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Mirrors > Home > ILE Home > Th. List > zfz1isolemsplit | GIF version |
Description: Lemma for zfz1iso 10577. 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 9074 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
2 | zfz1isolemsplit.xf | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
3 | zfz1isolemsplit.mx | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑋) | |
4 | diffisn 6780 | . . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (𝑋 ∖ {𝑀}) ∈ Fin) | |
5 | 2, 3, 4 | syl2anc 408 | . . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑀}) ∈ Fin) |
6 | hashcl 10520 | . . . . 5 ⊢ ((𝑋 ∖ {𝑀}) ∈ Fin → (♯‘(𝑋 ∖ {𝑀})) ∈ ℕ0) | |
7 | 5, 6 | syl 14 | . . . 4 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) ∈ ℕ0) |
8 | nn0uz 9353 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 1m1e0 8782 | . . . . . 6 ⊢ (1 − 1) = 0 | |
10 | 9 | fveq2i 5417 | . . . . 5 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0) |
11 | 8, 10 | eqtr4i 2161 | . . . 4 ⊢ ℕ0 = (ℤ≥‘(1 − 1)) |
12 | 7, 11 | eleqtrdi 2230 | . . 3 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) ∈ (ℤ≥‘(1 − 1))) |
13 | fzsuc2 9852 | . . 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 10558 | . . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1)) | |
16 | 2, 3, 15 | syl2anc 408 | . . . . 5 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1)) |
17 | 16 | oveq1d 5782 | . . . 4 ⊢ (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (((♯‘𝑋) − 1) + 1)) |
18 | hashcl 10520 | . . . . . . 7 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0) | |
19 | 2, 18 | syl 14 | . . . . . 6 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ0) |
20 | 19 | nn0cnd 9025 | . . . . 5 ⊢ (𝜑 → (♯‘𝑋) ∈ ℂ) |
21 | 1cnd 7775 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
22 | 20, 21 | npcand 8070 | . . . 4 ⊢ (𝜑 → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋)) |
23 | 17, 22 | eqtrd 2170 | . . 3 ⊢ (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (♯‘𝑋)) |
24 | 23 | oveq2d 5783 | . 2 ⊢ (𝜑 → (1...((♯‘(𝑋 ∖ {𝑀})) + 1)) = (1...(♯‘𝑋))) |
25 | 23 | sneqd 3535 | . . 3 ⊢ (𝜑 → {((♯‘(𝑋 ∖ {𝑀})) + 1)} = {(♯‘𝑋)}) |
26 | 25 | uneq2d 3225 | . 2 ⊢ (𝜑 → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {((♯‘(𝑋 ∖ {𝑀})) + 1)}) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})) |
27 | 14, 24, 26 | 3eqtr3d 2178 | 1 ⊢ (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∖ cdif 3063 ∪ cun 3064 {csn 3522 ‘cfv 5118 (class class class)co 5767 Fincfn 6627 0cc0 7613 1c1 7614 + caddc 7616 − cmin 7926 ℕ0cn0 8970 ℤcz 9047 ℤ≥cuz 9319 ...cfz 9783 ♯chash 10514 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-frec 6281 df-1o 6306 df-oadd 6310 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 df-ihash 10515 |
This theorem is referenced by: zfz1isolemiso 10575 zfz1isolem1 10576 |
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