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Mirrors > Home > MPE Home > Th. List > fac1 | Structured version Visualization version GIF version |
Description: The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Ref | Expression |
---|---|
fac1 | ⊢ (!‘1) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 11995 | . . 3 ⊢ 1 ∈ ℕ | |
2 | facnn 14000 | . . 3 ⊢ (1 ∈ ℕ → (!‘1) = (seq1( · , I )‘1)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (!‘1) = (seq1( · , I )‘1) |
4 | 1z 12361 | . . 3 ⊢ 1 ∈ ℤ | |
5 | seq1 13745 | . . 3 ⊢ (1 ∈ ℤ → (seq1( · , I )‘1) = ( I ‘1)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (seq1( · , I )‘1) = ( I ‘1) |
7 | fvi 6841 | . . 3 ⊢ (1 ∈ ℕ → ( I ‘1) = 1) | |
8 | 1, 7 | ax-mp 5 | . 2 ⊢ ( I ‘1) = 1 |
9 | 3, 6, 8 | 3eqtri 2772 | 1 ⊢ (!‘1) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2110 I cid 5489 ‘cfv 6432 1c1 10883 · cmul 10887 ℕcn 11984 ℤcz 12330 seqcseq 13732 !cfa 13998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-n0 12245 df-z 12331 df-uz 12594 df-seq 13733 df-fac 13999 |
This theorem is referenced by: facp1 14003 fac2 14004 faclbnd4lem1 14018 bcn1 14038 ege2le3 15810 ef4p 15833 efgt1p2 15834 efgt1p 15835 symg1hash 19008 dveflem 25154 logfacrlim2 26385 subfacval2 33158 subfacval3 33160 wallispi2lem2 43595 |
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