Theorem onmulscl 28670

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 6899	. . . 4 ⊢ (L‘𝐴) ∈ V
2		fvex 6899	. . . 4 ⊢ (L‘𝐵) ∈ V
3	1, 2	ab2rexex 7992	. . 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 28269	. . . . . . . . . 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 28647	. . . . . . . . . 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 28527	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ (L‘𝐴) ∧ 𝑧 ∈ (L‘𝐵))) → (𝑦 ·s 𝐵) ∈ No)
12		onno 28647	. . . . . . . . . 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 28269	. . . . . . . . . 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 28527	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ (L‘𝐴) ∧ 𝑧 ∈ (L‘𝐵))) → (𝐴 ·s 𝑧) ∈ No)
19	11, 18	addscld 28372	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ (L‘𝐴) ∧ 𝑧 ∈ (L‘𝐵))) → ((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) ∈ No)
20	7, 17	mulscld 28527	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ (L‘𝐴) ∧ 𝑧 ∈ (L‘𝐵))) → (𝑦 ·s 𝑧) ∈ No)
21	19, 20	subscld 28455	. . . . 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 2848	. . . . 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 3219	. . 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 3992	. 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 28265	. . . . . 6 ⊢ (L‘𝐴) ⊆ No
27	1	elpw 4560	. . . . . 6 ⊢ ((L‘𝐴) ∈ 𝒫 No ↔ (L‘𝐴) ⊆ No)
28	26, 27	mpbir 234	. . . . 5 ⊢ (L‘𝐴) ∈ 𝒫 No
29		nulsgts 28168	. . . . 5 ⊢ ((L‘𝐴) ∈ 𝒫 No → (L‘𝐴) <<s ∅)
30	28, 29	mp1i 14	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (L‘𝐴) <<s ∅)
31		leftssno 28265	. . . . . 6 ⊢ (L‘𝐵) ⊆ No
32	2	elpw 4560	. . . . . 6 ⊢ ((L‘𝐵) ∈ 𝒫 No ↔ (L‘𝐵) ⊆ No)
33	31, 32	mpbir 234	. . . . 5 ⊢ (L‘𝐵) ∈ 𝒫 No
34		nulsgts 28168	. . . . 5 ⊢ ((L‘𝐵) ∈ 𝒫 No → (L‘𝐵) <<s ∅)
35	33, 34	mp1i 14	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (L‘𝐵) <<s ∅)
36		oncutleft 28655	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 = ((L‘𝐴) |s ∅))
37	36	adantr 486	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 = ((L‘𝐴) |s ∅))
38		oncutleft 28655	. . . . 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 28542	. . 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 4307	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) −s (𝑦 ·s 𝑧))
42	41	abf 4363	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) −s (𝑦 ·s 𝑧))} = ∅
43	42	uneq2i 4164	. . . . 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 4343	. . . . 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 2783	. . . 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 4307	. . . . . . . . 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 3090	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ (L‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) −s (𝑦 ·s 𝑧))
49	48	abf 4363	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ (L‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) −s (𝑦 ·s 𝑧))} = ∅
50		rex0 4307	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ (L‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) −s (𝑦 ·s 𝑧))
51	50	abf 4363	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ (L‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) −s (𝑦 ·s 𝑧))} = ∅
52	49, 51	uneq12i 4165	. . . . 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 4343	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
54	52, 53	eqtri 2783	. . . 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 7433	. . 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 2811	. 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 28651	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ·s 𝐵) ∈ Ons)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  𝒫 cpw 4556   class class class wbr 5102  ‘cfv 6539  (class class class)co 7421  Nocsur 28003   <<s cslts 28149   |s ccuts 28151  Lcleft 28217   +_s cadds 28351   −_s csubs 28412   ·_s cmuls 28498  On_scons 28643

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-1o 8480  df-2o 8481  df-nadd 8679  df-no 28006  df-lts 28007  df-bday 28008  df-les 28108  df-slts 28150  df-cuts 28152  df-0s 28199  df-made 28219  df-old 28220  df-left 28222  df-right 28223  df-norec 28330  df-norec2 28341  df-adds 28352  df-negs 28413  df-subs 28414  df-muls 28499  df-ons 28644

This theorem is used by: (None)