Theorem ackvalsucsucval 49744

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 12627	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
2		ackvalsuc1 49735	. . 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 6892	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (Ack‘𝑀) ∈ V)
5	1	adantl 487	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ0)
6		eqidd 2762	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) = ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)))
7		itcovalsucov 49724	. . . . 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 1398	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1)) = ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))))
9	8	fveq1d 6879	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1) = (((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)))‘1))
10		ackfnnn0 49741	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀) Fn ℕ0)
11	10	adantr 486	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (Ack‘𝑀) Fn ℕ0)
12		nn0ex 12593	. . . . . . . . 9 ⊢ ℕ0 ∈ V
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ℕ0 ∈ V)
14		ackendofnn0 49740	. . . . . . . . 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 49725	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘𝑁):ℕ0⟶ℕ0)
18	17	ffnd 6702	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘𝑁) Fn ℕ0)
19	17	frnd 6710	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ran ((IterComp‘(Ack‘𝑀))‘𝑁) ⊆ ℕ0)
20		fnco 6649	. . . . . 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 1398	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘𝑁)) Fn ℕ0)
22		eqidd 2762	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘𝑁) = ((IterComp‘(Ack‘𝑀))‘𝑁))
23		itcovalsucov 49724	. . . . . . 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 1398	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) = ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘𝑁)))
25	24	fneq1d 6624	. . . . 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 12603	. . . 4 ⊢ 1 ∈ ℕ0
28		fvco2 6974	. . . 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 2796	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1) = ((Ack‘𝑀)‘(((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)))
31		ackvalsuc1 49735	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1))
32	31	eqcomd 2767	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1) = ((Ack‘(𝑀 + 1))‘𝑁))
33	32	fveq2d 6881	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘𝑀)‘(((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))
34	3, 30, 33	3eqtrd 2800	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 2145  Vcvv 3451   ⊆ wss 3899  ran crn 5652   ∘ ccom 5655   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  1c1 11182   + caddc 11184  ℕ₀cn0 12587  IterCompcitco 49713  Ackcack 49714

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  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 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-n0 12588  df-z 12675  df-uz 12947  df-seq 14125  df-itco 49715  df-ack 49716

This theorem is used by:  ackval41a  49750  ackval42  49752