Theorem k3n0j0lt 36061

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 35908	. . . . . 6 ⊢ 9o ∈ On
2		j0iso 36045	. . . . . . . 8 ⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)
3		isof1o 7362	. . . . . . . 8 ⊢ (𝐽0 Isom 𝑅0, E ((On × On), On) → 𝐽0:(On × On)⟶1-1ontoOn)
4		f1of 6854	. . . . . . . 8 ⊢ (𝐽0:(On × On)⟶1-1ontoOn → 𝐽0:(On × On)⟶On)
5	2, 3, 4	mp2b 10	. . . . . . 7 ⊢ 𝐽0:(On × On)⟶On
6		0elon 6444	. . . . . . 7 ⊢ ∅ ∈ On
7	5, 6	f0cli 7130	. . . . . 6 ⊢ (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ On
8		omcl 8582	. . . . . 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 36058	. . . . . 6 ⊢ (𝐾3‘𝐴) ∈ 9o
11		onelon 6413	. . . . . 6 ⊢ ((9o ∈ On ∧ (𝐾3‘𝐴) ∈ 9o) → (𝐾3‘𝐴) ∈ On)
12	1, 10, 11	mp2an 705	. . . . 5 ⊢ (𝐾3‘𝐴) ∈ On
13		oaord1 8597	. . . . 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 36059	. . . . 5 ⊢ (𝐴 ∈ On → 𝐴 = (𝐽‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩))
17		k1on 36056	. . . . . . 7 ⊢ (𝐾1‘𝐴) ∈ On
18	17	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐾1‘𝐴) ∈ On)
19		k2on 36057	. . . . . . 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 7486	. . . . . 6 ⊢ (𝐴 ∈ On → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)) ∈ V)
23		opeq1 4845	. . . . . . . . . 10 ⊢ ((𝐾1‘𝐴) = 𝑥 → ⟨(𝐾1‘𝐴), 𝑦⟩ = ⟨𝑥, 𝑦⟩)
24	23	fveq2d 6920	. . . . . . . . 9 ⊢ ((𝐾1‘𝐴) = 𝑥 → (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩) = (𝐽0‘⟨𝑥, 𝑦⟩))
25	24	oveq2d 7467	. . . . . . . 8 ⊢ ((𝐾1‘𝐴) = 𝑥 → (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)) = (9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)))
26	25	oveq1d 7466	. . . . . . 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 4846	. . . . . . . . . 10 ⊢ ((𝐾2‘𝐴) = 𝑦 → ⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩ = ⟨(𝐾1‘𝐴), 𝑦⟩)
29	28	fveq2d 6920	. . . . . . . . 9 ⊢ ((𝐾2‘𝐴) = 𝑦 → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) = (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩))
30	29	oveq2d 7467	. . . . . . . 8 ⊢ ((𝐾2‘𝐴) = 𝑦 → (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) = (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)))
31	30	oveq1d 7466	. . . . . . 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 7459	. . . . . . 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 36033	. . . . . 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 7722	. . . . 5 ⊢ (𝐴 ∈ On → (𝐽‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩) = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)))
37	16, 36	eqtrd 2798	. . . 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 2867	. 2 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ 𝐴)
40	10	ne0ii 4293	. . . . . 6 ⊢ 9o ≠ ∅
41		on0eln0 6446	. . . . . . 7 ⊢ (9o ∈ On → (∅ ∈ 9o ↔ 9o ≠ ∅))
42	1, 41	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ 9o ↔ 9o ≠ ∅)
43	40, 42	mpbir 234	. . . . 5 ⊢ ∅ ∈ 9o
44		omword2 8620	. . . . 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 6437	. . . . 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 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ⟨cop 4840  ⟨cotp 4863   E cep 5557   × cxp 5656  Oncon0 6388  ⟶wf 6562  ⟶1-1ontowf1o 6565  ‘cfv 6566   Isom wiso 6567  (class class class)co 7451   +_o coa 8511   ·_o comu 8512  9_oc9o 35898  𝑅₀cr0 36022  𝐽₀ccj0 36023  𝐽cj 36024  𝐾₁ck1 36025  𝐾₂ck2 36026  𝐾₃ck3 36027

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-mpt3 7716  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-3o 8516  df-4o 8517  df-oadd 8518  df-omul 8519  df-oi 9541  df-5o 35899  df-6o 35900  df-7o 35901  df-8o 35902  df-9o 35903  df-lexo 36030  df-r0 36031  df-j0 36032  df-j 36033  df-k1 36034  df-k2 36035  df-k3 36036

This theorem is used by:  k3n0k1lt  36064  k3n0k2lt  36065