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Mirrors > Home > ILE Home > Th. List > fac4 | GIF version |
Description: The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
fac4 | ⊢ (!‘4) = ;24 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn0 9109 | . . 3 ⊢ 3 ∈ ℕ0 | |
2 | facp1 10608 | . . 3 ⊢ (3 ∈ ℕ0 → (!‘(3 + 1)) = ((!‘3) · (3 + 1))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (!‘(3 + 1)) = ((!‘3) · (3 + 1)) |
4 | 3p1e4 8969 | . . 3 ⊢ (3 + 1) = 4 | |
5 | 4 | fveq2i 5472 | . 2 ⊢ (!‘(3 + 1)) = (!‘4) |
6 | fac3 10610 | . . . 4 ⊢ (!‘3) = 6 | |
7 | 6, 4 | oveq12i 5837 | . . 3 ⊢ ((!‘3) · (3 + 1)) = (6 · 4) |
8 | 6t4e24 9401 | . . 3 ⊢ (6 · 4) = ;24 | |
9 | 7, 8 | eqtri 2178 | . 2 ⊢ ((!‘3) · (3 + 1)) = ;24 |
10 | 3, 5, 9 | 3eqtr3i 2186 | 1 ⊢ (!‘4) = ;24 |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∈ wcel 2128 ‘cfv 5171 (class class class)co 5825 1c1 7734 + caddc 7736 · cmul 7738 2c2 8885 3c3 8886 4c4 8887 6c6 8889 ℕ0cn0 9091 ;cdc 9296 !cfa 10603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-ltadd 7849 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-frec 6339 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-inn 8835 df-2 8893 df-3 8894 df-4 8895 df-5 8896 df-6 8897 df-7 8898 df-8 8899 df-9 8900 df-n0 9092 df-z 9169 df-dec 9297 df-uz 9441 df-seqfrec 10349 df-fac 10604 |
This theorem is referenced by: ex-fac 13346 |
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