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Mirrors > Home > ILE Home > Th. List > fac2 | GIF version |
Description: The factorial of 2. (Contributed by NM, 17-Mar-2005.) |
Ref | Expression |
---|---|
fac2 | ⊢ (!‘2) = 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 9008 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | fveq2i 5537 | . 2 ⊢ (!‘2) = (!‘(1 + 1)) |
3 | 1nn0 9222 | . . . 4 ⊢ 1 ∈ ℕ0 | |
4 | facp1 10742 | . . . 4 ⊢ (1 ∈ ℕ0 → (!‘(1 + 1)) = ((!‘1) · (1 + 1))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (!‘(1 + 1)) = ((!‘1) · (1 + 1)) |
6 | fac1 10741 | . . . . 5 ⊢ (!‘1) = 1 | |
7 | 1p1e2 9066 | . . . . 5 ⊢ (1 + 1) = 2 | |
8 | 6, 7 | oveq12i 5908 | . . . 4 ⊢ ((!‘1) · (1 + 1)) = (1 · 2) |
9 | 2cn 9020 | . . . . 5 ⊢ 2 ∈ ℂ | |
10 | 9 | mullidi 7990 | . . . 4 ⊢ (1 · 2) = 2 |
11 | 8, 10 | eqtri 2210 | . . 3 ⊢ ((!‘1) · (1 + 1)) = 2 |
12 | 5, 11 | eqtri 2210 | . 2 ⊢ (!‘(1 + 1)) = 2 |
13 | 2, 12 | eqtri 2210 | 1 ⊢ (!‘2) = 2 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 ‘cfv 5235 (class class class)co 5896 1c1 7842 + caddc 7844 · cmul 7846 2c2 9000 ℕ0cn0 9206 !cfa 10737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-ltadd 7957 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-frec 6416 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-inn 8950 df-2 9008 df-n0 9207 df-z 9284 df-uz 9559 df-seqfrec 10477 df-fac 10738 |
This theorem is referenced by: fac3 10744 bcn2 10776 4bc2eq6 10786 ef4p 11734 efgt1p2 11735 dveflem 14644 |
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