Theorem ackvalsucsucval 48677

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 12482	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
2		ackvalsuc1 48668	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ (𝑁 + 1) ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1))
3	1, 2	sylan2 593	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1))
4		fvexd 6873	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (Ack‘𝑀) ∈ V)
5	1	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ0)
6		eqidd 2730	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) = ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)))
7		itcovalsucov 48657	. . . . 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 1373	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1)) = ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))))
9	8	fveq1d 6860	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1) = (((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)))‘1))
10		ackfnnn0 48674	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀) Fn ℕ0)
11	10	adantr 480	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (Ack‘𝑀) Fn ℕ0)
12		nn0ex 12448	. . . . . . . . 9 ⊢ ℕ0 ∈ V
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ℕ0 ∈ V)
14		ackendofnn0 48673	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀):ℕ0⟶ℕ0)
15	14	adantr 480	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (Ack‘𝑀):ℕ0⟶ℕ0)
16		simpr 484	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
17	13, 15, 16	itcovalendof 48658	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘𝑁):ℕ0⟶ℕ0)
18	17	ffnd 6689	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘𝑁) Fn ℕ0)
19	17	frnd 6696	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ran ((IterComp‘(Ack‘𝑀))‘𝑁) ⊆ ℕ0)
20		fnco 6636	. . . . . 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 1373	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘𝑁)) Fn ℕ0)
22		eqidd 2730	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘𝑁) = ((IterComp‘(Ack‘𝑀))‘𝑁))
23		itcovalsucov 48657	. . . . . . 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 1373	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) = ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘𝑁)))
25	24	fneq1d 6611	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) Fn ℕ0 ↔ ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘𝑁)) Fn ℕ0))
26	21, 25	mpbird 257	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) Fn ℕ0)
27		1nn0 12458	. . . 4 ⊢ 1 ∈ ℕ0
28		fvco2 6958	. . . 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 586	. . 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 2764	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1) = ((Ack‘𝑀)‘(((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)))
31		ackvalsuc1 48668	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1))
32	31	eqcomd 2735	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1) = ((Ack‘(𝑀 + 1))‘𝑁))
33	32	fveq2d 6862	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘𝑀)‘(((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))
34	3, 30, 33	3eqtrd 2768	1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ⊆ wss 3914  ran crn 5639   ∘ ccom 5642   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  1c1 11069   + caddc 11071  ℕ₀cn0 12442  IterCompcitco 48646  Ackcack 48647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-z 12530  df-uz 12794  df-seq 13967  df-itco 48648  df-ack 48649

This theorem is referenced by:  ackval41a  48683  ackval42  48685