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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 12473 | . 2 โข (๐ โ โ0 โ (๐ โ โ โจ ๐ = 0)) | |
2 | peano2nn 12223 | . . . . 5 โข (๐ โ โ โ (๐ + 1) โ โ) | |
3 | facnn 14236 | . . . . 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 7435 | . . . . . . 7 โข (๐ + 1) โ V | |
6 | fvi 6958 | . . . . . . 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 7413 | . . . . 5 โข ((seq1( ยท , I )โ๐) ยท ( I โ(๐ + 1))) = ((seq1( ยท , I )โ๐) ยท (๐ + 1)) |
9 | seqp1 13982 | . . . . . 6 โข (๐ โ (โคโฅโ1) โ (seq1( ยท , I )โ(๐ + 1)) = ((seq1( ยท , I )โ๐) ยท ( I โ(๐ + 1)))) | |
10 | nnuz 12864 | . . . . . 6 โข โ = (โคโฅโ1) | |
11 | 9, 10 | eleq2s 2843 | . . . . 5 โข (๐ โ โ โ (seq1( ยท , I )โ(๐ + 1)) = ((seq1( ยท , I )โ๐) ยท ( I โ(๐ + 1)))) |
12 | facnn 14236 | . . . . . 6 โข (๐ โ โ โ (!โ๐) = (seq1( ยท , I )โ๐)) | |
13 | 12 | oveq1d 7417 | . . . . 5 โข (๐ โ โ โ ((!โ๐) ยท (๐ + 1)) = ((seq1( ยท , I )โ๐) ยท (๐ + 1))) |
14 | 8, 11, 13 | 3eqtr4a 2790 | . . . 4 โข (๐ โ โ โ (seq1( ยท , I )โ(๐ + 1)) = ((!โ๐) ยท (๐ + 1))) |
15 | 4, 14 | eqtrd 2764 | . . 3 โข (๐ โ โ โ (!โ(๐ + 1)) = ((!โ๐) ยท (๐ + 1))) |
16 | 0p1e1 12333 | . . . . . 6 โข (0 + 1) = 1 | |
17 | 16 | fveq2i 6885 | . . . . 5 โข (!โ(0 + 1)) = (!โ1) |
18 | fac1 14238 | . . . . 5 โข (!โ1) = 1 | |
19 | 17, 18 | eqtri 2752 | . . . 4 โข (!โ(0 + 1)) = 1 |
20 | fvoveq1 7425 | . . . 4 โข (๐ = 0 โ (!โ(๐ + 1)) = (!โ(0 + 1))) | |
21 | fveq2 6882 | . . . . . 6 โข (๐ = 0 โ (!โ๐) = (!โ0)) | |
22 | oveq1 7409 | . . . . . 6 โข (๐ = 0 โ (๐ + 1) = (0 + 1)) | |
23 | 21, 22 | oveq12d 7420 | . . . . 5 โข (๐ = 0 โ ((!โ๐) ยท (๐ + 1)) = ((!โ0) ยท (0 + 1))) |
24 | fac0 14237 | . . . . . . 7 โข (!โ0) = 1 | |
25 | 24, 16 | oveq12i 7414 | . . . . . 6 โข ((!โ0) ยท (0 + 1)) = (1 ยท 1) |
26 | 1t1e1 12373 | . . . . . 6 โข (1 ยท 1) = 1 | |
27 | 25, 26 | eqtri 2752 | . . . . 5 โข ((!โ0) ยท (0 + 1)) = 1 |
28 | 23, 27 | eqtrdi 2780 | . . . 4 โข (๐ = 0 โ ((!โ๐) ยท (๐ + 1)) = 1) |
29 | 19, 20, 28 | 3eqtr4a 2790 | . . 3 โข (๐ = 0 โ (!โ(๐ + 1)) = ((!โ๐) ยท (๐ + 1))) |
30 | 15, 29 | jaoi 854 | . 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 844 = wceq 1533 โ wcel 2098 Vcvv 3466 I cid 5564 โcfv 6534 (class class class)co 7402 0cc0 11107 1c1 11108 + caddc 11110 ยท cmul 11112 โcn 12211 โ0cn0 12471 โคโฅcuz 12821 seqcseq 13967 !cfa 14234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-seq 13968 df-fac 14235 |
This theorem is referenced by: fac2 14240 fac3 14241 fac4 14242 facnn2 14243 faccl 14244 facdiv 14248 facwordi 14250 faclbnd 14251 faclbnd6 14260 facubnd 14261 bcm1k 14276 bcp1n 14277 4bc2eq6 14290 efcllem 16023 ef01bndlem 16130 eirrlem 16150 dvdsfac 16272 prmfac1 16661 pcfac 16837 2expltfac 17031 aaliou3lem2 26221 aaliou3lem8 26223 dvtaylp 26247 advlogexp 26530 facgam 26939 bcmono 27151 ex-fac 30199 subfacval2 34696 subfaclim 34697 faclim 35239 faclim2 35241 lcmineqlem18 41418 facp2 41494 fac2xp3 41555 factwoffsmonot 41558 bccp1k 43650 binomcxplemwb 43657 wallispi2lem2 45334 stirlinglem4 45339 etransclem24 45520 etransclem28 45524 etransclem38 45534 |
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