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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 11900 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | peano2nn 11650 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
3 | facnn 13636 | . . . . 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 7189 | . . . . . . 7 ⊢ (𝑁 + 1) ∈ V | |
6 | fvi 6740 | . . . . . . 7 ⊢ ((𝑁 + 1) ∈ V → ( I ‘(𝑁 + 1)) = (𝑁 + 1)) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 ⊢ ( I ‘(𝑁 + 1)) = (𝑁 + 1) |
8 | 7 | oveq2i 7167 | . . . . 5 ⊢ ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1))) = ((seq1( · , I )‘𝑁) · (𝑁 + 1)) |
9 | seqp1 13385 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘1) → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1)))) | |
10 | nnuz 12282 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
11 | 9, 10 | eleq2s 2931 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1)))) |
12 | facnn 13636 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) | |
13 | 12 | oveq1d 7171 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) · (𝑁 + 1)) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
14 | 8, 11, 13 | 3eqtr4a 2882 | . . . 4 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
15 | 4, 14 | eqtrd 2856 | . . 3 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
16 | 0p1e1 11760 | . . . . . 6 ⊢ (0 + 1) = 1 | |
17 | 16 | fveq2i 6673 | . . . . 5 ⊢ (!‘(0 + 1)) = (!‘1) |
18 | fac1 13638 | . . . . 5 ⊢ (!‘1) = 1 | |
19 | 17, 18 | eqtri 2844 | . . . 4 ⊢ (!‘(0 + 1)) = 1 |
20 | fvoveq1 7179 | . . . 4 ⊢ (𝑁 = 0 → (!‘(𝑁 + 1)) = (!‘(0 + 1))) | |
21 | fveq2 6670 | . . . . . 6 ⊢ (𝑁 = 0 → (!‘𝑁) = (!‘0)) | |
22 | oveq1 7163 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
23 | 21, 22 | oveq12d 7174 | . . . . 5 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = ((!‘0) · (0 + 1))) |
24 | fac0 13637 | . . . . . . 7 ⊢ (!‘0) = 1 | |
25 | 24, 16 | oveq12i 7168 | . . . . . 6 ⊢ ((!‘0) · (0 + 1)) = (1 · 1) |
26 | 1t1e1 11800 | . . . . . 6 ⊢ (1 · 1) = 1 | |
27 | 25, 26 | eqtri 2844 | . . . . 5 ⊢ ((!‘0) · (0 + 1)) = 1 |
28 | 23, 27 | syl6eq 2872 | . . . 4 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = 1) |
29 | 19, 20, 28 | 3eqtr4a 2882 | . . 3 ⊢ (𝑁 = 0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
30 | 15, 29 | jaoi 853 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
31 | 1, 30 | sylbi 219 | 1 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1537 ∈ wcel 2114 Vcvv 3494 I cid 5459 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ℕcn 11638 ℕ0cn0 11898 ℤ≥cuz 12244 seqcseq 13370 !cfa 13634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 df-fac 13635 |
This theorem is referenced by: fac2 13640 fac3 13641 fac4 13642 facnn2 13643 faccl 13644 facdiv 13648 facwordi 13650 faclbnd 13651 faclbnd6 13660 facubnd 13661 bcm1k 13676 bcp1n 13677 4bc2eq6 13690 efcllem 15431 ef01bndlem 15537 eirrlem 15557 dvdsfac 15676 prmfac1 16063 pcfac 16235 2expltfac 16426 aaliou3lem2 24932 aaliou3lem8 24934 dvtaylp 24958 advlogexp 25238 facgam 25643 bcmono 25853 ex-fac 28230 subfacval2 32434 subfaclim 32435 faclim 32978 faclim2 32980 fac2xp3 39114 facp2 39115 factwoffsmonot 39118 bccp1k 40693 binomcxplemwb 40700 wallispi2lem2 42377 stirlinglem4 42382 etransclem24 42563 etransclem28 42567 etransclem38 42577 |
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