Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 11992 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
4 | peano2zm 12013 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
5 | id 22 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
6 | zre 11973 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
7 | 6 | lem1d 11561 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ≤ 𝑀) |
8 | 4, 5, 7 | 3jca 1120 | . . . . . . . 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 12237 | . . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘(𝑀 − 1)) ↔ ((𝑀 − 1) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 − 1) ≤ 𝑀)) | |
11 | 9, 10 | sylibr 235 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) |
12 | 1, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) |
13 | fzss2 12935 | . . . . 5 ⊢ (𝑀 ∈ (ℤ≥‘(𝑀 − 1)) → (0...(𝑀 − 1)) ⊆ (0...𝑀)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (0...(𝑀 − 1)) ⊆ (0...𝑀)) |
15 | fzossfz 13044 | . . . . . 6 ⊢ (1..^𝑀) ⊆ (1...𝑀) | |
16 | iccpartipre.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (1..^𝑀)) | |
17 | 15, 16 | sseldi 3962 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) |
18 | elfzoelz 13026 | . . . . . . 7 ⊢ (𝐼 ∈ (1..^𝑀) → 𝐼 ∈ ℤ) | |
19 | 16, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
20 | 1 | nnzd 12074 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
21 | elfzm1b 12973 | . . . . . 6 ⊢ ((𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐼 ∈ (1...𝑀) ↔ (𝐼 − 1) ∈ (0...(𝑀 − 1)))) | |
22 | 19, 20, 21 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝐼 ∈ (1...𝑀) ↔ (𝐼 − 1) ∈ (0...(𝑀 − 1)))) |
23 | 17, 22 | mpbid 233 | . . . 4 ⊢ (𝜑 → (𝐼 − 1) ∈ (0...(𝑀 − 1))) |
24 | 14, 23 | sseldd 3965 | . . 3 ⊢ (𝜑 → (𝐼 − 1) ∈ (0...𝑀)) |
25 | 1, 2, 24 | iccpartxr 43456 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) ∈ ℝ*) |
26 | 1eluzge0 12280 | . . . . . 6 ⊢ 1 ∈ (ℤ≥‘0) | |
27 | fzoss1 13052 | . . . . . 6 ⊢ (1 ∈ (ℤ≥‘0) → (1..^𝑀) ⊆ (0..^𝑀)) | |
28 | 26, 27 | mp1i 13 | . . . . 5 ⊢ (𝜑 → (1..^𝑀) ⊆ (0..^𝑀)) |
29 | fzossfz 13044 | . . . . 5 ⊢ (0..^𝑀) ⊆ (0...𝑀) | |
30 | 28, 29 | sstrdi 3976 | . . . 4 ⊢ (𝜑 → (1..^𝑀) ⊆ (0...𝑀)) |
31 | 30, 16 | sseldd 3965 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
32 | 1, 2, 31 | iccpartxr 43456 | . 2 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*) |
33 | 28, 16 | sseldd 3965 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) |
34 | fzofzp1 13122 | . . . 4 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) | |
35 | 33, 34 | syl 17 | . . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) |
36 | 1, 2, 35 | iccpartxr 43456 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 + 1)) ∈ ℝ*) |
37 | 1, 2, 17 | iccpartgtprec 43457 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼)) |
38 | iccpartimp 43454 | . . . 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 1363 | . . 3 ⊢ (𝜑 → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1)))) |
40 | 39 | simprd 496 | . 2 ⊢ (𝜑 → (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))) |
41 | xrre2 12551 | . 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 1370 | 1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 ∈ wcel 2105 ⊆ wss 3933 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 − cmin 10858 ℕcn 11626 ℤcz 11969 ℤ≥cuz 12231 ...cfz 12880 ..^cfzo 13021 RePartciccp 43450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-iccp 43451 |
This theorem is referenced by: iccpartiltu 43459 iccpartigtl 43460 iccpartgt 43464 bgoldbtbndlem3 43849 |
Copyright terms: Public domain | W3C validator |