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Mirrors > Home > MPE Home > Th. List > facp1 | Structured version Visualization version GIF version |
Description: The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Ref | Expression |
---|---|
facp1 | ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 11887 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | peano2nn 11637 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
3 | facnn 13631 | . . . . 5 ⊢ ((𝑁 + 1) ∈ ℕ → (!‘(𝑁 + 1)) = (seq1( · , I )‘(𝑁 + 1))) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 + 1)) = (seq1( · , I )‘(𝑁 + 1))) |
5 | ovex 7168 | . . . . . . 7 ⊢ (𝑁 + 1) ∈ V | |
6 | fvi 6715 | . . . . . . 7 ⊢ ((𝑁 + 1) ∈ V → ( I ‘(𝑁 + 1)) = (𝑁 + 1)) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 ⊢ ( I ‘(𝑁 + 1)) = (𝑁 + 1) |
8 | 7 | oveq2i 7146 | . . . . 5 ⊢ ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1))) = ((seq1( · , I )‘𝑁) · (𝑁 + 1)) |
9 | seqp1 13379 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘1) → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1)))) | |
10 | nnuz 12269 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
11 | 9, 10 | eleq2s 2908 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1)))) |
12 | facnn 13631 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) | |
13 | 12 | oveq1d 7150 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) · (𝑁 + 1)) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
14 | 8, 11, 13 | 3eqtr4a 2859 | . . . 4 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
15 | 4, 14 | eqtrd 2833 | . . 3 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
16 | 0p1e1 11747 | . . . . . 6 ⊢ (0 + 1) = 1 | |
17 | 16 | fveq2i 6648 | . . . . 5 ⊢ (!‘(0 + 1)) = (!‘1) |
18 | fac1 13633 | . . . . 5 ⊢ (!‘1) = 1 | |
19 | 17, 18 | eqtri 2821 | . . . 4 ⊢ (!‘(0 + 1)) = 1 |
20 | fvoveq1 7158 | . . . 4 ⊢ (𝑁 = 0 → (!‘(𝑁 + 1)) = (!‘(0 + 1))) | |
21 | fveq2 6645 | . . . . . 6 ⊢ (𝑁 = 0 → (!‘𝑁) = (!‘0)) | |
22 | oveq1 7142 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
23 | 21, 22 | oveq12d 7153 | . . . . 5 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = ((!‘0) · (0 + 1))) |
24 | fac0 13632 | . . . . . . 7 ⊢ (!‘0) = 1 | |
25 | 24, 16 | oveq12i 7147 | . . . . . 6 ⊢ ((!‘0) · (0 + 1)) = (1 · 1) |
26 | 1t1e1 11787 | . . . . . 6 ⊢ (1 · 1) = 1 | |
27 | 25, 26 | eqtri 2821 | . . . . 5 ⊢ ((!‘0) · (0 + 1)) = 1 |
28 | 23, 27 | eqtrdi 2849 | . . . 4 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = 1) |
29 | 19, 20, 28 | 3eqtr4a 2859 | . . 3 ⊢ (𝑁 = 0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
30 | 15, 29 | jaoi 854 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
31 | 1, 30 | sylbi 220 | 1 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1538 ∈ wcel 2111 Vcvv 3441 I cid 5424 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ℕcn 11625 ℕ0cn0 11885 ℤ≥cuz 12231 seqcseq 13364 !cfa 13629 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13365 df-fac 13630 |
This theorem is referenced by: fac2 13635 fac3 13636 fac4 13637 facnn2 13638 faccl 13639 facdiv 13643 facwordi 13645 faclbnd 13646 faclbnd6 13655 facubnd 13656 bcm1k 13671 bcp1n 13672 4bc2eq6 13685 efcllem 15423 ef01bndlem 15529 eirrlem 15549 dvdsfac 15668 prmfac1 16053 pcfac 16225 2expltfac 16418 aaliou3lem2 24939 aaliou3lem8 24941 dvtaylp 24965 advlogexp 25246 facgam 25651 bcmono 25861 ex-fac 28236 subfacval2 32547 subfaclim 32548 faclim 33091 faclim2 33093 lcmineqlem18 39334 facp2 39347 fac2xp3 39385 factwoffsmonot 39388 bccp1k 41045 binomcxplemwb 41052 wallispi2lem2 42714 stirlinglem4 42719 etransclem24 42900 etransclem28 42904 etransclem38 42914 |
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