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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ackval1 | Structured version Visualization version GIF version |
Description: The Ackermann function at 1. (Contributed by AV, 4-May-2024.) |
Ref | Expression |
---|---|
ackval1 | ⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1e0p1 12773 | . . 3 ⊢ 1 = (0 + 1) | |
2 | 1 | fveq2i 6910 | . 2 ⊢ (Ack‘1) = (Ack‘(0 + 1)) |
3 | 0nn0 12539 | . . 3 ⊢ 0 ∈ ℕ0 | |
4 | ackvalsuc1mpt 48528 | . . 3 ⊢ (0 ∈ ℕ0 → (Ack‘(0 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1))) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (Ack‘(0 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1)) |
6 | peano2nn0 12564 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0) | |
7 | 1nn0 12540 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
8 | ackval0 48530 | . . . . . . . 8 ⊢ (Ack‘0) = (𝑖 ∈ ℕ0 ↦ (𝑖 + 1)) | |
9 | 8 | itcovalpc 48522 | . . . . . . 7 ⊢ (((𝑛 + 1) ∈ ℕ0 ∧ 1 ∈ ℕ0) → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1))))) |
10 | 6, 7, 9 | sylancl 586 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1))))) |
11 | nn0cn 12534 | . . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ ℕ0 → (𝑛 + 1) ∈ ℂ) | |
12 | 6, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ) |
13 | 12 | mullidd 11277 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 → (1 · (𝑛 + 1)) = (𝑛 + 1)) |
14 | 13 | oveq2d 7447 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → (𝑖 + (1 · (𝑛 + 1))) = (𝑖 + (𝑛 + 1))) |
15 | 14 | mpteq2dv 5250 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1)))) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))) |
16 | 10, 15 | eqtrd 2775 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))) |
17 | 16 | fveq1d 6909 | . . . 4 ⊢ (𝑛 ∈ ℕ0 → (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1) = ((𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))‘1)) |
18 | eqidd 2736 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))) | |
19 | oveq1 7438 | . . . . . 6 ⊢ (𝑖 = 1 → (𝑖 + (𝑛 + 1)) = (1 + (𝑛 + 1))) | |
20 | 19 | adantl 481 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑖 = 1) → (𝑖 + (𝑛 + 1)) = (1 + (𝑛 + 1))) |
21 | 7 | a1i 11 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → 1 ∈ ℕ0) |
22 | ovexd 7466 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) ∈ V) | |
23 | 18, 20, 21, 22 | fvmptd 7023 | . . . 4 ⊢ (𝑛 ∈ ℕ0 → ((𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))‘1) = (1 + (𝑛 + 1))) |
24 | 1cnd 11254 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → 1 ∈ ℂ) | |
25 | nn0cn 12534 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℂ) | |
26 | peano2cn 11431 | . . . . . . 7 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ) | |
27 | 25, 26 | syl 17 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ) |
28 | 24, 27 | addcomd 11461 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) = ((𝑛 + 1) + 1)) |
29 | 25, 24, 24 | addassd 11281 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((𝑛 + 1) + 1) = (𝑛 + (1 + 1))) |
30 | 1p1e2 12389 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
31 | 30 | oveq2i 7442 | . . . . . 6 ⊢ (𝑛 + (1 + 1)) = (𝑛 + 2) |
32 | 31 | a1i 11 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + (1 + 1)) = (𝑛 + 2)) |
33 | 28, 29, 32 | 3eqtrd 2779 | . . . 4 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) = (𝑛 + 2)) |
34 | 17, 23, 33 | 3eqtrd 2779 | . . 3 ⊢ (𝑛 ∈ ℕ0 → (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1) = (𝑛 + 2)) |
35 | 34 | mpteq2ia 5251 | . 2 ⊢ (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) |
36 | 2, 5, 35 | 3eqtri 2767 | 1 ⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 2c2 12319 ℕ0cn0 12524 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-inf2 9679 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-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-seq 14040 df-itco 48509 df-ack 48510 |
This theorem is referenced by: ackval2 48532 ackval1012 48540 |
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