![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 | โข (๐ โ โ0 โ (!โ(๐ + 1)) = ((!โ๐) ยท (๐ + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 12420 | . 2 โข (๐ โ โ0 โ (๐ โ โ โจ ๐ = 0)) | |
2 | peano2nn 12170 | . . . . 5 โข (๐ โ โ โ (๐ + 1) โ โ) | |
3 | facnn 14181 | . . . . 5 โข ((๐ + 1) โ โ โ (!โ(๐ + 1)) = (seq1( ยท , I )โ(๐ + 1))) | |
4 | 2, 3 | syl 17 | . . . 4 โข (๐ โ โ โ (!โ(๐ + 1)) = (seq1( ยท , I )โ(๐ + 1))) |
5 | ovex 7391 | . . . . . . 7 โข (๐ + 1) โ V | |
6 | fvi 6918 | . . . . . . 7 โข ((๐ + 1) โ V โ ( I โ(๐ + 1)) = (๐ + 1)) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 โข ( I โ(๐ + 1)) = (๐ + 1) |
8 | 7 | oveq2i 7369 | . . . . 5 โข ((seq1( ยท , I )โ๐) ยท ( I โ(๐ + 1))) = ((seq1( ยท , I )โ๐) ยท (๐ + 1)) |
9 | seqp1 13927 | . . . . . 6 โข (๐ โ (โคโฅโ1) โ (seq1( ยท , I )โ(๐ + 1)) = ((seq1( ยท , I )โ๐) ยท ( I โ(๐ + 1)))) | |
10 | nnuz 12811 | . . . . . 6 โข โ = (โคโฅโ1) | |
11 | 9, 10 | eleq2s 2852 | . . . . 5 โข (๐ โ โ โ (seq1( ยท , I )โ(๐ + 1)) = ((seq1( ยท , I )โ๐) ยท ( I โ(๐ + 1)))) |
12 | facnn 14181 | . . . . . 6 โข (๐ โ โ โ (!โ๐) = (seq1( ยท , I )โ๐)) | |
13 | 12 | oveq1d 7373 | . . . . 5 โข (๐ โ โ โ ((!โ๐) ยท (๐ + 1)) = ((seq1( ยท , I )โ๐) ยท (๐ + 1))) |
14 | 8, 11, 13 | 3eqtr4a 2799 | . . . 4 โข (๐ โ โ โ (seq1( ยท , I )โ(๐ + 1)) = ((!โ๐) ยท (๐ + 1))) |
15 | 4, 14 | eqtrd 2773 | . . 3 โข (๐ โ โ โ (!โ(๐ + 1)) = ((!โ๐) ยท (๐ + 1))) |
16 | 0p1e1 12280 | . . . . . 6 โข (0 + 1) = 1 | |
17 | 16 | fveq2i 6846 | . . . . 5 โข (!โ(0 + 1)) = (!โ1) |
18 | fac1 14183 | . . . . 5 โข (!โ1) = 1 | |
19 | 17, 18 | eqtri 2761 | . . . 4 โข (!โ(0 + 1)) = 1 |
20 | fvoveq1 7381 | . . . 4 โข (๐ = 0 โ (!โ(๐ + 1)) = (!โ(0 + 1))) | |
21 | fveq2 6843 | . . . . . 6 โข (๐ = 0 โ (!โ๐) = (!โ0)) | |
22 | oveq1 7365 | . . . . . 6 โข (๐ = 0 โ (๐ + 1) = (0 + 1)) | |
23 | 21, 22 | oveq12d 7376 | . . . . 5 โข (๐ = 0 โ ((!โ๐) ยท (๐ + 1)) = ((!โ0) ยท (0 + 1))) |
24 | fac0 14182 | . . . . . . 7 โข (!โ0) = 1 | |
25 | 24, 16 | oveq12i 7370 | . . . . . 6 โข ((!โ0) ยท (0 + 1)) = (1 ยท 1) |
26 | 1t1e1 12320 | . . . . . 6 โข (1 ยท 1) = 1 | |
27 | 25, 26 | eqtri 2761 | . . . . 5 โข ((!โ0) ยท (0 + 1)) = 1 |
28 | 23, 27 | eqtrdi 2789 | . . . 4 โข (๐ = 0 โ ((!โ๐) ยท (๐ + 1)) = 1) |
29 | 19, 20, 28 | 3eqtr4a 2799 | . . 3 โข (๐ = 0 โ (!โ(๐ + 1)) = ((!โ๐) ยท (๐ + 1))) |
30 | 15, 29 | jaoi 856 | . 2 โข ((๐ โ โ โจ ๐ = 0) โ (!โ(๐ + 1)) = ((!โ๐) ยท (๐ + 1))) |
31 | 1, 30 | sylbi 216 | 1 โข (๐ โ โ0 โ (!โ(๐ + 1)) = ((!โ๐) ยท (๐ + 1))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โจ wo 846 = wceq 1542 โ wcel 2107 Vcvv 3444 I cid 5531 โcfv 6497 (class class class)co 7358 0cc0 11056 1c1 11057 + caddc 11059 ยท cmul 11061 โcn 12158 โ0cn0 12418 โคโฅcuz 12768 seqcseq 13912 !cfa 14179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-seq 13913 df-fac 14180 |
This theorem is referenced by: fac2 14185 fac3 14186 fac4 14187 facnn2 14188 faccl 14189 facdiv 14193 facwordi 14195 faclbnd 14196 faclbnd6 14205 facubnd 14206 bcm1k 14221 bcp1n 14222 4bc2eq6 14235 efcllem 15965 ef01bndlem 16071 eirrlem 16091 dvdsfac 16213 prmfac1 16602 pcfac 16776 2expltfac 16970 aaliou3lem2 25719 aaliou3lem8 25721 dvtaylp 25745 advlogexp 26026 facgam 26431 bcmono 26641 ex-fac 29437 subfacval2 33838 subfaclim 33839 faclim 34375 faclim2 34377 lcmineqlem18 40549 facp2 40597 fac2xp3 40658 factwoffsmonot 40661 bccp1k 42709 binomcxplemwb 42716 wallispi2lem2 44399 stirlinglem4 44404 etransclem24 44585 etransclem28 44589 etransclem38 44599 |
Copyright terms: Public domain | W3C validator |