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Mirrors > Home > ILE Home > Th. List > ex-bc | GIF version |
Description: Example for df-bc 10622. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-bc | ⊢ (5C3) = ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 8895 | . . 3 ⊢ 5 = (4 + 1) | |
2 | 1 | oveq1i 5834 | . 2 ⊢ (5C3) = ((4 + 1)C3) |
3 | 4bc3eq4 10647 | . . . 4 ⊢ (4C3) = 4 | |
4 | 3m1e2 8953 | . . . . . 6 ⊢ (3 − 1) = 2 | |
5 | 4 | oveq2i 5835 | . . . . 5 ⊢ (4C(3 − 1)) = (4C2) |
6 | 4bc2eq6 10648 | . . . . 5 ⊢ (4C2) = 6 | |
7 | 5, 6 | eqtri 2178 | . . . 4 ⊢ (4C(3 − 1)) = 6 |
8 | 3, 7 | oveq12i 5836 | . . 3 ⊢ ((4C3) + (4C(3 − 1))) = (4 + 6) |
9 | 4nn0 9109 | . . . 4 ⊢ 4 ∈ ℕ0 | |
10 | 3z 9196 | . . . 4 ⊢ 3 ∈ ℤ | |
11 | bcpasc 10640 | . . . 4 ⊢ ((4 ∈ ℕ0 ∧ 3 ∈ ℤ) → ((4C3) + (4C(3 − 1))) = ((4 + 1)C3)) | |
12 | 9, 10, 11 | mp2an 423 | . . 3 ⊢ ((4C3) + (4C(3 − 1))) = ((4 + 1)C3) |
13 | 6cn 8915 | . . . 4 ⊢ 6 ∈ ℂ | |
14 | 4cn 8911 | . . . 4 ⊢ 4 ∈ ℂ | |
15 | 6p4e10 9366 | . . . 4 ⊢ (6 + 4) = ;10 | |
16 | 13, 14, 15 | addcomli 8020 | . . 3 ⊢ (4 + 6) = ;10 |
17 | 8, 12, 16 | 3eqtr3i 2186 | . 2 ⊢ ((4 + 1)C3) = ;10 |
18 | 2, 17 | eqtri 2178 | 1 ⊢ (5C3) = ;10 |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∈ wcel 2128 (class class class)co 5824 0cc0 7732 1c1 7733 + caddc 7735 − cmin 8046 2c2 8884 3c3 8885 4c4 8886 5c5 8887 6c6 8888 ℕ0cn0 9090 ℤcz 9167 ;cdc 9295 Ccbc 10621 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-frec 6338 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-5 8895 df-6 8896 df-7 8897 df-8 8898 df-9 8899 df-n0 9091 df-z 9168 df-dec 9296 df-uz 9440 df-q 9529 df-rp 9561 df-fz 9913 df-seqfrec 10345 df-fac 10600 df-bc 10622 |
This theorem is referenced by: (None) |
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