Theorem ackvalsuc1mpt 49166

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 49148	. . 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 6835	. 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 2737	. . . 4 ⊢ (𝑀 + 1) = (𝑀 + 1)
6	1	eqcomi 2746	. . . . . 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 6835	. . . . 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 2738	. . . . 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 480	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → (𝑀 + 1) ≠ 0)
13		neeq1 2995	. . . . . . . . . 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 2938	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑖 = (𝑀 + 1)) → ¬ 𝑖 = 0)
17	16	iffalsed 4478	. . . . . 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 2772	. . . . 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 6949	. . . 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 2738	. . . 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 6834	. . . . . . . 8 ⊢ (𝑓 = (Ack‘𝑀) → (IterComp‘𝑓) = (IterComp‘(Ack‘𝑀)))
25	24	fveq1d 6836	. . . . . . 7 ⊢ (𝑓 = (Ack‘𝑀) → ((IterComp‘𝑓)‘(𝑛 + 1)) = ((IterComp‘(Ack‘𝑀))‘(𝑛 + 1)))
26	25	fveq1d 6836	. . . . . 6 ⊢ (𝑓 = (Ack‘𝑀) → (((IterComp‘𝑓)‘(𝑛 + 1))‘1) = (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1))
27	26	mpteq2dv 5180	. . . . 5 ⊢ (𝑓 = (Ack‘𝑀) → (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))
28	27	ad2antrl 729	. . . 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 6849	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀) ∈ V)
30		ovexd 7395	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ V)
31		nn0ex 12434	. . . . . 6 ⊢ ℕ0 ∈ V
32	31	mptex 7171	. . . . 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 7512	. . 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 2772	. 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 2784	1 ⊢ (𝑀 ∈ ℕ0 → (Ack‘(𝑀 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3430  ifcif 4467   ↦ cmpt 5167  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  0cc0 11029  1c1 11030   + caddc 11032  ℕ₀cn0 12428  seqcseq 13954  IterCompcitco 49145  Ackcack 49146

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 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  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 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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  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 49148

This theorem is referenced by:  ackvalsuc1  49167  ackval1  49169  ackval2  49170  ackval3  49171  ackendofnn0  49172