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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 10980 | . . . 4 ⊢ 0 ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 0 ∈ V) |
3 | 1ex 10982 | . . . 4 ⊢ 1 ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 1 ∈ V) |
5 | df-fac 13999 | . . . 4 ⊢ ! = ({〈0, 1〉} ∪ seq1( · , I )) | |
6 | nnuz 12632 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
7 | dfn2 12257 | . . . . . . . 8 ⊢ ℕ = (ℕ0 ∖ {0}) | |
8 | 6, 7 | eqtr3i 2770 | . . . . . . 7 ⊢ (ℤ≥‘1) = (ℕ0 ∖ {0}) |
9 | 8 | reseq2i 5887 | . . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = (seq1( · , I ) ↾ (ℕ0 ∖ {0})) |
10 | 1z 12361 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
11 | seqfn 13744 | . . . . . . 7 ⊢ (1 ∈ ℤ → seq1( · , I ) Fn (ℤ≥‘1)) | |
12 | fnresdm 6549 | . . . . . . 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 2770 | . . . . 5 ⊢ (seq1( · , I ) ↾ (ℕ0 ∖ {0})) = seq1( · , I ) |
15 | 14 | uneq2i 4099 | . . . 4 ⊢ ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) = ({〈0, 1〉} ∪ seq1( · , I )) |
16 | 5, 15 | eqtr4i 2771 | . . 3 ⊢ ! = ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) |
17 | id 22 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 𝑁 ∈ (ℕ0 ∖ {0})) | |
18 | 2, 4, 16, 17 | fvsnun2 7052 | . 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 1542 ∈ wcel 2110 Vcvv 3431 ∖ cdif 3889 ∪ cun 3890 {csn 4567 〈cop 4573 I cid 5489 ↾ cres 5592 Fn wfn 6427 ‘cfv 6432 0cc0 10882 1c1 10883 · cmul 10887 ℕcn 11984 ℕ0cn0 12244 ℤcz 12330 ℤ≥cuz 12593 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: fac1 14002 facp1 14003 bcval5 14043 fprodfac 15694 logfac 25767 wilthlem3 26230 |
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