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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 8804 | . . 3 ⊢ 3 = (2 + 1) | |
2 | 1 | fveq2i 5432 | . 2 ⊢ (!‘3) = (!‘(2 + 1)) |
3 | 2nn0 9018 | . . 3 ⊢ 2 ∈ ℕ0 | |
4 | facp1 10508 | . . 3 ⊢ (2 ∈ ℕ0 → (!‘(2 + 1)) = ((!‘2) · (2 + 1))) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (!‘(2 + 1)) = ((!‘2) · (2 + 1)) |
6 | fac2 10509 | . . . 4 ⊢ (!‘2) = 2 | |
7 | 2p1e3 8877 | . . . 4 ⊢ (2 + 1) = 3 | |
8 | 6, 7 | oveq12i 5794 | . . 3 ⊢ ((!‘2) · (2 + 1)) = (2 · 3) |
9 | 2cn 8815 | . . . 4 ⊢ 2 ∈ ℂ | |
10 | 3cn 8819 | . . . 4 ⊢ 3 ∈ ℂ | |
11 | 9, 10 | mulcomi 7796 | . . 3 ⊢ (2 · 3) = (3 · 2) |
12 | 3t2e6 8900 | . . 3 ⊢ (3 · 2) = 6 | |
13 | 8, 11, 12 | 3eqtri 2165 | . 2 ⊢ ((!‘2) · (2 + 1)) = 6 |
14 | 2, 5, 13 | 3eqtri 2165 | 1 ⊢ (!‘3) = 6 |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 ‘cfv 5131 (class class class)co 5782 1c1 7645 + caddc 7647 · cmul 7649 2c2 8795 3c3 8796 6c6 8799 ℕ0cn0 9001 !cfa 10503 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-n0 9002 df-z 9079 df-uz 9351 df-seqfrec 10250 df-fac 10504 |
This theorem is referenced by: fac4 10511 4bc2eq6 10552 ef4p 11437 ef01bndlem 11499 |
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