Theorem onmulscl 28212

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 6841	. . . 4 ⊢ ( L ‘𝐴) ∈ V
2		fvex 6841	. . . 4 ⊢ ( L ‘𝐵) ∈ V
3	1, 2	ab2rexex 7917	. . 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 27827	. . . . . . . . . . 11 ⊢ ( L ‘𝐴) ⊆ No
6	5	sseli 3926	. . . . . . . . . 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 28193	. . . . . . . . . 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 28075	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝑦 ·s 𝐵) ∈ No )
13		onsno 28193	. . . . . . . . . 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 27827	. . . . . . . . . . 11 ⊢ ( L ‘𝐵) ⊆ No
17	16	sseli 3926	. . . . . . . . . 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 28075	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝐴 ·s 𝑧) ∈ No )
21	12, 20	addscld 27924	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → ((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) ∈ No )
22	8, 19	mulscld 28075	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝑦 ·s 𝑧) ∈ No )
23	21, 22	subscld 28004	. . . . 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 2821	. . . . 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 3190	. . 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 4016	. 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 4553	. . . . . 6 ⊢ (( L ‘𝐴) ∈ 𝒫 No ↔ ( L ‘𝐴) ⊆ No )
29	5, 28	mpbir 231	. . . . 5 ⊢ ( L ‘𝐴) ∈ 𝒫 No
30		nulssgt 27740	. . . . 5 ⊢ (( L ‘𝐴) ∈ 𝒫 No → ( L ‘𝐴) <<s ∅)
31	29, 30	mp1i 13	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐴) <<s ∅)
32	2	elpw 4553	. . . . . 6 ⊢ (( L ‘𝐵) ∈ 𝒫 No ↔ ( L ‘𝐵) ⊆ No )
33	16, 32	mpbir 231	. . . . 5 ⊢ ( L ‘𝐵) ∈ 𝒫 No
34		nulssgt 27740	. . . . 5 ⊢ (( L ‘𝐵) ∈ 𝒫 No → ( L ‘𝐵) <<s ∅)
35	33, 34	mp1i 13	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐵) <<s ∅)
36		onscutleft 28201	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
37	36	adantr 480	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 = (( L ‘𝐴) |s ∅))
38		onscutleft 28201	. . . . 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 28090	. . 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 4309	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
42	41	abf 4355	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} = ∅
43	42	uneq2i 4114	. . . . 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 2756	. . . 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 4309	. . . . . . . . 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 3061	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
49	48	abf 4355	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} = ∅
50		rex0 4309	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
51	50	abf 4355	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} = ∅
52	49, 51	uneq12i 4115	. . . . 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 2756	. . . 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 7364	. . 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 2784	. 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 28197	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ·s 𝐵) ∈ Ons)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  ∃wrex 3057  Vcvv 3437   ∪ cun 3896   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4549   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352   No csur 27579   <<s csslt 27721   |s cscut 27723   L cleft 27787   +_s cadds 27903   -_s csubs 27963   ·_s cmuls 28046  On_scons 28189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-ot 4584  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-1o 8391  df-2o 8392  df-nadd 8587  df-no 27582  df-slt 27583  df-bday 27584  df-sle 27685  df-sslt 27722  df-scut 27724  df-0s 27769  df-made 27789  df-old 27790  df-left 27792  df-right 27793  df-norec 27882  df-norec2 27893  df-adds 27904  df-negs 27964  df-subs 27965  df-muls 28047  df-ons 28190

This theorem is referenced by: (None)