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Mirrors > Home > MPE Home > Th. List > fac0 | Structured version Visualization version GIF version |
Description: The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Ref | Expression |
---|---|
fac0 | ⊢ (!‘0) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 11154 | . . . 4 ⊢ 0 ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → 0 ∈ V) |
3 | 1ex 11156 | . . . 4 ⊢ 1 ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → 1 ∈ V) |
5 | df-fac 14180 | . . . 4 ⊢ ! = ({⟨0, 1⟩} ∪ seq1( · , I )) | |
6 | nnuz 12811 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
7 | dfn2 12431 | . . . . . . . 8 ⊢ ℕ = (ℕ0 ∖ {0}) | |
8 | 6, 7 | eqtr3i 2763 | . . . . . . 7 ⊢ (ℤ≥‘1) = (ℕ0 ∖ {0}) |
9 | 8 | reseq2i 5935 | . . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = (seq1( · , I ) ↾ (ℕ0 ∖ {0})) |
10 | 1z 12538 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
11 | seqfn 13924 | . . . . . . 7 ⊢ (1 ∈ ℤ → seq1( · , I ) Fn (ℤ≥‘1)) | |
12 | fnresdm 6621 | . . . . . . 7 ⊢ (seq1( · , I ) Fn (ℤ≥‘1) → (seq1( · , I ) ↾ (ℤ≥‘1)) = seq1( · , I )) | |
13 | 10, 11, 12 | mp2b 10 | . . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = seq1( · , I ) |
14 | 9, 13 | eqtr3i 2763 | . . . . 5 ⊢ (seq1( · , I ) ↾ (ℕ0 ∖ {0})) = seq1( · , I ) |
15 | 14 | uneq2i 4121 | . . . 4 ⊢ ({⟨0, 1⟩} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) = ({⟨0, 1⟩} ∪ seq1( · , I )) |
16 | 5, 15 | eqtr4i 2764 | . . 3 ⊢ ! = ({⟨0, 1⟩} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) |
17 | 2, 4, 16 | fvsnun1 7129 | . 2 ⊢ (⊤ → (!‘0) = 1) |
18 | 17 | mptru 1549 | 1 ⊢ (!‘0) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 Vcvv 3444 ∖ cdif 3908 ∪ cun 3909 {csn 4587 ⟨cop 4593 I cid 5531 ↾ cres 5636 Fn wfn 6492 ‘cfv 6497 0cc0 11056 1c1 11057 · cmul 11061 ℕcn 12158 ℕ0cn0 12418 ℤcz 12504 ℤ≥cuz 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: facp1 14184 faccl 14189 facwordi 14195 faclbnd 14196 faclbnd4lem3 14201 facubnd 14206 bcn0 14216 bcval5 14224 hashf1 14362 fprodfac 15861 fallfacfac 15933 ef0lem 15966 ege2le3 15977 eft0val 15999 prmfac1 16602 pcfac 16776 tayl0 25737 logfac 25972 advlogexp 26026 facgam 26431 logexprlim 26589 subfacval2 33838 faclim 34375 bccn0 42711 mccl 43925 dvnxpaek 44269 dvnprodlem3 44275 etransclem14 44575 etransclem24 44585 etransclem25 44586 etransclem35 44596 |
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