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Mirrors > Home > ILE Home > Th. List > frecfzen2 | GIF version |
Description: The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.) |
Ref | Expression |
---|---|
frecfzennn.1 | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) |
Ref | Expression |
---|---|
frecfzen2 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) ≈ (◡𝐺‘((𝑁 + 1) − 𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 9355 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
2 | eluzelz 9359 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
3 | 1z 9104 | . . . . 5 ⊢ 1 ∈ ℤ | |
4 | zsubcl 9119 | . . . . 5 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (1 − 𝑀) ∈ ℤ) | |
5 | 3, 1, 4 | sylancr 411 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (1 − 𝑀) ∈ ℤ) |
6 | fzen 9854 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (1 − 𝑀) ∈ ℤ) → (𝑀...𝑁) ≈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀)))) | |
7 | 1, 2, 5, 6 | syl3anc 1217 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) ≈ ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀)))) |
8 | 1 | zcnd 9198 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℂ) |
9 | ax-1cn 7737 | . . . . 5 ⊢ 1 ∈ ℂ | |
10 | pncan3 7994 | . . . . 5 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑀 + (1 − 𝑀)) = 1) | |
11 | 8, 9, 10 | sylancl 410 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀 + (1 − 𝑀)) = 1) |
12 | zcn 9083 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
13 | zcn 9083 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
14 | addsubass 7996 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑀 ∈ ℂ) → ((𝑁 + 1) − 𝑀) = (𝑁 + (1 − 𝑀))) | |
15 | 9, 14 | mp3an2 1304 | . . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → ((𝑁 + 1) − 𝑀) = (𝑁 + (1 − 𝑀))) |
16 | 12, 13, 15 | syl2an 287 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 + 1) − 𝑀) = (𝑁 + (1 − 𝑀))) |
17 | 2, 1, 16 | syl2anc 409 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑁 + 1) − 𝑀) = (𝑁 + (1 − 𝑀))) |
18 | 17 | eqcomd 2146 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + (1 − 𝑀)) = ((𝑁 + 1) − 𝑀)) |
19 | 11, 18 | oveq12d 5800 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀 + (1 − 𝑀))...(𝑁 + (1 − 𝑀))) = (1...((𝑁 + 1) − 𝑀))) |
20 | 7, 19 | breqtrd 3962 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) ≈ (1...((𝑁 + 1) − 𝑀))) |
21 | peano2uz 9405 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) | |
22 | uznn0sub 9381 | . . 3 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) → ((𝑁 + 1) − 𝑀) ∈ ℕ0) | |
23 | frecfzennn.1 | . . . 4 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
24 | 23 | frecfzennn 10230 | . . 3 ⊢ (((𝑁 + 1) − 𝑀) ∈ ℕ0 → (1...((𝑁 + 1) − 𝑀)) ≈ (◡𝐺‘((𝑁 + 1) − 𝑀))) |
25 | 21, 22, 24 | 3syl 17 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (1...((𝑁 + 1) − 𝑀)) ≈ (◡𝐺‘((𝑁 + 1) − 𝑀))) |
26 | entr 6686 | . 2 ⊢ (((𝑀...𝑁) ≈ (1...((𝑁 + 1) − 𝑀)) ∧ (1...((𝑁 + 1) − 𝑀)) ≈ (◡𝐺‘((𝑁 + 1) − 𝑀))) → (𝑀...𝑁) ≈ (◡𝐺‘((𝑁 + 1) − 𝑀))) | |
27 | 20, 25, 26 | syl2anc 409 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) ≈ (◡𝐺‘((𝑁 + 1) − 𝑀))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 ↦ cmpt 3997 ◡ccnv 4546 ‘cfv 5131 (class class class)co 5782 freccfrec 6295 ≈ cen 6640 ℂcc 7642 0cc0 7644 1c1 7645 + caddc 7647 − cmin 7957 ℕ0cn0 9001 ℤcz 9078 ℤ≥cuz 9350 ...cfz 9821 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-1o 6321 df-er 6437 df-en 6643 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-fz 9822 |
This theorem is referenced by: fzfig 10234 |
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