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Mirrors > Home > ILE Home > Th. List > zfz1isolemsplit | GIF version |
Description: Lemma for zfz1iso 10584. Removing one element from an integer range. (Contributed by Jim Kingdon, 8-Sep-2022.) |
Ref | Expression |
---|---|
zfz1isolemsplit.xf | ⊢ (𝜑 → 𝑋 ∈ Fin) |
zfz1isolemsplit.mx | ⊢ (𝜑 → 𝑀 ∈ 𝑋) |
Ref | Expression |
---|---|
zfz1isolemsplit | ⊢ (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1zzd 9081 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
2 | zfz1isolemsplit.xf | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
3 | zfz1isolemsplit.mx | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑋) | |
4 | diffisn 6787 | . . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (𝑋 ∖ {𝑀}) ∈ Fin) | |
5 | 2, 3, 4 | syl2anc 408 | . . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑀}) ∈ Fin) |
6 | hashcl 10527 | . . . . 5 ⊢ ((𝑋 ∖ {𝑀}) ∈ Fin → (♯‘(𝑋 ∖ {𝑀})) ∈ ℕ0) | |
7 | 5, 6 | syl 14 | . . . 4 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) ∈ ℕ0) |
8 | nn0uz 9360 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 1m1e0 8789 | . . . . . 6 ⊢ (1 − 1) = 0 | |
10 | 9 | fveq2i 5424 | . . . . 5 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0) |
11 | 8, 10 | eqtr4i 2163 | . . . 4 ⊢ ℕ0 = (ℤ≥‘(1 − 1)) |
12 | 7, 11 | eleqtrdi 2232 | . . 3 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) ∈ (ℤ≥‘(1 − 1))) |
13 | fzsuc2 9859 | . . 3 ⊢ ((1 ∈ ℤ ∧ (♯‘(𝑋 ∖ {𝑀})) ∈ (ℤ≥‘(1 − 1))) → (1...((♯‘(𝑋 ∖ {𝑀})) + 1)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {((♯‘(𝑋 ∖ {𝑀})) + 1)})) | |
14 | 1, 12, 13 | syl2anc 408 | . 2 ⊢ (𝜑 → (1...((♯‘(𝑋 ∖ {𝑀})) + 1)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {((♯‘(𝑋 ∖ {𝑀})) + 1)})) |
15 | hashdifsn 10565 | . . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1)) | |
16 | 2, 3, 15 | syl2anc 408 | . . . . 5 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1)) |
17 | 16 | oveq1d 5789 | . . . 4 ⊢ (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (((♯‘𝑋) − 1) + 1)) |
18 | hashcl 10527 | . . . . . . 7 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0) | |
19 | 2, 18 | syl 14 | . . . . . 6 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ0) |
20 | 19 | nn0cnd 9032 | . . . . 5 ⊢ (𝜑 → (♯‘𝑋) ∈ ℂ) |
21 | 1cnd 7782 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
22 | 20, 21 | npcand 8077 | . . . 4 ⊢ (𝜑 → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋)) |
23 | 17, 22 | eqtrd 2172 | . . 3 ⊢ (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (♯‘𝑋)) |
24 | 23 | oveq2d 5790 | . 2 ⊢ (𝜑 → (1...((♯‘(𝑋 ∖ {𝑀})) + 1)) = (1...(♯‘𝑋))) |
25 | 23 | sneqd 3540 | . . 3 ⊢ (𝜑 → {((♯‘(𝑋 ∖ {𝑀})) + 1)} = {(♯‘𝑋)}) |
26 | 25 | uneq2d 3230 | . 2 ⊢ (𝜑 → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {((♯‘(𝑋 ∖ {𝑀})) + 1)}) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})) |
27 | 14, 24, 26 | 3eqtr3d 2180 | 1 ⊢ (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∖ cdif 3068 ∪ cun 3069 {csn 3527 ‘cfv 5123 (class class class)co 5774 Fincfn 6634 0cc0 7620 1c1 7621 + caddc 7623 − cmin 7933 ℕ0cn0 8977 ℤcz 9054 ℤ≥cuz 9326 ...cfz 9790 ♯chash 10521 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 df-ihash 10522 |
This theorem is referenced by: zfz1isolemiso 10582 zfz1isolem1 10583 |
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