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 11276 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
4 | peano2zm 11297 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
5 | id 22 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
6 | zre 11258 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
7 | 6 | lem1d 10836 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ≤ 𝑀) |
8 | 4, 5, 7 | 3jca 1235 | . . . . . . . 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 11569 | . . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘(𝑀 − 1)) ↔ ((𝑀 − 1) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 − 1) ≤ 𝑀)) | |
11 | 9, 10 | sylibr 223 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) |
12 | 1, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) |
13 | fzss2 12252 | . . . . 5 ⊢ (𝑀 ∈ (ℤ≥‘(𝑀 − 1)) → (0...(𝑀 − 1)) ⊆ (0...𝑀)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (0...(𝑀 − 1)) ⊆ (0...𝑀)) |
15 | fzossfz 12357 | . . . . . 6 ⊢ (1..^𝑀) ⊆ (1...𝑀) | |
16 | iccpartipre.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (1..^𝑀)) | |
17 | 15, 16 | sseldi 3566 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) |
18 | elfzoelz 12339 | . . . . . . 7 ⊢ (𝐼 ∈ (1..^𝑀) → 𝐼 ∈ ℤ) | |
19 | 16, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
20 | 1 | nnzd 11357 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
21 | elfzm1b 12287 | . . . . . 6 ⊢ ((𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐼 ∈ (1...𝑀) ↔ (𝐼 − 1) ∈ (0...(𝑀 − 1)))) | |
22 | 19, 20, 21 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (𝐼 ∈ (1...𝑀) ↔ (𝐼 − 1) ∈ (0...(𝑀 − 1)))) |
23 | 17, 22 | mpbid 221 | . . . 4 ⊢ (𝜑 → (𝐼 − 1) ∈ (0...(𝑀 − 1))) |
24 | 14, 23 | sseldd 3569 | . . 3 ⊢ (𝜑 → (𝐼 − 1) ∈ (0...𝑀)) |
25 | 1, 2, 24 | iccpartxr 39957 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) ∈ ℝ*) |
26 | 1eluzge0 11608 | . . . . . 6 ⊢ 1 ∈ (ℤ≥‘0) | |
27 | fzoss1 12364 | . . . . . 6 ⊢ (1 ∈ (ℤ≥‘0) → (1..^𝑀) ⊆ (0..^𝑀)) | |
28 | 26, 27 | mp1i 13 | . . . . 5 ⊢ (𝜑 → (1..^𝑀) ⊆ (0..^𝑀)) |
29 | fzossfz 12357 | . . . . 5 ⊢ (0..^𝑀) ⊆ (0...𝑀) | |
30 | 28, 29 | syl6ss 3580 | . . . 4 ⊢ (𝜑 → (1..^𝑀) ⊆ (0...𝑀)) |
31 | 30, 16 | sseldd 3569 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
32 | 1, 2, 31 | iccpartxr 39957 | . 2 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*) |
33 | 28, 16 | sseldd 3569 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) |
34 | fzofzp1 12431 | . . . 4 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) | |
35 | 33, 34 | syl 17 | . . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) |
36 | 1, 2, 35 | iccpartxr 39957 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 + 1)) ∈ ℝ*) |
37 | 1, 2, 17 | iccpartgtprec 39958 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼)) |
38 | iccpartimp 39955 | . . . 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 1318 | . . 3 ⊢ (𝜑 → (𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1)))) |
40 | 39 | simprd 478 | . 2 ⊢ (𝜑 → (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))) |
41 | xrre2 11875 | . 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 1326 | 1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℤcz 11254 ℤ≥cuz 11563 ...cfz 12197 ..^cfzo 12334 RePartciccp 39951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-iccp 39952 |
This theorem is referenced by: iccpartiltu 39960 iccpartigtl 39961 iccpartgt 39965 bgoldbtbndlem3 40223 |
Copyright terms: Public domain | W3C validator |