Theorem ackvalsucsucval 50015

Description: The Ackermann function at the successors. This is the third equation of Péter's definition of the Ackermann function. (Contributed by AV, 8-May-2024.)

Assertion

Ref	Expression
ackvalsucsucval	⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))

Proof of Theorem ackvalsucsucval

Step	Hyp	Ref	Expression
1		peano2nn0 12676	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
2		ackvalsuc1 50006	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ (𝑁 + 1) ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1))
3	1, 2	sylan2 605	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1))
4		fvexd 6926	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (Ack‘𝑀) ∈ V)
5	1	adantl 487	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ0)
6		eqidd 2764	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) = ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)))
7		itcovalsucov 49995	. . . . 5 ⊢ (((Ack‘𝑀) ∈ V ∧ (𝑁 + 1) ∈ ℕ0 ∧ ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) = ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))) → ((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1)) = ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))))
8	4, 5, 6, 7	syl3anc 1399	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1)) = ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))))
9	8	fveq1d 6913	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1) = (((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)))‘1))
10		ackfnnn0 50012	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀) Fn ℕ0)
11	10	adantr 486	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (Ack‘𝑀) Fn ℕ0)
12		nn0ex 12642	. . . . . . . . 9 ⊢ ℕ0 ∈ V
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ℕ0 ∈ V)
14		ackendofnn0 50011	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀):ℕ0⟶ℕ0)
15	14	adantr 486	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (Ack‘𝑀):ℕ0⟶ℕ0)
16		simpr 490	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
17	13, 15, 16	itcovalendof 49996	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘𝑁):ℕ0⟶ℕ0)
18	17	ffnd 6735	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘𝑁) Fn ℕ0)
19	17	frnd 6743	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ran ((IterComp‘(Ack‘𝑀))‘𝑁) ⊆ ℕ0)
20		fnco 6682	. . . . . 6 ⊢ (((Ack‘𝑀) Fn ℕ0 ∧ ((IterComp‘(Ack‘𝑀))‘𝑁) Fn ℕ0 ∧ ran ((IterComp‘(Ack‘𝑀))‘𝑁) ⊆ ℕ0) → ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘𝑁)) Fn ℕ0)
21	11, 18, 19, 20	syl3anc 1399	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘𝑁)) Fn ℕ0)
22		eqidd 2764	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘𝑁) = ((IterComp‘(Ack‘𝑀))‘𝑁))
23		itcovalsucov 49995	. . . . . . 7 ⊢ (((Ack‘𝑀) ∈ V ∧ 𝑁 ∈ ℕ0 ∧ ((IterComp‘(Ack‘𝑀))‘𝑁) = ((IterComp‘(Ack‘𝑀))‘𝑁)) → ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) = ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘𝑁)))
24	4, 16, 22, 23	syl3anc 1399	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) = ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘𝑁)))
25	24	fneq1d 6657	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) Fn ℕ0 ↔ ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘𝑁)) Fn ℕ0))
26	21, 25	mpbird 260	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) Fn ℕ0)
27		1nn0 12652	. . . 4 ⊢ 1 ∈ ℕ0
28		fvco2 7008	. . . 4 ⊢ ((((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) Fn ℕ0 ∧ 1 ∈ ℕ0) → (((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)))‘1) = ((Ack‘𝑀)‘(((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)))
29	26, 27, 28	sylancl 598	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)))‘1) = ((Ack‘𝑀)‘(((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)))
30	9, 29	eqtrd 2798	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1) = ((Ack‘𝑀)‘(((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)))
31		ackvalsuc1 50006	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1))
32	31	eqcomd 2769	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1) = ((Ack‘(𝑀 + 1))‘𝑁))
33	32	fveq2d 6915	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘𝑀)‘(((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))
34	3, 30, 33	3eqtrd 2802	1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  ran crn 5657   ∘ ccom 5660   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446  1c1 11231   + caddc 11233  ℕ₀cn0 12636  IterCompcitco 49984  Ackcack 49985

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 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-inf2 9672  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

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-nel 3066  df-ral 3081  df-rex 3091  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 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-er 8748  df-en 9005  df-dom 9006  df-sdom 9007  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-n0 12637  df-z 12724  df-uz 12996  df-seq 14175  df-itco 49986  df-ack 49987

This theorem is used by:  ackval41a  50021  ackval42  50023