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Mirrors > Home > MPE Home > Th. List > ex-fac | Structured version Visualization version GIF version |
Description: Example for df-fac 13626. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-fac | ⊢ (!‘5) = ;;120 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 11695 | . . . 4 ⊢ 5 = (4 + 1) | |
2 | 1 | fveq2i 6666 | . . 3 ⊢ (!‘5) = (!‘(4 + 1)) |
3 | 4nn0 11908 | . . . 4 ⊢ 4 ∈ ℕ0 | |
4 | facp1 13630 | . . . 4 ⊢ (4 ∈ ℕ0 → (!‘(4 + 1)) = ((!‘4) · (4 + 1))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (!‘(4 + 1)) = ((!‘4) · (4 + 1)) |
6 | 2, 5 | eqtri 2842 | . 2 ⊢ (!‘5) = ((!‘4) · (4 + 1)) |
7 | fac4 13633 | . . . 4 ⊢ (!‘4) = ;24 | |
8 | 4p1e5 11775 | . . . 4 ⊢ (4 + 1) = 5 | |
9 | 7, 8 | oveq12i 7160 | . . 3 ⊢ ((!‘4) · (4 + 1)) = (;24 · 5) |
10 | 5nn0 11909 | . . . 4 ⊢ 5 ∈ ℕ0 | |
11 | 2nn0 11906 | . . . 4 ⊢ 2 ∈ ℕ0 | |
12 | eqid 2819 | . . . 4 ⊢ ;24 = ;24 | |
13 | 0nn0 11904 | . . . 4 ⊢ 0 ∈ ℕ0 | |
14 | 1nn0 11905 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
15 | 5cn 11717 | . . . . . 6 ⊢ 5 ∈ ℂ | |
16 | 2cn 11704 | . . . . . 6 ⊢ 2 ∈ ℂ | |
17 | 5t2e10 12190 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
18 | 15, 16, 17 | mulcomli 10642 | . . . . 5 ⊢ (2 · 5) = ;10 |
19 | 16 | addid2i 10820 | . . . . 5 ⊢ (0 + 2) = 2 |
20 | 14, 13, 11, 18, 19 | decaddi 12150 | . . . 4 ⊢ ((2 · 5) + 2) = ;12 |
21 | 4cn 11714 | . . . . 5 ⊢ 4 ∈ ℂ | |
22 | 5t4e20 12192 | . . . . 5 ⊢ (5 · 4) = ;20 | |
23 | 15, 21, 22 | mulcomli 10642 | . . . 4 ⊢ (4 · 5) = ;20 |
24 | 10, 11, 3, 12, 13, 11, 20, 23 | decmul1c 12155 | . . 3 ⊢ (;24 · 5) = ;;120 |
25 | 9, 24 | eqtri 2842 | . 2 ⊢ ((!‘4) · (4 + 1)) = ;;120 |
26 | 6, 25 | eqtri 2842 | 1 ⊢ (!‘5) = ;;120 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 0cc0 10529 1c1 10530 + caddc 10532 · cmul 10534 2c2 11684 4c4 11686 5c5 11687 ℕ0cn0 11889 ;cdc 12090 !cfa 13625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-seq 13362 df-fac 13626 |
This theorem is referenced by: (None) |
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