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Mirrors > Home > ILE Home > Th. List > ex-fac | GIF version |
Description: Example for df-fac 10723. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-fac | ⊢ (!‘5) = ;;120 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 8998 | . . . 4 ⊢ 5 = (4 + 1) | |
2 | 1 | fveq2i 5532 | . . 3 ⊢ (!‘5) = (!‘(4 + 1)) |
3 | 4nn0 9212 | . . . 4 ⊢ 4 ∈ ℕ0 | |
4 | facp1 10727 | . . . 4 ⊢ (4 ∈ ℕ0 → (!‘(4 + 1)) = ((!‘4) · (4 + 1))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (!‘(4 + 1)) = ((!‘4) · (4 + 1)) |
6 | 2, 5 | eqtri 2209 | . 2 ⊢ (!‘5) = ((!‘4) · (4 + 1)) |
7 | fac4 10730 | . . . 4 ⊢ (!‘4) = ;24 | |
8 | 4p1e5 9072 | . . . 4 ⊢ (4 + 1) = 5 | |
9 | 7, 8 | oveq12i 5902 | . . 3 ⊢ ((!‘4) · (4 + 1)) = (;24 · 5) |
10 | 5nn0 9213 | . . . 4 ⊢ 5 ∈ ℕ0 | |
11 | 2nn0 9210 | . . . 4 ⊢ 2 ∈ ℕ0 | |
12 | eqid 2188 | . . . 4 ⊢ ;24 = ;24 | |
13 | 0nn0 9208 | . . . 4 ⊢ 0 ∈ ℕ0 | |
14 | 1nn0 9209 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
15 | 5cn 9016 | . . . . . 6 ⊢ 5 ∈ ℂ | |
16 | 2cn 9007 | . . . . . 6 ⊢ 2 ∈ ℂ | |
17 | 5t2e10 9500 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
18 | 15, 16, 17 | mulcomli 7981 | . . . . 5 ⊢ (2 · 5) = ;10 |
19 | 16 | addid2i 8117 | . . . . 5 ⊢ (0 + 2) = 2 |
20 | 14, 13, 11, 18, 19 | decaddi 9460 | . . . 4 ⊢ ((2 · 5) + 2) = ;12 |
21 | 4cn 9014 | . . . . 5 ⊢ 4 ∈ ℂ | |
22 | 5t4e20 9502 | . . . . 5 ⊢ (5 · 4) = ;20 | |
23 | 15, 21, 22 | mulcomli 7981 | . . . 4 ⊢ (4 · 5) = ;20 |
24 | 10, 11, 3, 12, 13, 11, 20, 23 | decmul1c 9465 | . . 3 ⊢ (;24 · 5) = ;;120 |
25 | 9, 24 | eqtri 2209 | . 2 ⊢ ((!‘4) · (4 + 1)) = ;;120 |
26 | 6, 25 | eqtri 2209 | 1 ⊢ (!‘5) = ;;120 |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 ∈ wcel 2159 ‘cfv 5230 (class class class)co 5890 0cc0 7828 1c1 7829 + caddc 7831 · cmul 7833 2c2 8987 4c4 8989 5c5 8990 ℕ0cn0 9193 ;cdc 9401 !cfa 10722 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-coll 4132 ax-sep 4135 ax-nul 4143 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-iinf 4601 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-addcom 7928 ax-mulcom 7929 ax-addass 7930 ax-mulass 7931 ax-distr 7932 ax-i2m1 7933 ax-0lt1 7934 ax-1rid 7935 ax-0id 7936 ax-rnegex 7937 ax-cnre 7939 ax-pre-ltirr 7940 ax-pre-ltwlin 7941 ax-pre-lttrn 7942 ax-pre-ltadd 7944 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rab 2476 df-v 2753 df-sbc 2977 df-csb 3072 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-iun 3902 df-br 4018 df-opab 4079 df-mpt 4080 df-tr 4116 df-id 4307 df-iord 4380 df-on 4382 df-ilim 4383 df-suc 4385 df-iom 4604 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-f1 5235 df-fo 5236 df-f1o 5237 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-1st 6158 df-2nd 6159 df-recs 6323 df-frec 6409 df-pnf 8011 df-mnf 8012 df-xr 8013 df-ltxr 8014 df-le 8015 df-sub 8147 df-neg 8148 df-inn 8937 df-2 8995 df-3 8996 df-4 8997 df-5 8998 df-6 8999 df-7 9000 df-8 9001 df-9 9002 df-n0 9194 df-z 9271 df-dec 9402 df-uz 9546 df-seqfrec 10463 df-fac 10723 |
This theorem is referenced by: (None) |
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