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Mirrors > Home > MPE Home > Th. List > Mathboxes > facp2 | Structured version Visualization version GIF version |
Description: The factorial of a successor's successor. (Contributed by metakunt, 19-Apr-2024.) |
Ref | Expression |
---|---|
facp2 | ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 2)) = ((!‘𝑁) · ((𝑁 + 1) · (𝑁 + 2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0cn 11901 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
2 | ax-1cn 10588 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
3 | addass 10617 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1))) | |
4 | 2, 2, 3 | mp3an23 1448 | . . . . . . . 8 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1))) |
5 | 1, 4 | syl 17 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1))) |
6 | df-2 11694 | . . . . . . . . . 10 ⊢ 2 = (1 + 1) | |
7 | 6 | oveq2i 7160 | . . . . . . . . 9 ⊢ (𝑁 + 2) = (𝑁 + (1 + 1)) |
8 | 7 | eqcomi 2829 | . . . . . . . 8 ⊢ (𝑁 + (1 + 1)) = (𝑁 + 2) |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + (1 + 1)) = (𝑁 + 2)) |
10 | 5, 9 | eqtrd 2855 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + 2)) |
11 | 10 | fveq2d 6667 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (!‘((𝑁 + 1) + 1)) = (!‘(𝑁 + 2))) |
12 | peano2nn0 11931 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
13 | facp1 13635 | . . . . . 6 ⊢ ((𝑁 + 1) ∈ ℕ0 → (!‘((𝑁 + 1) + 1)) = ((!‘(𝑁 + 1)) · ((𝑁 + 1) + 1))) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (!‘((𝑁 + 1) + 1)) = ((!‘(𝑁 + 1)) · ((𝑁 + 1) + 1))) |
15 | 11, 14 | eqtr3d 2857 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 2)) = ((!‘(𝑁 + 1)) · ((𝑁 + 1) + 1))) |
16 | 10 | oveq2d 7165 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((!‘(𝑁 + 1)) · ((𝑁 + 1) + 1)) = ((!‘(𝑁 + 1)) · (𝑁 + 2))) |
17 | 15, 16 | eqtrd 2855 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 2)) = ((!‘(𝑁 + 1)) · (𝑁 + 2))) |
18 | facp1 13635 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) | |
19 | 18 | oveq1d 7164 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((!‘(𝑁 + 1)) · (𝑁 + 2)) = (((!‘𝑁) · (𝑁 + 1)) · (𝑁 + 2))) |
20 | 17, 19 | eqtrd 2855 | . 2 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 2)) = (((!‘𝑁) · (𝑁 + 1)) · (𝑁 + 2))) |
21 | faccl 13640 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
22 | nncn 11639 | . . . 4 ⊢ ((!‘𝑁) ∈ ℕ → (!‘𝑁) ∈ ℂ) | |
23 | 21, 22 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℂ) |
24 | nn0cn 11901 | . . . 4 ⊢ ((𝑁 + 1) ∈ ℕ0 → (𝑁 + 1) ∈ ℂ) | |
25 | 12, 24 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ) |
26 | 2cn 11706 | . . . . 5 ⊢ 2 ∈ ℂ | |
27 | addcl 10612 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ) → (𝑁 + 2) ∈ ℂ) | |
28 | 26, 27 | mpan2 689 | . . . 4 ⊢ (𝑁 ∈ ℂ → (𝑁 + 2) ∈ ℂ) |
29 | 1, 28 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 2) ∈ ℂ) |
30 | mulass 10618 | . . 3 ⊢ (((!‘𝑁) ∈ ℂ ∧ (𝑁 + 1) ∈ ℂ ∧ (𝑁 + 2) ∈ ℂ) → (((!‘𝑁) · (𝑁 + 1)) · (𝑁 + 2)) = ((!‘𝑁) · ((𝑁 + 1) · (𝑁 + 2)))) | |
31 | 23, 25, 29, 30 | syl3anc 1366 | . 2 ⊢ (𝑁 ∈ ℕ0 → (((!‘𝑁) · (𝑁 + 1)) · (𝑁 + 2)) = ((!‘𝑁) · ((𝑁 + 1) · (𝑁 + 2)))) |
32 | 20, 31 | eqtrd 2855 | 1 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 2)) = ((!‘𝑁) · ((𝑁 + 1) · (𝑁 + 2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 ℂcc 10528 1c1 10531 + caddc 10533 · cmul 10535 ℕcn 11631 2c2 11686 ℕ0cn0 11891 !cfa 13630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-seq 13367 df-fac 13631 |
This theorem is referenced by: (None) |
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