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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 11902 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | peano2nn 11652 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
3 | facnn 13638 | . . . . 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 7191 | . . . . . . 7 ⊢ (𝑁 + 1) ∈ V | |
6 | fvi 6742 | . . . . . . 7 ⊢ ((𝑁 + 1) ∈ V → ( I ‘(𝑁 + 1)) = (𝑁 + 1)) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 ⊢ ( I ‘(𝑁 + 1)) = (𝑁 + 1) |
8 | 7 | oveq2i 7169 | . . . . 5 ⊢ ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1))) = ((seq1( · , I )‘𝑁) · (𝑁 + 1)) |
9 | seqp1 13387 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘1) → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1)))) | |
10 | nnuz 12284 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
11 | 9, 10 | eleq2s 2933 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1)))) |
12 | facnn 13638 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) | |
13 | 12 | oveq1d 7173 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) · (𝑁 + 1)) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
14 | 8, 11, 13 | 3eqtr4a 2884 | . . . 4 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
15 | 4, 14 | eqtrd 2858 | . . 3 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
16 | 0p1e1 11762 | . . . . . 6 ⊢ (0 + 1) = 1 | |
17 | 16 | fveq2i 6675 | . . . . 5 ⊢ (!‘(0 + 1)) = (!‘1) |
18 | fac1 13640 | . . . . 5 ⊢ (!‘1) = 1 | |
19 | 17, 18 | eqtri 2846 | . . . 4 ⊢ (!‘(0 + 1)) = 1 |
20 | fvoveq1 7181 | . . . 4 ⊢ (𝑁 = 0 → (!‘(𝑁 + 1)) = (!‘(0 + 1))) | |
21 | fveq2 6672 | . . . . . 6 ⊢ (𝑁 = 0 → (!‘𝑁) = (!‘0)) | |
22 | oveq1 7165 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
23 | 21, 22 | oveq12d 7176 | . . . . 5 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = ((!‘0) · (0 + 1))) |
24 | fac0 13639 | . . . . . . 7 ⊢ (!‘0) = 1 | |
25 | 24, 16 | oveq12i 7170 | . . . . . 6 ⊢ ((!‘0) · (0 + 1)) = (1 · 1) |
26 | 1t1e1 11802 | . . . . . 6 ⊢ (1 · 1) = 1 | |
27 | 25, 26 | eqtri 2846 | . . . . 5 ⊢ ((!‘0) · (0 + 1)) = 1 |
28 | 23, 27 | syl6eq 2874 | . . . 4 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = 1) |
29 | 19, 20, 28 | 3eqtr4a 2884 | . . 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 3496 I cid 5461 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 ℕcn 11640 ℕ0cn0 11900 ℤ≥cuz 12246 seqcseq 13372 !cfa 13636 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-seq 13373 df-fac 13637 |
This theorem is referenced by: fac2 13642 fac3 13643 fac4 13644 facnn2 13645 faccl 13646 facdiv 13650 facwordi 13652 faclbnd 13653 faclbnd6 13662 facubnd 13663 bcm1k 13678 bcp1n 13679 4bc2eq6 13692 efcllem 15433 ef01bndlem 15539 eirrlem 15559 dvdsfac 15678 prmfac1 16065 pcfac 16237 2expltfac 16428 aaliou3lem2 24934 aaliou3lem8 24936 dvtaylp 24960 advlogexp 25240 facgam 25645 bcmono 25855 ex-fac 28232 subfacval2 32436 subfaclim 32437 faclim 32980 faclim2 32982 fac2xp3 39101 facp2 39102 factwoffsmonot 39105 bccp1k 40680 binomcxplemwb 40687 wallispi2lem2 42364 stirlinglem4 42369 etransclem24 42550 etransclem28 42554 etransclem38 42564 |
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