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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 12632 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
4 | peano2zm 12658 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
5 | id 22 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
6 | zre 12615 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
7 | 6 | lem1d 12199 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ≤ 𝑀) |
8 | 4, 5, 7 | 3jca 1127 | . . . . . . . 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 12882 | . . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘(𝑀 − 1)) ↔ ((𝑀 − 1) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 − 1) ≤ 𝑀)) | |
11 | 9, 10 | sylibr 234 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) |
12 | 1, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) |
13 | fzss2 13601 | . . . . 5 ⊢ (𝑀 ∈ (ℤ≥‘(𝑀 − 1)) → (0...(𝑀 − 1)) ⊆ (0...𝑀)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (0...(𝑀 − 1)) ⊆ (0...𝑀)) |
15 | fzossfz 13715 | . . . . . 6 ⊢ (1..^𝑀) ⊆ (1...𝑀) | |
16 | iccpartipre.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (1..^𝑀)) | |
17 | 15, 16 | sselid 3993 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) |
18 | elfzoelz 13696 | . . . . . . 7 ⊢ (𝐼 ∈ (1..^𝑀) → 𝐼 ∈ ℤ) | |
19 | 16, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
20 | 1 | nnzd 12638 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
21 | elfzm1b 13639 | . . . . . 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 232 | . . . 4 ⊢ (𝜑 → (𝐼 − 1) ∈ (0...(𝑀 − 1))) |
24 | 14, 23 | sseldd 3996 | . . 3 ⊢ (𝜑 → (𝐼 − 1) ∈ (0...𝑀)) |
25 | 1, 2, 24 | iccpartxr 47344 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) ∈ ℝ*) |
26 | 1eluzge0 12932 | . . . . . 6 ⊢ 1 ∈ (ℤ≥‘0) | |
27 | fzoss1 13723 | . . . . . 6 ⊢ (1 ∈ (ℤ≥‘0) → (1..^𝑀) ⊆ (0..^𝑀)) | |
28 | 26, 27 | mp1i 13 | . . . . 5 ⊢ (𝜑 → (1..^𝑀) ⊆ (0..^𝑀)) |
29 | fzossfz 13715 | . . . . 5 ⊢ (0..^𝑀) ⊆ (0...𝑀) | |
30 | 28, 29 | sstrdi 4008 | . . . 4 ⊢ (𝜑 → (1..^𝑀) ⊆ (0...𝑀)) |
31 | 30, 16 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
32 | 1, 2, 31 | iccpartxr 47344 | . 2 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*) |
33 | 28, 16 | sseldd 3996 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) |
34 | fzofzp1 13800 | . . . 4 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) | |
35 | 33, 34 | syl 17 | . . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) |
36 | 1, 2, 35 | iccpartxr 47344 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 + 1)) ∈ ℝ*) |
37 | 1, 2, 17 | iccpartgtprec 47345 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼)) |
38 | iccpartimp 47342 | . . . 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 1370 | . . 3 ⊢ (𝜑 → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1)))) |
40 | 39 | simprd 495 | . 2 ⊢ (𝜑 → (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))) |
41 | xrre2 13209 | . 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 1377 | 1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 ⊆ wss 3963 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 ℕcn 12264 ℤcz 12611 ℤ≥cuz 12876 ...cfz 13544 ..^cfzo 13691 RePartciccp 47338 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-iccp 47339 |
This theorem is referenced by: iccpartiltu 47347 iccpartigtl 47348 iccpartgt 47352 bgoldbtbndlem3 47732 |
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