Theorem onmulscl 28286

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 6855	. . . 4 ⊢ ( L ‘𝐴) ∈ V
2		fvex 6855	. . . 4 ⊢ ( L ‘𝐵) ∈ V
3	1, 2	ab2rexex 7933	. . 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 27885	. . . . . . . . . 10 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ No )
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵)) → 𝑦 ∈ No )
7	6	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → 𝑦 ∈ No )
8		onno 28263	. . . . . . . . . 10 ⊢ (𝐵 ∈ Ons → 𝐵 ∈ No )
9	8	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐵 ∈ No )
10	9	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → 𝐵 ∈ No )
11	7, 10	mulscld 28143	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝑦 ·s 𝐵) ∈ No )
12		onno 28263	. . . . . . . . . 10 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
13	12	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 ∈ No )
14	13	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → 𝐴 ∈ No )
15		leftno 27885	. . . . . . . . . 10 ⊢ (𝑧 ∈ ( L ‘𝐵) → 𝑧 ∈ No )
16	15	adantl 481	. . . . . . . . 9 ⊢ ((𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵)) → 𝑧 ∈ No )
17	16	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → 𝑧 ∈ No )
18	14, 17	mulscld 28143	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝐴 ·s 𝑧) ∈ No )
19	11, 18	addscld 27988	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → ((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) ∈ No )
20	7, 17	mulscld 28143	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ (𝑦 ∈ ( L ‘𝐴) ∧ 𝑧 ∈ ( L ‘𝐵))) → (𝑦 ·s 𝑧) ∈ No )
21	19, 20	subscld 28071	. . . . 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 2825	. . . . 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 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 ))
24	23	rexlimdvva 3195	. . 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 4021	. 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 27881	. . . . . 6 ⊢ ( L ‘𝐴) ⊆ No
27	1	elpw 4560	. . . . . 6 ⊢ (( L ‘𝐴) ∈ 𝒫 No ↔ ( L ‘𝐴) ⊆ No )
28	26, 27	mpbir 231	. . . . 5 ⊢ ( L ‘𝐴) ∈ 𝒫 No
29		nulsgts 27784	. . . . 5 ⊢ (( L ‘𝐴) ∈ 𝒫 No → ( L ‘𝐴) <<s ∅)
30	28, 29	mp1i 13	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐴) <<s ∅)
31		leftssno 27881	. . . . . 6 ⊢ ( L ‘𝐵) ⊆ No
32	2	elpw 4560	. . . . . 6 ⊢ (( L ‘𝐵) ∈ 𝒫 No ↔ ( L ‘𝐵) ⊆ No )
33	31, 32	mpbir 231	. . . . 5 ⊢ ( L ‘𝐵) ∈ 𝒫 No
34		nulsgts 27784	. . . . 5 ⊢ (( L ‘𝐵) ∈ 𝒫 No → ( L ‘𝐵) <<s ∅)
35	33, 34	mp1i 13	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐵) <<s ∅)
36		oncutleft 28271	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
37	36	adantr 480	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 = (( L ‘𝐴) |s ∅))
38		oncutleft 28271	. . . . 5 ⊢ (𝐵 ∈ Ons → 𝐵 = (( L ‘𝐵) |s ∅))
39	38	adantl 481	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐵 = (( L ‘𝐵) |s ∅))
40	30, 35, 37, 39	mulsunif 28158	. . 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 4314	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
42	41	abf 4360	. . . . . 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 4348	. . . . 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 2760	. . . 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 4314	. . . . . . . . 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 3066	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
49	48	abf 4360	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)∃𝑧 ∈ ∅ 𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))} = ∅
50		rex0 4314	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ ∃𝑧 ∈ ( L ‘𝐵)𝑥 = (((𝑦 ·s 𝐵) +s (𝐴 ·s 𝑧)) -s (𝑦 ·s 𝑧))
51	50	abf 4360	. . . . . 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 4348	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
54	52, 53	eqtri 2760	. . . 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 7380	. . 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 2788	. 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 28267	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ·s 𝐵) ∈ Ons)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368   No csur 27619   <<s cslts 27765   |s ccuts 27767   L cleft 27833   +_s cadds 27967   -_s csubs 28028   ·_s cmuls 28114  On_scons 28259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-1o 8407  df-2o 8408  df-nadd 8604  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029  df-subs 28030  df-muls 28115  df-ons 28260

This theorem is referenced by: (None)