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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 12245 | . . 3 ⊢ 1 ∈ ℕ | |
2 | facnn 14258 | . . 3 ⊢ (1 ∈ ℕ → (!‘1) = (seq1( · , I )‘1)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (!‘1) = (seq1( · , I )‘1) |
4 | 1z 12614 | . . 3 ⊢ 1 ∈ ℤ | |
5 | seq1 14003 | . . 3 ⊢ (1 ∈ ℤ → (seq1( · , I )‘1) = ( I ‘1)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (seq1( · , I )‘1) = ( I ‘1) |
7 | fvi 6968 | . . 3 ⊢ (1 ∈ ℕ → ( I ‘1) = 1) | |
8 | 1, 7 | ax-mp 5 | . 2 ⊢ ( I ‘1) = 1 |
9 | 3, 6, 8 | 3eqtri 2759 | 1 ⊢ (!‘1) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 I cid 5569 ‘cfv 6542 1c1 11131 · cmul 11135 ℕcn 12234 ℤcz 12580 seqcseq 13990 !cfa 14256 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845 df-seq 13991 df-fac 14257 |
This theorem is referenced by: facp1 14261 fac2 14262 faclbnd4lem1 14276 bcn1 14296 ege2le3 16058 ef4p 16081 efgt1p2 16082 efgt1p 16083 symg1hash 19335 dveflem 25898 logfacrlim2 27146 subfacval2 34733 subfacval3 34735 wallispi2lem2 45383 |
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