Theorem ackvalsuc1mpt 49169

Description: The Ackermann function at a successor of the first argument as a mapping of the second argument. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 4-May-2024.)

Assertion

Ref	Expression
ackvalsuc1mpt	⊢ (𝑀 ∈ ℕ0 → (Ack‘(𝑀 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))

Distinct variable group:   𝑛,𝑀

Proof of Theorem ackvalsuc1mpt

Dummy variables 𝑓 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ack 49151	. . 3 ⊢ Ack = seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖)))
2	1	fveq1i 6828	. 2 ⊢ (Ack‘(𝑀 + 1)) = (seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖)))‘(𝑀 + 1))
3		nn0uz 12817	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
4		id 22	. . . 4 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℕ0)
5		eqid 2739	. . . 4 ⊢ (𝑀 + 1) = (𝑀 + 1)
6	1	eqcomi 2748	. . . . . 6 ⊢ seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖))) = Ack
7	6	fveq1i 6828	. . . . 5 ⊢ (seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖)))‘𝑀) = (Ack‘𝑀)
8	7	a1i 11	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖)))‘𝑀) = (Ack‘𝑀))
9		eqidd 2740	. . . . 5 ⊢ (𝑀 ∈ ℕ0 → (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖)))
10		nn0p1gt0 12457	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ0 → 0 < (𝑀 + 1))
11	10	gt0ne0d 11705	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ≠ 0)
12	11	adantr 481	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → (𝑀 + 1) ≠ 0)
13		neeq1 2996	. . . . . . . . . 10 ⊢ (𝑖 = (𝑀 + 1) → (𝑖 ≠ 0 ↔ (𝑀 + 1) ≠ 0))
14	13	adantl 482	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → (𝑖 ≠ 0 ↔ (𝑀 + 1) ≠ 0))
15	12, 14	mpbird 258	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → 𝑖 ≠ 0)
16	15	neneqd 2939	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → ¬ 𝑖 = 0)
17	16	iffalsed 4465	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖) = 𝑖)
18		simpr 485	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → 𝑖 = (𝑀 + 1))
19	17, 18	eqtrd 2774	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖) = (𝑀 + 1))
20		peano2nn0 12468	. . . . 5 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ0)
21	9, 19, 20, 20	fvmptd 6943	. . . 4 ⊢ (𝑀 ∈ ℕ0 → ((𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖))‘(𝑀 + 1)) = (𝑀 + 1))
22	3, 4, 5, 8, 21	seqp1d 13971	. . 3 ⊢ (𝑀 ∈ ℕ0 → (seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖)))‘(𝑀 + 1)) = ((Ack‘𝑀)(𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1)))(𝑀 + 1)))
23		eqidd 2740	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))) = (𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))))
24		fveq2 6827	. . . . . . . 8 ⊢ (𝑓 = (Ack‘𝑀) → (IterComp‘𝑓) = (IterComp‘(Ack‘𝑀)))
25	24	fveq1d 6829	. . . . . . 7 ⊢ (𝑓 = (Ack‘𝑀) → ((IterComp‘𝑓)‘(𝑛 + 1)) = ((IterComp‘(Ack‘𝑀))‘(𝑛 + 1)))
26	25	fveq1d 6829	. . . . . 6 ⊢ (𝑓 = (Ack‘𝑀) → (((IterComp‘𝑓)‘(𝑛 + 1))‘1) = (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1))
27	26	mpteq2dv 5166	. . . . 5 ⊢ (𝑓 = (Ack‘𝑀) → (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))
28	27	ad2antrl 734	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ (𝑓 = (Ack‘𝑀) ∧ 𝑗 = (𝑀 + 1))) → (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))
29		fvexd 6842	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀) ∈ V)
30		ovexd 7391	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ V)
31		nn0ex 12434	. . . . . 6 ⊢ ℕ0 ∈ V
32	31	mptex 7167	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)) ∈ V
33	32	a1i 11	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)) ∈ V)
34	23, 28, 29, 30, 33	ovmpod 7508	. . 3 ⊢ (𝑀 ∈ ℕ0 → ((Ack‘𝑀)(𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1)))(𝑀 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))
35	22, 34	eqtrd 2774	. 2 ⊢ (𝑀 ∈ ℕ0 → (seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖)))‘(𝑀 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))
36	2, 35	eqtrid 2786	1 ⊢ (𝑀 ∈ ℕ0 → (Ack‘(𝑀 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  Vcvv 3431  ifcif 4454   ↦ cmpt 5153  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  0cc0 11029  1c1 11030   + caddc 11032  ℕ₀cn0 12428  seqcseq 13954  IterCompcitco 49148  Ackcack 49149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-seq 13955  df-ack 49151

This theorem is referenced by:  ackvalsuc1  49170  ackval1  49172  ackval2  49173  ackval3  49174  ackendofnn0  49175