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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 12519 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
4 | peano2zm 12545 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
5 | id 22 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
6 | zre 12502 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
7 | 6 | lem1d 12087 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ≤ 𝑀) |
8 | 4, 5, 7 | 3jca 1128 | . . . . . . . 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 12768 | . . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘(𝑀 − 1)) ↔ ((𝑀 − 1) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 − 1) ≤ 𝑀)) | |
11 | 9, 10 | sylibr 233 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) |
12 | 1, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) |
13 | fzss2 13480 | . . . . 5 ⊢ (𝑀 ∈ (ℤ≥‘(𝑀 − 1)) → (0...(𝑀 − 1)) ⊆ (0...𝑀)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (0...(𝑀 − 1)) ⊆ (0...𝑀)) |
15 | fzossfz 13590 | . . . . . 6 ⊢ (1..^𝑀) ⊆ (1...𝑀) | |
16 | iccpartipre.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (1..^𝑀)) | |
17 | 15, 16 | sselid 3942 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) |
18 | elfzoelz 13571 | . . . . . . 7 ⊢ (𝐼 ∈ (1..^𝑀) → 𝐼 ∈ ℤ) | |
19 | 16, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
20 | 1 | nnzd 12525 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
21 | elfzm1b 13518 | . . . . . 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 231 | . . . 4 ⊢ (𝜑 → (𝐼 − 1) ∈ (0...(𝑀 − 1))) |
24 | 14, 23 | sseldd 3945 | . . 3 ⊢ (𝜑 → (𝐼 − 1) ∈ (0...𝑀)) |
25 | 1, 2, 24 | iccpartxr 45583 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) ∈ ℝ*) |
26 | 1eluzge0 12816 | . . . . . 6 ⊢ 1 ∈ (ℤ≥‘0) | |
27 | fzoss1 13598 | . . . . . 6 ⊢ (1 ∈ (ℤ≥‘0) → (1..^𝑀) ⊆ (0..^𝑀)) | |
28 | 26, 27 | mp1i 13 | . . . . 5 ⊢ (𝜑 → (1..^𝑀) ⊆ (0..^𝑀)) |
29 | fzossfz 13590 | . . . . 5 ⊢ (0..^𝑀) ⊆ (0...𝑀) | |
30 | 28, 29 | sstrdi 3956 | . . . 4 ⊢ (𝜑 → (1..^𝑀) ⊆ (0...𝑀)) |
31 | 30, 16 | sseldd 3945 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
32 | 1, 2, 31 | iccpartxr 45583 | . 2 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*) |
33 | 28, 16 | sseldd 3945 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) |
34 | fzofzp1 13668 | . . . 4 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) | |
35 | 33, 34 | syl 17 | . . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) |
36 | 1, 2, 35 | iccpartxr 45583 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 + 1)) ∈ ℝ*) |
37 | 1, 2, 17 | iccpartgtprec 45584 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼)) |
38 | iccpartimp 45581 | . . . 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 1371 | . . 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 13088 | . 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 1378 | 1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 ⊆ wss 3910 class class class wbr 5105 ‘cfv 6496 (class class class)co 7356 ↑m cmap 8764 ℝcr 11049 0cc0 11050 1c1 11051 + caddc 11053 ℝ*cxr 11187 < clt 11188 ≤ cle 11189 − cmin 11384 ℕcn 12152 ℤcz 12498 ℤ≥cuz 12762 ...cfz 13423 ..^cfzo 13566 RePartciccp 45577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-map 8766 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-n0 12413 df-z 12499 df-uz 12763 df-fz 13424 df-fzo 13567 df-iccp 45578 |
This theorem is referenced by: iccpartiltu 45586 iccpartigtl 45587 iccpartgt 45591 bgoldbtbndlem3 45971 |
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