Theorem ackvalsucsucval 49164

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 12477	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
2		ackvalsuc1 49155	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ (𝑁 + 1) ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1))
3	1, 2	sylan2 594	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1))
4		fvexd 6855	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (Ack‘𝑀) ∈ V)
5	1	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ0)
6		eqidd 2737	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) = ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)))
7		itcovalsucov 49144	. . . . 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 1374	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1)) = ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))))
9	8	fveq1d 6842	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1) = (((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)))‘1))
10		ackfnnn0 49161	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀) Fn ℕ0)
11	10	adantr 480	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (Ack‘𝑀) Fn ℕ0)
12		nn0ex 12443	. . . . . . . . 9 ⊢ ℕ0 ∈ V
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ℕ0 ∈ V)
14		ackendofnn0 49160	. . . . . . . . 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 49145	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘𝑁):ℕ0⟶ℕ0)
18	17	ffnd 6669	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘𝑁) Fn ℕ0)
19	17	frnd 6676	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ran ((IterComp‘(Ack‘𝑀))‘𝑁) ⊆ ℕ0)
20		fnco 6616	. . . . . 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 1374	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘𝑁)) Fn ℕ0)
22		eqidd 2737	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘𝑁) = ((IterComp‘(Ack‘𝑀))‘𝑁))
23		itcovalsucov 49144	. . . . . . 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 1374	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘(Ack‘𝑀))‘(𝑁 + 1)) = ((Ack‘𝑀) ∘ ((IterComp‘(Ack‘𝑀))‘𝑁)))
25	24	fneq1d 6591	. . . . 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 12453	. . . 4 ⊢ 1 ∈ ℕ0
28		fvco2 6937	. . . 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 587	. . 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 2771	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘((𝑁 + 1) + 1))‘1) = ((Ack‘𝑀)‘(((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)))
31		ackvalsuc1 49155	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1))
32	31	eqcomd 2742	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1) = ((Ack‘(𝑀 + 1))‘𝑁))
33	32	fveq2d 6844	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘𝑀)‘(((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))
34	3, 30, 33	3eqtrd 2775	1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ⊆ wss 3889  ran crn 5632   ∘ ccom 5635   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  1c1 11039   + caddc 11041  ℕ₀cn0 12437  IterCompcitco 49133  Ackcack 49134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789  df-seq 13964  df-itco 49135  df-ack 49136

This theorem is referenced by:  ackval41a  49170  ackval42  49172