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Mirrors > Home > ILE Home > Th. List > fac3 | GIF version |
Description: The factorial of 3. (Contributed by NM, 17-Mar-2005.) |
Ref | Expression |
---|---|
fac3 | ⊢ (!‘3) = 6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3 8931 | . . 3 ⊢ 3 = (2 + 1) | |
2 | 1 | fveq2i 5497 | . 2 ⊢ (!‘3) = (!‘(2 + 1)) |
3 | 2nn0 9145 | . . 3 ⊢ 2 ∈ ℕ0 | |
4 | facp1 10657 | . . 3 ⊢ (2 ∈ ℕ0 → (!‘(2 + 1)) = ((!‘2) · (2 + 1))) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (!‘(2 + 1)) = ((!‘2) · (2 + 1)) |
6 | fac2 10658 | . . . 4 ⊢ (!‘2) = 2 | |
7 | 2p1e3 9004 | . . . 4 ⊢ (2 + 1) = 3 | |
8 | 6, 7 | oveq12i 5863 | . . 3 ⊢ ((!‘2) · (2 + 1)) = (2 · 3) |
9 | 2cn 8942 | . . . 4 ⊢ 2 ∈ ℂ | |
10 | 3cn 8946 | . . . 4 ⊢ 3 ∈ ℂ | |
11 | 9, 10 | mulcomi 7919 | . . 3 ⊢ (2 · 3) = (3 · 2) |
12 | 3t2e6 9027 | . . 3 ⊢ (3 · 2) = 6 | |
13 | 8, 11, 12 | 3eqtri 2195 | . 2 ⊢ ((!‘2) · (2 + 1)) = 6 |
14 | 2, 5, 13 | 3eqtri 2195 | 1 ⊢ (!‘3) = 6 |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 ‘cfv 5196 (class class class)co 5851 1c1 7768 + caddc 7770 · cmul 7772 2c2 8922 3c3 8923 6c6 8926 ℕ0cn0 9128 !cfa 10652 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-ltadd 7883 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-inn 8872 df-2 8930 df-3 8931 df-4 8932 df-5 8933 df-6 8934 df-n0 9129 df-z 9206 df-uz 9481 df-seqfrec 10395 df-fac 10653 |
This theorem is referenced by: fac4 10660 4bc2eq6 10701 ef4p 11650 ef01bndlem 11712 |
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