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Mirrors > Home > ILE Home > Th. List > zfz1isolemsplit | GIF version |
Description: Lemma for zfz1iso 10754. 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 9218 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
2 | zfz1isolemsplit.xf | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
3 | zfz1isolemsplit.mx | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑋) | |
4 | diffisn 6859 | . . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (𝑋 ∖ {𝑀}) ∈ Fin) | |
5 | 2, 3, 4 | syl2anc 409 | . . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑀}) ∈ Fin) |
6 | hashcl 10694 | . . . . 5 ⊢ ((𝑋 ∖ {𝑀}) ∈ Fin → (♯‘(𝑋 ∖ {𝑀})) ∈ ℕ0) | |
7 | 5, 6 | syl 14 | . . . 4 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) ∈ ℕ0) |
8 | nn0uz 9500 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 1m1e0 8926 | . . . . . 6 ⊢ (1 − 1) = 0 | |
10 | 9 | fveq2i 5489 | . . . . 5 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0) |
11 | 8, 10 | eqtr4i 2189 | . . . 4 ⊢ ℕ0 = (ℤ≥‘(1 − 1)) |
12 | 7, 11 | eleqtrdi 2259 | . . 3 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) ∈ (ℤ≥‘(1 − 1))) |
13 | fzsuc2 10014 | . . 3 ⊢ ((1 ∈ ℤ ∧ (♯‘(𝑋 ∖ {𝑀})) ∈ (ℤ≥‘(1 − 1))) → (1...((♯‘(𝑋 ∖ {𝑀})) + 1)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {((♯‘(𝑋 ∖ {𝑀})) + 1)})) | |
14 | 1, 12, 13 | syl2anc 409 | . 2 ⊢ (𝜑 → (1...((♯‘(𝑋 ∖ {𝑀})) + 1)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {((♯‘(𝑋 ∖ {𝑀})) + 1)})) |
15 | hashdifsn 10732 | . . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1)) | |
16 | 2, 3, 15 | syl2anc 409 | . . . . 5 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1)) |
17 | 16 | oveq1d 5857 | . . . 4 ⊢ (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (((♯‘𝑋) − 1) + 1)) |
18 | hashcl 10694 | . . . . . . 7 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0) | |
19 | 2, 18 | syl 14 | . . . . . 6 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ0) |
20 | 19 | nn0cnd 9169 | . . . . 5 ⊢ (𝜑 → (♯‘𝑋) ∈ ℂ) |
21 | 1cnd 7915 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
22 | 20, 21 | npcand 8213 | . . . 4 ⊢ (𝜑 → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋)) |
23 | 17, 22 | eqtrd 2198 | . . 3 ⊢ (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (♯‘𝑋)) |
24 | 23 | oveq2d 5858 | . 2 ⊢ (𝜑 → (1...((♯‘(𝑋 ∖ {𝑀})) + 1)) = (1...(♯‘𝑋))) |
25 | 23 | sneqd 3589 | . . 3 ⊢ (𝜑 → {((♯‘(𝑋 ∖ {𝑀})) + 1)} = {(♯‘𝑋)}) |
26 | 25 | uneq2d 3276 | . 2 ⊢ (𝜑 → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {((♯‘(𝑋 ∖ {𝑀})) + 1)}) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})) |
27 | 14, 24, 26 | 3eqtr3d 2206 | 1 ⊢ (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1343 ∈ wcel 2136 ∖ cdif 3113 ∪ cun 3114 {csn 3576 ‘cfv 5188 (class class class)co 5842 Fincfn 6706 0cc0 7753 1c1 7754 + caddc 7756 − cmin 8069 ℕ0cn0 9114 ℤcz 9191 ℤ≥cuz 9466 ...cfz 9944 ♯chash 10688 |
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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-addcom 7853 ax-addass 7855 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-0id 7861 ax-rnegex 7862 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-frec 6359 df-1o 6384 df-oadd 6388 df-er 6501 df-en 6707 df-dom 6708 df-fin 6709 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-inn 8858 df-n0 9115 df-z 9192 df-uz 9467 df-fz 9945 df-ihash 10689 |
This theorem is referenced by: zfz1isolemiso 10752 zfz1isolem1 10753 |
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