Theorem onmulscl 28657

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 6896	. . . 4 ⊢ ( L ‘𝐴) ∈ V
2		fvex 6896	. . . 4 ⊢ ( L ‘𝐵) ∈ V
3	1, 2	ab2rexex 7989	. . 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 28256	. . . . . . . . . 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 28634	. . . . . . . . . 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 28514	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝑦 ·s 𝐵) ∈ No )
12		onno 28634	. . . . . . . . . 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 28256	. . . . . . . . . 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 28514	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝐴 ·s 𝑧) ∈ No )
19	11, 18	addscld 28359	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → ((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) ∈ No )
20	7, 17	mulscld 28514	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝑦 ·s 𝑧) ∈ No )
21	19, 20	subscld 28442	. . . . 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 2849	. . . . 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 3220	. . 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 4015	. 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 28252	. . . . . 6 ⊢ ( L ‘𝐴) ⊆ No
27	1	elpw 4561	. . . . . 6 ⊢ (( L ‘𝐴) ∈ 𝒫 No ↔ ( L ‘𝐴) ⊆ No )
28	26, 27	mpbir 234	. . . . 5 ⊢ ( L ‘𝐴) ∈ 𝒫 No
29		nulsgts 28155	. . . . 5 ⊢ (( L ‘𝐴) ∈ 𝒫 No → ( L ‘𝐴) <<s ∅)
30	28, 29	mp1i 14	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐴) <<s ∅)
31		leftssno 28252	. . . . . 6 ⊢ ( L ‘𝐵) ⊆ No
32	2	elpw 4561	. . . . . 6 ⊢ (( L ‘𝐵) ∈ 𝒫 No ↔ ( L ‘𝐵) ⊆ No )
33	31, 32	mpbir 234	. . . . 5 ⊢ ( L ‘𝐵) ∈ 𝒫 No
34		nulsgts 28155	. . . . 5 ⊢ (( L ‘𝐵) ∈ 𝒫 No → ( L ‘𝐵) <<s ∅)
35	33, 34	mp1i 14	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐵) <<s ∅)
36		oncutleft 28642	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
37	36	adantr 486	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 = (( L ‘𝐴) |s ∅))
38		oncutleft 28642	. . . . 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 28529	. . 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 4308	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
42	41	abf 4364	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} = ∅
43	42	uneq2i 4112	. . . . 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 4344	. . . . 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 2784	. . . 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 4308	. . . . . . . . 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 3091	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
49	48	abf 4364	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} = ∅
50		rex0 4308	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
51	50	abf 4364	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} = ∅
52	49, 51	uneq12i 4113	. . . . 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 4344	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
54	52, 53	eqtri 2784	. . . 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 7430	. . 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 2812	. 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 28638	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ·s 𝐵) ∈ Ons)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739  ∃wrex 3087  Vcvv 3451   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557   class class class wbr 5103  ‘cfv 6537  (class class class)co 7418   No csur 27990   <<s cslts 28136   |s ccuts 28138   L cleft 28204   +_s cadds 28338   -_s csubs 28399   ·_s cmuls 28485  On_scons 28630

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-1o 8469  df-2o 8470  df-nadd 8668  df-no 27993  df-lts 27994  df-bday 27995  df-les 28095  df-slts 28137  df-cuts 28139  df-0s 28186  df-made 28206  df-old 28207  df-left 28209  df-right 28210  df-norec 28317  df-norec2 28328  df-adds 28339  df-negs 28400  df-subs 28401  df-muls 28486  df-ons 28631

This theorem is used by: (None)