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Mirrors > Home > ILE Home > Th. List > ex-fac | GIF version |
Description: Example for df-fac 10195. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-fac | ⊢ (!‘5) = ;;120 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 8545 | . . . 4 ⊢ 5 = (4 + 1) | |
2 | 1 | fveq2i 5321 | . . 3 ⊢ (!‘5) = (!‘(4 + 1)) |
3 | 4nn0 8753 | . . . 4 ⊢ 4 ∈ ℕ0 | |
4 | facp1 10199 | . . . 4 ⊢ (4 ∈ ℕ0 → (!‘(4 + 1)) = ((!‘4) · (4 + 1))) | |
5 | 3, 4 | ax-mp 7 | . . 3 ⊢ (!‘(4 + 1)) = ((!‘4) · (4 + 1)) |
6 | 2, 5 | eqtri 2109 | . 2 ⊢ (!‘5) = ((!‘4) · (4 + 1)) |
7 | fac4 10202 | . . . 4 ⊢ (!‘4) = ;24 | |
8 | 4p1e5 8613 | . . . 4 ⊢ (4 + 1) = 5 | |
9 | 7, 8 | oveq12i 5678 | . . 3 ⊢ ((!‘4) · (4 + 1)) = (;24 · 5) |
10 | 5nn0 8754 | . . . 4 ⊢ 5 ∈ ℕ0 | |
11 | 2nn0 8751 | . . . 4 ⊢ 2 ∈ ℕ0 | |
12 | eqid 2089 | . . . 4 ⊢ ;24 = ;24 | |
13 | 0nn0 8749 | . . . 4 ⊢ 0 ∈ ℕ0 | |
14 | 1nn0 8750 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
15 | 5cn 8563 | . . . . . 6 ⊢ 5 ∈ ℂ | |
16 | 2cn 8554 | . . . . . 6 ⊢ 2 ∈ ℂ | |
17 | 5t2e10 9037 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
18 | 15, 16, 17 | mulcomli 7556 | . . . . 5 ⊢ (2 · 5) = ;10 |
19 | 16 | addid2i 7686 | . . . . 5 ⊢ (0 + 2) = 2 |
20 | 14, 13, 11, 18, 19 | decaddi 8997 | . . . 4 ⊢ ((2 · 5) + 2) = ;12 |
21 | 4cn 8561 | . . . . 5 ⊢ 4 ∈ ℂ | |
22 | 5t4e20 9039 | . . . . 5 ⊢ (5 · 4) = ;20 | |
23 | 15, 21, 22 | mulcomli 7556 | . . . 4 ⊢ (4 · 5) = ;20 |
24 | 10, 11, 3, 12, 13, 11, 20, 23 | decmul1c 9002 | . . 3 ⊢ (;24 · 5) = ;;120 |
25 | 9, 24 | eqtri 2109 | . 2 ⊢ ((!‘4) · (4 + 1)) = ;;120 |
26 | 6, 25 | eqtri 2109 | 1 ⊢ (!‘5) = ;;120 |
Colors of variables: wff set class |
Syntax hints: = wceq 1290 ∈ wcel 1439 ‘cfv 5028 (class class class)co 5666 0cc0 7411 1c1 7412 + caddc 7414 · cmul 7416 2c2 8534 4c4 8536 5c5 8537 ℕ0cn0 8734 ;cdc 8938 !cfa 10194 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-ltadd 7522 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-frec 6170 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-inn 8484 df-2 8542 df-3 8543 df-4 8544 df-5 8545 df-6 8546 df-7 8547 df-8 8548 df-9 8549 df-n0 8735 df-z 8812 df-dec 8939 df-uz 9081 df-iseq 9914 df-fac 10195 |
This theorem is referenced by: (None) |
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