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Mirrors > Home > MPE Home > Th. List > ex-fac | Structured version Visualization version GIF version |
Description: Example for df-fac 13101. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-fac | ⊢ (!‘5) = ;;120 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 11120 | . . . 4 ⊢ 5 = (4 + 1) | |
2 | 1 | fveq2i 6232 | . . 3 ⊢ (!‘5) = (!‘(4 + 1)) |
3 | 4nn0 11349 | . . . 4 ⊢ 4 ∈ ℕ0 | |
4 | facp1 13105 | . . . 4 ⊢ (4 ∈ ℕ0 → (!‘(4 + 1)) = ((!‘4) · (4 + 1))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (!‘(4 + 1)) = ((!‘4) · (4 + 1)) |
6 | 2, 5 | eqtri 2673 | . 2 ⊢ (!‘5) = ((!‘4) · (4 + 1)) |
7 | fac4 13108 | . . . 4 ⊢ (!‘4) = ;24 | |
8 | 4p1e5 11192 | . . . 4 ⊢ (4 + 1) = 5 | |
9 | 7, 8 | oveq12i 6702 | . . 3 ⊢ ((!‘4) · (4 + 1)) = (;24 · 5) |
10 | 5nn0 11350 | . . . 4 ⊢ 5 ∈ ℕ0 | |
11 | 2nn0 11347 | . . . 4 ⊢ 2 ∈ ℕ0 | |
12 | eqid 2651 | . . . 4 ⊢ ;24 = ;24 | |
13 | 0nn0 11345 | . . . 4 ⊢ 0 ∈ ℕ0 | |
14 | 1nn0 11346 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
15 | 5cn 11138 | . . . . . 6 ⊢ 5 ∈ ℂ | |
16 | 2cn 11129 | . . . . . 6 ⊢ 2 ∈ ℂ | |
17 | 5t2e10 11672 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
18 | 15, 16, 17 | mulcomli 10085 | . . . . 5 ⊢ (2 · 5) = ;10 |
19 | 16 | addid2i 10262 | . . . . 5 ⊢ (0 + 2) = 2 |
20 | 14, 13, 11, 18, 19 | decaddi 11617 | . . . 4 ⊢ ((2 · 5) + 2) = ;12 |
21 | 4cn 11136 | . . . . 5 ⊢ 4 ∈ ℂ | |
22 | 5t4e20 11675 | . . . . 5 ⊢ (5 · 4) = ;20 | |
23 | 15, 21, 22 | mulcomli 10085 | . . . 4 ⊢ (4 · 5) = ;20 |
24 | 10, 11, 3, 12, 13, 11, 20, 23 | decmul1c 11625 | . . 3 ⊢ (;24 · 5) = ;;120 |
25 | 9, 24 | eqtri 2673 | . 2 ⊢ ((!‘4) · (4 + 1)) = ;;120 |
26 | 6, 25 | eqtri 2673 | 1 ⊢ (!‘5) = ;;120 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 0cc0 9974 1c1 9975 + caddc 9977 · cmul 9979 2c2 11108 4c4 11110 5c5 11111 ℕ0cn0 11330 ;cdc 11531 !cfa 13100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-seq 12842 df-fac 13101 |
This theorem is referenced by: (None) |
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