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| Mirrors > Home > ILE Home > Th. List > zfz1isolemsplit | GIF version | ||
| Description: Lemma for zfz1iso 10986. 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 9399 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
| 2 | zfz1isolemsplit.xf | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
| 3 | zfz1isolemsplit.mx | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑋) | |
| 4 | diffisn 6990 | . . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (𝑋 ∖ {𝑀}) ∈ Fin) | |
| 5 | 2, 3, 4 | syl2anc 411 | . . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑀}) ∈ Fin) |
| 6 | hashcl 10926 | . . . . 5 ⊢ ((𝑋 ∖ {𝑀}) ∈ Fin → (♯‘(𝑋 ∖ {𝑀})) ∈ ℕ0) | |
| 7 | 5, 6 | syl 14 | . . . 4 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) ∈ ℕ0) |
| 8 | nn0uz 9683 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
| 9 | 1m1e0 9105 | . . . . . 6 ⊢ (1 − 1) = 0 | |
| 10 | 9 | fveq2i 5579 | . . . . 5 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0) |
| 11 | 8, 10 | eqtr4i 2229 | . . . 4 ⊢ ℕ0 = (ℤ≥‘(1 − 1)) |
| 12 | 7, 11 | eleqtrdi 2298 | . . 3 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) ∈ (ℤ≥‘(1 − 1))) |
| 13 | fzsuc2 10201 | . . 3 ⊢ ((1 ∈ ℤ ∧ (♯‘(𝑋 ∖ {𝑀})) ∈ (ℤ≥‘(1 − 1))) → (1...((♯‘(𝑋 ∖ {𝑀})) + 1)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {((♯‘(𝑋 ∖ {𝑀})) + 1)})) | |
| 14 | 1, 12, 13 | syl2anc 411 | . 2 ⊢ (𝜑 → (1...((♯‘(𝑋 ∖ {𝑀})) + 1)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {((♯‘(𝑋 ∖ {𝑀})) + 1)})) |
| 15 | hashdifsn 10964 | . . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1)) | |
| 16 | 2, 3, 15 | syl2anc 411 | . . . . 5 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1)) |
| 17 | 16 | oveq1d 5959 | . . . 4 ⊢ (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (((♯‘𝑋) − 1) + 1)) |
| 18 | hashcl 10926 | . . . . . . 7 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0) | |
| 19 | 2, 18 | syl 14 | . . . . . 6 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ0) |
| 20 | 19 | nn0cnd 9350 | . . . . 5 ⊢ (𝜑 → (♯‘𝑋) ∈ ℂ) |
| 21 | 1cnd 8088 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
| 22 | 20, 21 | npcand 8387 | . . . 4 ⊢ (𝜑 → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋)) |
| 23 | 17, 22 | eqtrd 2238 | . . 3 ⊢ (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (♯‘𝑋)) |
| 24 | 23 | oveq2d 5960 | . 2 ⊢ (𝜑 → (1...((♯‘(𝑋 ∖ {𝑀})) + 1)) = (1...(♯‘𝑋))) |
| 25 | 23 | sneqd 3646 | . . 3 ⊢ (𝜑 → {((♯‘(𝑋 ∖ {𝑀})) + 1)} = {(♯‘𝑋)}) |
| 26 | 25 | uneq2d 3327 | . 2 ⊢ (𝜑 → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {((♯‘(𝑋 ∖ {𝑀})) + 1)}) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})) |
| 27 | 14, 24, 26 | 3eqtr3d 2246 | 1 ⊢ (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 = wceq 1373 ∈ wcel 2176 ∖ cdif 3163 ∪ cun 3164 {csn 3633 ‘cfv 5271 (class class class)co 5944 Fincfn 6827 0cc0 7925 1c1 7926 + caddc 7928 − cmin 8243 ℕ0cn0 9295 ℤcz 9372 ℤ≥cuz 9648 ...cfz 10130 ♯chash 10920 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-addcom 8025 ax-addass 8027 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-0id 8033 ax-rnegex 8034 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 |
| This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-irdg 6456 df-frec 6477 df-1o 6502 df-oadd 6506 df-er 6620 df-en 6828 df-dom 6829 df-fin 6830 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-inn 9037 df-n0 9296 df-z 9373 df-uz 9649 df-fz 10131 df-ihash 10921 |
| This theorem is referenced by: zfz1isolemiso 10984 zfz1isolem1 10985 |
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