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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 12481 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | peano2nn 12231 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
3 | facnn 14242 | . . . . 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 7445 | . . . . . . 7 ⊢ (𝑁 + 1) ∈ V | |
6 | fvi 6967 | . . . . . . 7 ⊢ ((𝑁 + 1) ∈ V → ( I ‘(𝑁 + 1)) = (𝑁 + 1)) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 ⊢ ( I ‘(𝑁 + 1)) = (𝑁 + 1) |
8 | 7 | oveq2i 7423 | . . . . 5 ⊢ ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1))) = ((seq1( · , I )‘𝑁) · (𝑁 + 1)) |
9 | seqp1 13988 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘1) → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1)))) | |
10 | nnuz 12872 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
11 | 9, 10 | eleq2s 2850 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1)))) |
12 | facnn 14242 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) | |
13 | 12 | oveq1d 7427 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) · (𝑁 + 1)) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
14 | 8, 11, 13 | 3eqtr4a 2797 | . . . 4 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
15 | 4, 14 | eqtrd 2771 | . . 3 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
16 | 0p1e1 12341 | . . . . . 6 ⊢ (0 + 1) = 1 | |
17 | 16 | fveq2i 6894 | . . . . 5 ⊢ (!‘(0 + 1)) = (!‘1) |
18 | fac1 14244 | . . . . 5 ⊢ (!‘1) = 1 | |
19 | 17, 18 | eqtri 2759 | . . . 4 ⊢ (!‘(0 + 1)) = 1 |
20 | fvoveq1 7435 | . . . 4 ⊢ (𝑁 = 0 → (!‘(𝑁 + 1)) = (!‘(0 + 1))) | |
21 | fveq2 6891 | . . . . . 6 ⊢ (𝑁 = 0 → (!‘𝑁) = (!‘0)) | |
22 | oveq1 7419 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
23 | 21, 22 | oveq12d 7430 | . . . . 5 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = ((!‘0) · (0 + 1))) |
24 | fac0 14243 | . . . . . . 7 ⊢ (!‘0) = 1 | |
25 | 24, 16 | oveq12i 7424 | . . . . . 6 ⊢ ((!‘0) · (0 + 1)) = (1 · 1) |
26 | 1t1e1 12381 | . . . . . 6 ⊢ (1 · 1) = 1 | |
27 | 25, 26 | eqtri 2759 | . . . . 5 ⊢ ((!‘0) · (0 + 1)) = 1 |
28 | 23, 27 | eqtrdi 2787 | . . . 4 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = 1) |
29 | 19, 20, 28 | 3eqtr4a 2797 | . . 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 216 | 1 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1540 ∈ wcel 2105 Vcvv 3473 I cid 5573 ‘cfv 6543 (class class class)co 7412 0cc0 11116 1c1 11117 + caddc 11119 · cmul 11121 ℕcn 12219 ℕ0cn0 12479 ℤ≥cuz 12829 seqcseq 13973 !cfa 14240 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-seq 13974 df-fac 14241 |
This theorem is referenced by: fac2 14246 fac3 14247 fac4 14248 facnn2 14249 faccl 14250 facdiv 14254 facwordi 14256 faclbnd 14257 faclbnd6 14266 facubnd 14267 bcm1k 14282 bcp1n 14283 4bc2eq6 14296 efcllem 16028 ef01bndlem 16134 eirrlem 16154 dvdsfac 16276 prmfac1 16665 pcfac 16839 2expltfac 17033 aaliou3lem2 26195 aaliou3lem8 26197 dvtaylp 26221 advlogexp 26503 facgam 26911 bcmono 27123 ex-fac 30137 subfacval2 34642 subfaclim 34643 faclim 35186 faclim2 35188 lcmineqlem18 41378 facp2 41426 fac2xp3 41487 factwoffsmonot 41490 bccp1k 43563 binomcxplemwb 43570 wallispi2lem2 45247 stirlinglem4 45252 etransclem24 45433 etransclem28 45437 etransclem38 45447 |
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