used by: (None)