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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 | โข (๐ โ โ0 โ (!โ(๐ + 1)) = ((!โ๐) ยท (๐ + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 12505 | . 2 โข (๐ โ โ0 โ (๐ โ โ โจ ๐ = 0)) | |
2 | peano2nn 12255 | . . . . 5 โข (๐ โ โ โ (๐ + 1) โ โ) | |
3 | facnn 14267 | . . . . 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 7453 | . . . . . . 7 โข (๐ + 1) โ V | |
6 | fvi 6974 | . . . . . . 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 7431 | . . . . 5 โข ((seq1( ยท , I )โ๐) ยท ( I โ(๐ + 1))) = ((seq1( ยท , I )โ๐) ยท (๐ + 1)) |
9 | seqp1 14014 | . . . . . 6 โข (๐ โ (โคโฅโ1) โ (seq1( ยท , I )โ(๐ + 1)) = ((seq1( ยท , I )โ๐) ยท ( I โ(๐ + 1)))) | |
10 | nnuz 12896 | . . . . . 6 โข โ = (โคโฅโ1) | |
11 | 9, 10 | eleq2s 2847 | . . . . 5 โข (๐ โ โ โ (seq1( ยท , I )โ(๐ + 1)) = ((seq1( ยท , I )โ๐) ยท ( I โ(๐ + 1)))) |
12 | facnn 14267 | . . . . . 6 โข (๐ โ โ โ (!โ๐) = (seq1( ยท , I )โ๐)) | |
13 | 12 | oveq1d 7435 | . . . . 5 โข (๐ โ โ โ ((!โ๐) ยท (๐ + 1)) = ((seq1( ยท , I )โ๐) ยท (๐ + 1))) |
14 | 8, 11, 13 | 3eqtr4a 2794 | . . . 4 โข (๐ โ โ โ (seq1( ยท , I )โ(๐ + 1)) = ((!โ๐) ยท (๐ + 1))) |
15 | 4, 14 | eqtrd 2768 | . . 3 โข (๐ โ โ โ (!โ(๐ + 1)) = ((!โ๐) ยท (๐ + 1))) |
16 | 0p1e1 12365 | . . . . . 6 โข (0 + 1) = 1 | |
17 | 16 | fveq2i 6900 | . . . . 5 โข (!โ(0 + 1)) = (!โ1) |
18 | fac1 14269 | . . . . 5 โข (!โ1) = 1 | |
19 | 17, 18 | eqtri 2756 | . . . 4 โข (!โ(0 + 1)) = 1 |
20 | fvoveq1 7443 | . . . 4 โข (๐ = 0 โ (!โ(๐ + 1)) = (!โ(0 + 1))) | |
21 | fveq2 6897 | . . . . . 6 โข (๐ = 0 โ (!โ๐) = (!โ0)) | |
22 | oveq1 7427 | . . . . . 6 โข (๐ = 0 โ (๐ + 1) = (0 + 1)) | |
23 | 21, 22 | oveq12d 7438 | . . . . 5 โข (๐ = 0 โ ((!โ๐) ยท (๐ + 1)) = ((!โ0) ยท (0 + 1))) |
24 | fac0 14268 | . . . . . . 7 โข (!โ0) = 1 | |
25 | 24, 16 | oveq12i 7432 | . . . . . 6 โข ((!โ0) ยท (0 + 1)) = (1 ยท 1) |
26 | 1t1e1 12405 | . . . . . 6 โข (1 ยท 1) = 1 | |
27 | 25, 26 | eqtri 2756 | . . . . 5 โข ((!โ0) ยท (0 + 1)) = 1 |
28 | 23, 27 | eqtrdi 2784 | . . . 4 โข (๐ = 0 โ ((!โ๐) ยท (๐ + 1)) = 1) |
29 | 19, 20, 28 | 3eqtr4a 2794 | . . 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 1534 โ wcel 2099 Vcvv 3471 I cid 5575 โcfv 6548 (class class class)co 7420 0cc0 11139 1c1 11140 + caddc 11142 ยท cmul 11144 โcn 12243 โ0cn0 12503 โคโฅcuz 12853 seqcseq 13999 !cfa 14265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-seq 14000 df-fac 14266 |
This theorem is referenced by: fac2 14271 fac3 14272 fac4 14273 facnn2 14274 faccl 14275 facdiv 14279 facwordi 14281 faclbnd 14282 faclbnd6 14291 facubnd 14292 bcm1k 14307 bcp1n 14308 4bc2eq6 14321 efcllem 16054 ef01bndlem 16161 eirrlem 16181 dvdsfac 16303 prmfac1 16692 pcfac 16868 2expltfac 17062 aaliou3lem2 26291 aaliou3lem8 26293 dvtaylp 26318 advlogexp 26602 facgam 27011 bcmono 27223 ex-fac 30274 subfacval2 34797 subfaclim 34798 faclim 35340 faclim2 35342 lcmineqlem18 41517 facp2 41615 fac2xp3 41691 factwoffsmonot 41694 bccp1k 43778 binomcxplemwb 43785 wallispi2lem2 45460 stirlinglem4 45465 etransclem24 45646 etransclem28 45650 etransclem38 45660 |
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