Theorem ackvalsuc1mpt 49336

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 49318	. . 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 6880	. 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 12896	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
4		id 23	. . . 4 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℕ0)
5		eqid 2769	. . . 4 ⊢ (𝑀 + 1) = (𝑀 + 1)
6	1	eqcomi 2778	. . . . . 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 6880	. . . . 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 2770	. . . . 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 12529	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ0 → 0 < (𝑀 + 1))
11	10	gt0ne0d 11774	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ≠ 0)
12	11	adantr 485	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → (𝑀 + 1) ≠ 0)
13		neeq1 3026	. . . . . . . . . 10 ⊢ (𝑖 = (𝑀 + 1) → (𝑖 ≠ 0 ↔ (𝑀 + 1) ≠ 0))
14	13	adantl 486	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → (𝑖 ≠ 0 ↔ (𝑀 + 1) ≠ 0))
15	12, 14	mpbird 260	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → 𝑖 ≠ 0)
16	15	neneqd 2969	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → ¬ 𝑖 = 0)
17	16	iffalsed 4500	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖) = 𝑖)
18		simpr 489	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → 𝑖 = (𝑀 + 1))
19	17, 18	eqtrd 2804	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖) = (𝑀 + 1))
20		peano2nn0 12540	. . . . 5 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ0)
21	9, 19, 20, 20	fvmptd 6995	. . . 4 ⊢ (𝑀 ∈ ℕ0 → ((𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖))‘(𝑀 + 1)) = (𝑀 + 1))
22	3, 4, 5, 8, 21	seqp1d 14050	. . 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 2770	. . . 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 6879	. . . . . . . 8 ⊢ (𝑓 = (Ack‘𝑀) → (IterComp‘𝑓) = (IterComp‘(Ack‘𝑀)))
25	24	fveq1d 6881	. . . . . . 7 ⊢ (𝑓 = (Ack‘𝑀) → ((IterComp‘𝑓)‘(𝑛 + 1)) = ((IterComp‘(Ack‘𝑀))‘(𝑛 + 1)))
26	25	fveq1d 6881	. . . . . 6 ⊢ (𝑓 = (Ack‘𝑀) → (((IterComp‘𝑓)‘(𝑛 + 1))‘1) = (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1))
27	26	mpteq2dv 5206	. . . . 5 ⊢ (𝑓 = (Ack‘𝑀) → (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))
28	27	ad2antrl 740	. . . 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 6894	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀) ∈ V)
30		ovexd 7443	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ V)
31		nn0ex 12506	. . . . . 6 ⊢ ℕ0 ∈ V
32	31	mptex 7219	. . . . 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 7560	. . 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 2804	. 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 2816	1 ⊢ (𝑀 ∈ ℕ0 → (Ack‘(𝑀 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463  ifcif 4489   ↦ cmpt 5193  ‘cfv 6533  (class class class)co 7408   ∈ cmpo 7410  0cc0 11096  1c1 11097   + caddc 11099  ℕ₀cn0 12500  seqcseq 14033  IterCompcitco 49315  Ackcack 49316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-n0 12501  df-z 12588  df-uz 12859  df-seq 14034  df-ack 49318

This theorem is referenced by:  ackvalsuc1  49337  ackval1  49339  ackval2  49340  ackval3  49341  ackendofnn0  49342