Theorem k3n0j0lt 35971

Description: A strict upper bound on the value of 𝐽₀ for nonzero values of 𝐾₃. (Contributed by BTernaryTau, 12-Oct-2026.)

Assertion

Ref	Expression
k3n0j0lt	⊢ ((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ 𝐴)

Proof of Theorem k3n0j0lt

Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		9on 35843	. . . . . 6 ⊢ 9o ∈ On
2		j0iso 35955	. . . . . . . 8 ⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)
3		isof1o 7335	. . . . . . . 8 ⊢ (𝐽0 Isom 𝑅0, E ((On × On), On) → 𝐽0:(On × On)–1-1-onto→On)
4		f1of 6829	. . . . . . . 8 ⊢ (𝐽0:(On × On)–1-1-onto→On → 𝐽0:(On × On)⟶On)
5	2, 3, 4	mp2b 10	. . . . . . 7 ⊢ 𝐽0:(On × On)⟶On
6		0elon 6421	. . . . . . 7 ⊢ ∅ ∈ On
7	5, 6	f0cli 7103	. . . . . 6 ⊢ (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ On
8		omcl 8552	. . . . . 6 ⊢ ((9o ∈ On ∧ (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ On) → (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ On)
9	1, 7, 8	mp2an 705	. . . . 5 ⊢ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ On
10		k39o 35968	. . . . . 6 ⊢ (𝐾3‘𝐴) ∈ 9o
11		onelon 6390	. . . . . 6 ⊢ ((9o ∈ On ∧ (𝐾3‘𝐴) ∈ 9o) → (𝐾3‘𝐴) ∈ On)
12	1, 10, 11	mp2an 705	. . . . 5 ⊢ (𝐾3‘𝐴) ∈ On
13		oaord1 8567	. . . . 5 ⊢ (((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ On ∧ (𝐾3‘𝐴) ∈ On) → (∅ ∈ (𝐾3‘𝐴) ↔ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴))))
14	9, 12, 13	mp2an 705	. . . 4 ⊢ (∅ ∈ (𝐾3‘𝐴) ↔ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)))
15	14	bilani 510	. . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)))
16		jk1k2k3 35969	. . . . 5 ⊢ (𝐴 ∈ On → 𝐴 = (𝐽‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩))
17		k1on 35966	. . . . . . 7 ⊢ (𝐾1‘𝐴) ∈ On
18	17	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐾1‘𝐴) ∈ On)
19		k2on 35967	. . . . . . 7 ⊢ (𝐾2‘𝐴) ∈ On
20	19	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐾2‘𝐴) ∈ On)
21	10	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐾3‘𝐴) ∈ 9o)
22		ovexd 7459	. . . . . 6 ⊢ (𝐴 ∈ On → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)) ∈ V)
23		opeq1 4839	. . . . . . . . . 10 ⊢ ((𝐾1‘𝐴) = 𝑥 → ⟨(𝐾1‘𝐴), 𝑦⟩ = ⟨𝑥, 𝑦⟩)
24	23	fveq2d 6894	. . . . . . . . 9 ⊢ ((𝐾1‘𝐴) = 𝑥 → (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩) = (𝐽0‘⟨𝑥, 𝑦⟩))
25	24	oveq2d 7440	. . . . . . . 8 ⊢ ((𝐾1‘𝐴) = 𝑥 → (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)) = (9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)))
26	25	oveq1d 7439	. . . . . . 7 ⊢ ((𝐾1‘𝐴) = 𝑥 → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)) +o 𝑛) = ((9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)) +o 𝑛))
27	26	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐾1‘𝐴) = 𝑥) → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)) +o 𝑛) = ((9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)) +o 𝑛))
28		opeq2 4840	. . . . . . . . . 10 ⊢ ((𝐾2‘𝐴) = 𝑦 → ⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩ = ⟨(𝐾1‘𝐴), 𝑦⟩)
29	28	fveq2d 6894	. . . . . . . . 9 ⊢ ((𝐾2‘𝐴) = 𝑦 → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) = (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩))
30	29	oveq2d 7440	. . . . . . . 8 ⊢ ((𝐾2‘𝐴) = 𝑦 → (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) = (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)))
31	30	oveq1d 7439	. . . . . . 7 ⊢ ((𝐾2‘𝐴) = 𝑦 → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o 𝑛) = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)) +o 𝑛))
32	31	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐾2‘𝐴) = 𝑦) → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o 𝑛) = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)) +o 𝑛))
33		oveq2 7432	. . . . . . 7 ⊢ ((𝐾3‘𝐴) = 𝑛 → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)) = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o 𝑛))
34	33	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) = 𝑛) → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)) = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o 𝑛))
35		df-j 35943	. . . . . 6 ⊢ 𝐽 = (𝑥 ∈ On, 𝑦 ∈ On, 𝑛 ∈ 9o ↦ ((9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)) +o 𝑛))
36	18, 20, 21, 22, 27, 32, 34, 35	mpt3fvot2d 7694	. . . . 5 ⊢ (𝐴 ∈ On → (𝐽‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩) = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)))
37	16, 36	eqtrd 2795	. . . 4 ⊢ (𝐴 ∈ On → 𝐴 = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)))
38	37	adantr 486	. . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → 𝐴 = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)))
39	15, 38	eleqtrrd 2863	. 2 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ 𝐴)
40	10	ne0ii 4289	. . . . . 6 ⊢ 9o ≠ ∅
41		on0eln0 6423	. . . . . . 7 ⊢ (9o ∈ On → (∅ ∈ 9o ↔ 9o ≠ ∅))
42	1, 41	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ 9o ↔ 9o ≠ ∅)
43	40, 42	mpbir 234	. . . . 5 ⊢ ∅ ∈ 9o
44		omword2 8590	. . . . 5 ⊢ ((((𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ On ∧ 9o ∈ On) ∧ ∅ ∈ 9o) → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ⊆ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)))
45	43, 44	mpan2 704	. . . 4 ⊢ (((𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ On ∧ 9o ∈ On) → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ⊆ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)))
46	7, 1, 45	mp2an 705	. . 3 ⊢ (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ⊆ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩))
47		ontr2 6414	. . . . 5 ⊢ (((𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ On ∧ 𝐴 ∈ On) → (((𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ⊆ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∧ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ 𝐴) → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ 𝐴))
48	7, 47	mpan 703	. . . 4 ⊢ (𝐴 ∈ On → (((𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ⊆ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∧ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ 𝐴) → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ 𝐴))
49	48	adantr 486	. . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → (((𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ⊆ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∧ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ 𝐴) → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ 𝐴))
50	46, 49	mpani 709	. 2 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ 𝐴 → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ 𝐴))
51	39, 50	mpd 16	1 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450   ⊆ wss 3896  ∅c0 4277  ⟨cop 4834  ⟨cotp 4857   E cep 5550   × cxp 5649  Oncon0 6365  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543   Isom wiso 6544  (class class class)co 7424   +_o coa 8481   ·_o comu 8482  9_oc9o 35833  𝑅₀cr0 35932  𝐽₀ccj0 35933  𝐽cj 35934  𝐾₁ck1 35935  𝐾₂ck2 35936  𝐾₃ck3 35937

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 5327  ax-pr 5391  ax-un 7755

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 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  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 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 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-mpt3 7688  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-2o 8485  df-3o 8486  df-4o 8487  df-oadd 8488  df-omul 8489  df-oi 9511  df-5o 35834  df-6o 35835  df-7o 35836  df-8o 35837  df-9o 35838  df-lexo 35940  df-r0 35941  df-j0 35942  df-j 35943  df-k1 35944  df-k2 35945  df-k3 35946

This theorem is used by:  k3n0k1lt  35974  k3n0k2lt  35975