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Mirrors > Home > ILE Home > Th. List > ex-bc | GIF version |
Description: Example for df-bc 10462. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-bc | ⊢ (5C3) = ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 8750 | . . 3 ⊢ 5 = (4 + 1) | |
2 | 1 | oveq1i 5752 | . 2 ⊢ (5C3) = ((4 + 1)C3) |
3 | 4bc3eq4 10487 | . . . 4 ⊢ (4C3) = 4 | |
4 | 3m1e2 8808 | . . . . . 6 ⊢ (3 − 1) = 2 | |
5 | 4 | oveq2i 5753 | . . . . 5 ⊢ (4C(3 − 1)) = (4C2) |
6 | 4bc2eq6 10488 | . . . . 5 ⊢ (4C2) = 6 | |
7 | 5, 6 | eqtri 2138 | . . . 4 ⊢ (4C(3 − 1)) = 6 |
8 | 3, 7 | oveq12i 5754 | . . 3 ⊢ ((4C3) + (4C(3 − 1))) = (4 + 6) |
9 | 4nn0 8964 | . . . 4 ⊢ 4 ∈ ℕ0 | |
10 | 3z 9051 | . . . 4 ⊢ 3 ∈ ℤ | |
11 | bcpasc 10480 | . . . 4 ⊢ ((4 ∈ ℕ0 ∧ 3 ∈ ℤ) → ((4C3) + (4C(3 − 1))) = ((4 + 1)C3)) | |
12 | 9, 10, 11 | mp2an 422 | . . 3 ⊢ ((4C3) + (4C(3 − 1))) = ((4 + 1)C3) |
13 | 6cn 8770 | . . . 4 ⊢ 6 ∈ ℂ | |
14 | 4cn 8766 | . . . 4 ⊢ 4 ∈ ℂ | |
15 | 6p4e10 9221 | . . . 4 ⊢ (6 + 4) = ;10 | |
16 | 13, 14, 15 | addcomli 7875 | . . 3 ⊢ (4 + 6) = ;10 |
17 | 8, 12, 16 | 3eqtr3i 2146 | . 2 ⊢ ((4 + 1)C3) = ;10 |
18 | 2, 17 | eqtri 2138 | 1 ⊢ (5C3) = ;10 |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∈ wcel 1465 (class class class)co 5742 0cc0 7588 1c1 7589 + caddc 7591 − cmin 7901 2c2 8739 3c3 8740 4c4 8741 5c5 8742 6c6 8743 ℕ0cn0 8945 ℤcz 9022 ;cdc 9150 Ccbc 10461 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-5 8750 df-6 8751 df-7 8752 df-8 8753 df-9 8754 df-n0 8946 df-z 9023 df-dec 9151 df-uz 9295 df-q 9380 df-rp 9410 df-fz 9759 df-seqfrec 10187 df-fac 10440 df-bc 10462 |
This theorem is referenced by: (None) |
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