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| Mirrors > Home > MPE Home > Th. List > facnn | Structured version Visualization version GIF version | ||
| Description: Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) | 
| Ref | Expression | 
|---|---|
| facnn | ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | c0ex 11255 | . . . 4 ⊢ 0 ∈ V | |
| 2 | 1 | a1i 11 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 0 ∈ V) | 
| 3 | 1ex 11257 | . . . 4 ⊢ 1 ∈ V | |
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 1 ∈ V) | 
| 5 | df-fac 14313 | . . . 4 ⊢ ! = ({〈0, 1〉} ∪ seq1( · , I )) | |
| 6 | nnuz 12921 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
| 7 | dfn2 12539 | . . . . . . . 8 ⊢ ℕ = (ℕ0 ∖ {0}) | |
| 8 | 6, 7 | eqtr3i 2767 | . . . . . . 7 ⊢ (ℤ≥‘1) = (ℕ0 ∖ {0}) | 
| 9 | 8 | reseq2i 5994 | . . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = (seq1( · , I ) ↾ (ℕ0 ∖ {0})) | 
| 10 | 1z 12647 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
| 11 | seqfn 14054 | . . . . . . 7 ⊢ (1 ∈ ℤ → seq1( · , I ) Fn (ℤ≥‘1)) | |
| 12 | fnresdm 6687 | . . . . . . 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 2767 | . . . . 5 ⊢ (seq1( · , I ) ↾ (ℕ0 ∖ {0})) = seq1( · , I ) | 
| 15 | 14 | uneq2i 4165 | . . . 4 ⊢ ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) = ({〈0, 1〉} ∪ seq1( · , I )) | 
| 16 | 5, 15 | eqtr4i 2768 | . . 3 ⊢ ! = ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) | 
| 17 | id 22 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 𝑁 ∈ (ℕ0 ∖ {0})) | |
| 18 | 2, 4, 16, 17 | fvsnun2 7203 | . 2 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → (!‘𝑁) = (seq1( · , I )‘𝑁)) | 
| 19 | 18, 7 | eleq2s 2859 | 1 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 {csn 4626 〈cop 4632 I cid 5577 ↾ cres 5687 Fn wfn 6556 ‘cfv 6561 0cc0 11155 1c1 11156 · cmul 11160 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 ℤ≥cuz 12878 seqcseq 14042 !cfa 14312 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-fac 14313 | 
| This theorem is referenced by: fac1 14316 facp1 14317 bcval5 14357 fprodfac 16009 logfac 26643 wilthlem3 27113 | 
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