Theorem ackvalsuc1mpt 45442

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 45424	. . 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 6652	. 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 12305	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
4		id 22	. . . 4 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℕ0)
5		eqid 2759	. . . 4 ⊢ (𝑀 + 1) = (𝑀 + 1)
6	1	eqcomi 2768	. . . . . 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 6652	. . . . 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 2760	. . . . 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 11948	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ0 → 0 < (𝑀 + 1))
11	10	gt0ne0d 11227	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ≠ 0)
12	11	adantr 485	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → (𝑀 + 1) ≠ 0)
13		neeq1 3011	. . . . . . . . . 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 2954	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → ¬ 𝑖 = 0)
17	16	iffalsed 4424	. . . . . 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 2794	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖) = (𝑀 + 1))
20		peano2nn0 11959	. . . . 5 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ0)
21	9, 19, 20, 20	fvmptd 6759	. . . 4 ⊢ (𝑀 ∈ ℕ0 → ((𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖))‘(𝑀 + 1)) = (𝑀 + 1))
22	3, 4, 5, 8, 21	seqp1d 13420	. . 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 2760	. . . 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 6651	. . . . . . . 8 ⊢ (𝑓 = (Ack‘𝑀) → (IterComp‘𝑓) = (IterComp‘(Ack‘𝑀)))
25	24	fveq1d 6653	. . . . . . 7 ⊢ (𝑓 = (Ack‘𝑀) → ((IterComp‘𝑓)‘(𝑛 + 1)) = ((IterComp‘(Ack‘𝑀))‘(𝑛 + 1)))
26	25	fveq1d 6653	. . . . . 6 ⊢ (𝑓 = (Ack‘𝑀) → (((IterComp‘𝑓)‘(𝑛 + 1))‘1) = (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1))
27	26	mpteq2dv 5121	. . . . 5 ⊢ (𝑓 = (Ack‘𝑀) → (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))
28	27	ad2antrl 728	. . . 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 6666	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀) ∈ V)
30		ovexd 7178	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ V)
31		nn0ex 11925	. . . . . 6 ⊢ ℕ0 ∈ V
32	31	mptex 6970	. . . . 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 7290	. . 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 2794	. 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	syl5eq 2806	1 ⊢ (𝑀 ∈ ℕ0 → (Ack‘(𝑀 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ≠ wne 2949  Vcvv 3407  ifcif 4413   ↦ cmpt 5105  ‘cfv 6328  (class class class)co 7143   ∈ cmpo 7145  0cc0 10560  1c1 10561   + caddc 10563  ℕ₀cn0 11919  seqcseq 13403  IterCompcitco 45421  Ackcack 45422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5149  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452  ax-cnex 10616  ax-resscn 10617  ax-1cn 10618  ax-icn 10619  ax-addcl 10620  ax-addrcl 10621  ax-mulcl 10622  ax-mulrcl 10623  ax-mulcom 10624  ax-addass 10625  ax-mulass 10626  ax-distr 10627  ax-i2m1 10628  ax-1ne0 10629  ax-1rid 10630  ax-rnegex 10631  ax-rrecex 10632  ax-cnre 10633  ax-pre-lttri 10634  ax-pre-lttrn 10635  ax-pre-ltadd 10636  ax-pre-mulgt0 10637

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-nel 3054  df-ral 3073  df-rex 3074  df-reu 3075  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7101  df-ov 7146  df-oprab 7147  df-mpo 7148  df-om 7573  df-2nd 7687  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-er 8292  df-en 8521  df-dom 8522  df-sdom 8523  df-pnf 10700  df-mnf 10701  df-xr 10702  df-ltxr 10703  df-le 10704  df-sub 10895  df-neg 10896  df-nn 11660  df-n0 11920  df-z 12006  df-uz 12268  df-seq 13404  df-ack 45424

This theorem is referenced by:  ackvalsuc1  45443  ackval1  45445  ackval2  45446  ackval3  45447  ackendofnn0  45448