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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 11239 | . . . 4 ⊢ 0 ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 0 ∈ V) |
3 | 1ex 11241 | . . . 4 ⊢ 1 ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 1 ∈ V) |
5 | df-fac 14266 | . . . 4 ⊢ ! = ({〈0, 1〉} ∪ seq1( · , I )) | |
6 | nnuz 12896 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
7 | dfn2 12516 | . . . . . . . 8 ⊢ ℕ = (ℕ0 ∖ {0}) | |
8 | 6, 7 | eqtr3i 2758 | . . . . . . 7 ⊢ (ℤ≥‘1) = (ℕ0 ∖ {0}) |
9 | 8 | reseq2i 5982 | . . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = (seq1( · , I ) ↾ (ℕ0 ∖ {0})) |
10 | 1z 12623 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
11 | seqfn 14011 | . . . . . . 7 ⊢ (1 ∈ ℤ → seq1( · , I ) Fn (ℤ≥‘1)) | |
12 | fnresdm 6674 | . . . . . . 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 2758 | . . . . 5 ⊢ (seq1( · , I ) ↾ (ℕ0 ∖ {0})) = seq1( · , I ) |
15 | 14 | uneq2i 4159 | . . . 4 ⊢ ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) = ({〈0, 1〉} ∪ seq1( · , I )) |
16 | 5, 15 | eqtr4i 2759 | . . 3 ⊢ ! = ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) |
17 | id 22 | . . 3 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → 𝑁 ∈ (ℕ0 ∖ {0})) | |
18 | 2, 4, 16, 17 | fvsnun2 7192 | . 2 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
19 | 18, 7 | eleq2s 2847 | 1 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∖ cdif 3944 ∪ cun 3945 {csn 4629 〈cop 4635 I cid 5575 ↾ cres 5680 Fn wfn 6543 ‘cfv 6548 0cc0 11139 1c1 11140 · cmul 11144 ℕcn 12243 ℕ0cn0 12503 ℤcz 12589 ℤ≥cuz 12853 seqcseq 13999 !cfa 14265 |
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 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-seq 14000 df-fac 14266 |
This theorem is referenced by: fac1 14269 facp1 14270 bcval5 14310 fprodfac 15950 logfac 26548 wilthlem3 27015 |
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