Theorem ackvalsuc1mpt 48528

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 48510	. . 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 6908	. 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 12918	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
4		id 22	. . . 4 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℕ0)
5		eqid 2735	. . . 4 ⊢ (𝑀 + 1) = (𝑀 + 1)
6	1	eqcomi 2744	. . . . . 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 6908	. . . . 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 2736	. . . . 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 12553	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ0 → 0 < (𝑀 + 1))
11	10	gt0ne0d 11825	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ≠ 0)
12	11	adantr 480	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → (𝑀 + 1) ≠ 0)
13		neeq1 3001	. . . . . . . . . 10 ⊢ (𝑖 = (𝑀 + 1) → (𝑖 ≠ 0 ↔ (𝑀 + 1) ≠ 0))
14	13	adantl 481	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → (𝑖 ≠ 0 ↔ (𝑀 + 1) ≠ 0))
15	12, 14	mpbird 257	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → 𝑖 ≠ 0)
16	15	neneqd 2943	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → ¬ 𝑖 = 0)
17	16	iffalsed 4542	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖) = 𝑖)
18		simpr 484	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → 𝑖 = (𝑀 + 1))
19	17, 18	eqtrd 2775	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖) = (𝑀 + 1))
20		peano2nn0 12564	. . . . 5 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ0)
21	9, 19, 20, 20	fvmptd 7023	. . . 4 ⊢ (𝑀 ∈ ℕ0 → ((𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖))‘(𝑀 + 1)) = (𝑀 + 1))
22	3, 4, 5, 8, 21	seqp1d 14056	. . 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 2736	. . . 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 6907	. . . . . . . 8 ⊢ (𝑓 = (Ack‘𝑀) → (IterComp‘𝑓) = (IterComp‘(Ack‘𝑀)))
25	24	fveq1d 6909	. . . . . . 7 ⊢ (𝑓 = (Ack‘𝑀) → ((IterComp‘𝑓)‘(𝑛 + 1)) = ((IterComp‘(Ack‘𝑀))‘(𝑛 + 1)))
26	25	fveq1d 6909	. . . . . 6 ⊢ (𝑓 = (Ack‘𝑀) → (((IterComp‘𝑓)‘(𝑛 + 1))‘1) = (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1))
27	26	mpteq2dv 5250	. . . . 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 6922	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀) ∈ V)
30		ovexd 7466	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ V)
31		nn0ex 12530	. . . . . 6 ⊢ ℕ0 ∈ V
32	31	mptex 7243	. . . . 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 7585	. . 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 2775	. 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 2787	1 ⊢ (𝑀 ∈ ℕ0 → (Ack‘(𝑀 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  Vcvv 3478  ifcif 4531   ↦ cmpt 5231  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  0cc0 11153  1c1 11154   + caddc 11156  ℕ₀cn0 12524  seqcseq 14039  IterCompcitco 48507  Ackcack 48508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-seq 14040  df-ack 48510

This theorem is referenced by:  ackvalsuc1  48529  ackval1  48531  ackval2  48532  ackval3  48533  ackendofnn0  48534