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Mirrors > Home > ILE Home > Th. List > ex-fac | GIF version |
Description: Example for df-fac 10440. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-fac | ⊢ (!‘5) = ;;120 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 8750 | . . . 4 ⊢ 5 = (4 + 1) | |
2 | 1 | fveq2i 5392 | . . 3 ⊢ (!‘5) = (!‘(4 + 1)) |
3 | 4nn0 8964 | . . . 4 ⊢ 4 ∈ ℕ0 | |
4 | facp1 10444 | . . . 4 ⊢ (4 ∈ ℕ0 → (!‘(4 + 1)) = ((!‘4) · (4 + 1))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (!‘(4 + 1)) = ((!‘4) · (4 + 1)) |
6 | 2, 5 | eqtri 2138 | . 2 ⊢ (!‘5) = ((!‘4) · (4 + 1)) |
7 | fac4 10447 | . . . 4 ⊢ (!‘4) = ;24 | |
8 | 4p1e5 8824 | . . . 4 ⊢ (4 + 1) = 5 | |
9 | 7, 8 | oveq12i 5754 | . . 3 ⊢ ((!‘4) · (4 + 1)) = (;24 · 5) |
10 | 5nn0 8965 | . . . 4 ⊢ 5 ∈ ℕ0 | |
11 | 2nn0 8962 | . . . 4 ⊢ 2 ∈ ℕ0 | |
12 | eqid 2117 | . . . 4 ⊢ ;24 = ;24 | |
13 | 0nn0 8960 | . . . 4 ⊢ 0 ∈ ℕ0 | |
14 | 1nn0 8961 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
15 | 5cn 8768 | . . . . . 6 ⊢ 5 ∈ ℂ | |
16 | 2cn 8759 | . . . . . 6 ⊢ 2 ∈ ℂ | |
17 | 5t2e10 9249 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
18 | 15, 16, 17 | mulcomli 7741 | . . . . 5 ⊢ (2 · 5) = ;10 |
19 | 16 | addid2i 7873 | . . . . 5 ⊢ (0 + 2) = 2 |
20 | 14, 13, 11, 18, 19 | decaddi 9209 | . . . 4 ⊢ ((2 · 5) + 2) = ;12 |
21 | 4cn 8766 | . . . . 5 ⊢ 4 ∈ ℂ | |
22 | 5t4e20 9251 | . . . . 5 ⊢ (5 · 4) = ;20 | |
23 | 15, 21, 22 | mulcomli 7741 | . . . 4 ⊢ (4 · 5) = ;20 |
24 | 10, 11, 3, 12, 13, 11, 20, 23 | decmul1c 9214 | . . 3 ⊢ (;24 · 5) = ;;120 |
25 | 9, 24 | eqtri 2138 | . 2 ⊢ ((!‘4) · (4 + 1)) = ;;120 |
26 | 6, 25 | eqtri 2138 | 1 ⊢ (!‘5) = ;;120 |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∈ wcel 1465 ‘cfv 5093 (class class class)co 5742 0cc0 7588 1c1 7589 + caddc 7591 · cmul 7593 2c2 8739 4c4 8741 5c5 8742 ℕ0cn0 8945 ;cdc 9150 !cfa 10439 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-5 8750 df-6 8751 df-7 8752 df-8 8753 df-9 8754 df-n0 8946 df-z 9023 df-dec 9151 df-uz 9295 df-seqfrec 10187 df-fac 10440 |
This theorem is referenced by: (None) |
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