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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpartipre | Structured version Visualization version GIF version |
Description: If there is a partition, then all intermediate points are real numbers. (Contributed by AV, 11-Jul-2020.) |
Ref | Expression |
---|---|
iccpartgtprec.m | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
iccpartgtprec.p | ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) |
iccpartipre.i | ⊢ (𝜑 → 𝐼 ∈ (1..^𝑀)) |
Ref | Expression |
---|---|
iccpartipre | ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccpartgtprec.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
2 | iccpartgtprec.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) | |
3 | nnz 12182 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
4 | peano2zm 12203 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
5 | id 22 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
6 | zre 12163 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
7 | 6 | lem1d 11748 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ≤ 𝑀) |
8 | 4, 5, 7 | 3jca 1130 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 − 1) ≤ 𝑀)) |
9 | 3, 8 | syl 17 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → ((𝑀 − 1) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 − 1) ≤ 𝑀)) |
10 | eluz2 12427 | . . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘(𝑀 − 1)) ↔ ((𝑀 − 1) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 − 1) ≤ 𝑀)) | |
11 | 9, 10 | sylibr 237 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) |
12 | 1, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) |
13 | fzss2 13135 | . . . . 5 ⊢ (𝑀 ∈ (ℤ≥‘(𝑀 − 1)) → (0...(𝑀 − 1)) ⊆ (0...𝑀)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (0...(𝑀 − 1)) ⊆ (0...𝑀)) |
15 | fzossfz 13244 | . . . . . 6 ⊢ (1..^𝑀) ⊆ (1...𝑀) | |
16 | iccpartipre.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (1..^𝑀)) | |
17 | 15, 16 | sseldi 3889 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) |
18 | elfzoelz 13226 | . . . . . . 7 ⊢ (𝐼 ∈ (1..^𝑀) → 𝐼 ∈ ℤ) | |
19 | 16, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
20 | 1 | nnzd 12264 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
21 | elfzm1b 13173 | . . . . . 6 ⊢ ((𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐼 ∈ (1...𝑀) ↔ (𝐼 − 1) ∈ (0...(𝑀 − 1)))) | |
22 | 19, 20, 21 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝐼 ∈ (1...𝑀) ↔ (𝐼 − 1) ∈ (0...(𝑀 − 1)))) |
23 | 17, 22 | mpbid 235 | . . . 4 ⊢ (𝜑 → (𝐼 − 1) ∈ (0...(𝑀 − 1))) |
24 | 14, 23 | sseldd 3892 | . . 3 ⊢ (𝜑 → (𝐼 − 1) ∈ (0...𝑀)) |
25 | 1, 2, 24 | iccpartxr 44498 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) ∈ ℝ*) |
26 | 1eluzge0 12471 | . . . . . 6 ⊢ 1 ∈ (ℤ≥‘0) | |
27 | fzoss1 13252 | . . . . . 6 ⊢ (1 ∈ (ℤ≥‘0) → (1..^𝑀) ⊆ (0..^𝑀)) | |
28 | 26, 27 | mp1i 13 | . . . . 5 ⊢ (𝜑 → (1..^𝑀) ⊆ (0..^𝑀)) |
29 | fzossfz 13244 | . . . . 5 ⊢ (0..^𝑀) ⊆ (0...𝑀) | |
30 | 28, 29 | sstrdi 3903 | . . . 4 ⊢ (𝜑 → (1..^𝑀) ⊆ (0...𝑀)) |
31 | 30, 16 | sseldd 3892 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
32 | 1, 2, 31 | iccpartxr 44498 | . 2 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*) |
33 | 28, 16 | sseldd 3892 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) |
34 | fzofzp1 13322 | . . . 4 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) | |
35 | 33, 34 | syl 17 | . . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) |
36 | 1, 2, 35 | iccpartxr 44498 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 + 1)) ∈ ℝ*) |
37 | 1, 2, 17 | iccpartgtprec 44499 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼)) |
38 | iccpartimp 44496 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝐼 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1)))) | |
39 | 1, 2, 33, 38 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1)))) |
40 | 39 | simprd 499 | . 2 ⊢ (𝜑 → (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))) |
41 | xrre2 12743 | . 2 ⊢ ((((𝑃‘(𝐼 − 1)) ∈ ℝ* ∧ (𝑃‘𝐼) ∈ ℝ* ∧ (𝑃‘(𝐼 + 1)) ∈ ℝ*) ∧ ((𝑃‘(𝐼 − 1)) < (𝑃‘𝐼) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1)))) → (𝑃‘𝐼) ∈ ℝ) | |
42 | 25, 32, 36, 37, 40, 41 | syl32anc 1380 | 1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2110 ⊆ wss 3857 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 ↑m cmap 8497 ℝcr 10711 0cc0 10712 1c1 10713 + caddc 10715 ℝ*cxr 10849 < clt 10850 ≤ cle 10851 − cmin 11045 ℕcn 11813 ℤcz 12159 ℤ≥cuz 12421 ...cfz 13078 ..^cfzo 13221 RePartciccp 44492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-fzo 13222 df-iccp 44493 |
This theorem is referenced by: iccpartiltu 44501 iccpartigtl 44502 iccpartgt 44506 bgoldbtbndlem3 44886 |
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