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Mirrors > Home > ILE Home > Th. List > zfz1isolemsplit | GIF version |
Description: Lemma for zfz1iso 10740. 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 9209 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
2 | zfz1isolemsplit.xf | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
3 | zfz1isolemsplit.mx | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑋) | |
4 | diffisn 6850 | . . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (𝑋 ∖ {𝑀}) ∈ Fin) | |
5 | 2, 3, 4 | syl2anc 409 | . . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑀}) ∈ Fin) |
6 | hashcl 10683 | . . . . 5 ⊢ ((𝑋 ∖ {𝑀}) ∈ Fin → (♯‘(𝑋 ∖ {𝑀})) ∈ ℕ0) | |
7 | 5, 6 | syl 14 | . . . 4 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) ∈ ℕ0) |
8 | nn0uz 9491 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 1m1e0 8917 | . . . . . 6 ⊢ (1 − 1) = 0 | |
10 | 9 | fveq2i 5483 | . . . . 5 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0) |
11 | 8, 10 | eqtr4i 2188 | . . . 4 ⊢ ℕ0 = (ℤ≥‘(1 − 1)) |
12 | 7, 11 | eleqtrdi 2257 | . . 3 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) ∈ (ℤ≥‘(1 − 1))) |
13 | fzsuc2 10004 | . . 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 10721 | . . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1)) | |
16 | 2, 3, 15 | syl2anc 409 | . . . . 5 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1)) |
17 | 16 | oveq1d 5851 | . . . 4 ⊢ (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (((♯‘𝑋) − 1) + 1)) |
18 | hashcl 10683 | . . . . . . 7 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0) | |
19 | 2, 18 | syl 14 | . . . . . 6 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ0) |
20 | 19 | nn0cnd 9160 | . . . . 5 ⊢ (𝜑 → (♯‘𝑋) ∈ ℂ) |
21 | 1cnd 7906 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
22 | 20, 21 | npcand 8204 | . . . 4 ⊢ (𝜑 → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋)) |
23 | 17, 22 | eqtrd 2197 | . . 3 ⊢ (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (♯‘𝑋)) |
24 | 23 | oveq2d 5852 | . 2 ⊢ (𝜑 → (1...((♯‘(𝑋 ∖ {𝑀})) + 1)) = (1...(♯‘𝑋))) |
25 | 23 | sneqd 3583 | . . 3 ⊢ (𝜑 → {((♯‘(𝑋 ∖ {𝑀})) + 1)} = {(♯‘𝑋)}) |
26 | 25 | uneq2d 3271 | . 2 ⊢ (𝜑 → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {((♯‘(𝑋 ∖ {𝑀})) + 1)}) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})) |
27 | 14, 24, 26 | 3eqtr3d 2205 | 1 ⊢ (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ∖ cdif 3108 ∪ cun 3109 {csn 3570 ‘cfv 5182 (class class class)co 5836 Fincfn 6697 0cc0 7744 1c1 7745 + caddc 7747 − cmin 8060 ℕ0cn0 9105 ℤcz 9182 ℤ≥cuz 9457 ...cfz 9935 ♯chash 10677 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-oadd 6379 df-er 6492 df-en 6698 df-dom 6699 df-fin 6700 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 df-ihash 10678 |
This theorem is referenced by: zfz1isolemiso 10738 zfz1isolem1 10739 |
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