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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpartgtprec | Structured version Visualization version GIF version |
Description: If there is a partition, then all intermediate points and the upper bound are strictly greater than the preceeding intermediate points or lower bound. (Contributed by AV, 11-Jul-2020.) |
Ref | Expression |
---|---|
iccpartgtprec.m | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
iccpartgtprec.p | ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) |
iccpartgtprec.i | ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) |
Ref | Expression |
---|---|
iccpartgtprec | ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccpartgtprec.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
2 | iccpartgtprec.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) | |
3 | iccpartgtprec.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) | |
4 | 1 | nnzd 12074 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
5 | fzval3 13094 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (1...𝑀) = (1..^(𝑀 + 1))) | |
6 | 5 | eleq2d 2895 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝐼 ∈ (1...𝑀) ↔ 𝐼 ∈ (1..^(𝑀 + 1)))) |
7 | 4, 6 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐼 ∈ (1...𝑀) ↔ 𝐼 ∈ (1..^(𝑀 + 1)))) |
8 | 3, 7 | mpbid 233 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (1..^(𝑀 + 1))) |
9 | 1 | nncnd 11642 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
10 | pncan1 11052 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℂ → ((𝑀 + 1) − 1) = 𝑀) | |
11 | 9, 10 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀) |
12 | 11 | eqcomd 2824 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 = ((𝑀 + 1) − 1)) |
13 | 12 | oveq2d 7161 | . . . . . . 7 ⊢ (𝜑 → (0..^𝑀) = (0..^((𝑀 + 1) − 1))) |
14 | 13 | eleq2d 2895 | . . . . . 6 ⊢ (𝜑 → ((𝐼 − 1) ∈ (0..^𝑀) ↔ (𝐼 − 1) ∈ (0..^((𝑀 + 1) − 1)))) |
15 | elfzelz 12896 | . . . . . . . 8 ⊢ (𝐼 ∈ (1...𝑀) → 𝐼 ∈ ℤ) | |
16 | 3, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
17 | 4 | peano2zd 12078 | . . . . . . 7 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ) |
18 | elfzom1b 13124 | . . . . . . 7 ⊢ ((𝐼 ∈ ℤ ∧ (𝑀 + 1) ∈ ℤ) → (𝐼 ∈ (1..^(𝑀 + 1)) ↔ (𝐼 − 1) ∈ (0..^((𝑀 + 1) − 1)))) | |
19 | 16, 17, 18 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐼 ∈ (1..^(𝑀 + 1)) ↔ (𝐼 − 1) ∈ (0..^((𝑀 + 1) − 1)))) |
20 | 14, 19 | bitr4d 283 | . . . . 5 ⊢ (𝜑 → ((𝐼 − 1) ∈ (0..^𝑀) ↔ 𝐼 ∈ (1..^(𝑀 + 1)))) |
21 | 8, 20 | mpbird 258 | . . . 4 ⊢ (𝜑 → (𝐼 − 1) ∈ (0..^𝑀)) |
22 | iccpartimp 43454 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ (𝐼 − 1) ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘(𝐼 − 1)) < (𝑃‘((𝐼 − 1) + 1)))) | |
23 | 1, 2, 21, 22 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘(𝐼 − 1)) < (𝑃‘((𝐼 − 1) + 1)))) |
24 | 23 | simprd 496 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘((𝐼 − 1) + 1))) |
25 | 16 | zcnd 12076 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
26 | npcan1 11053 | . . . . 5 ⊢ (𝐼 ∈ ℂ → ((𝐼 − 1) + 1) = 𝐼) | |
27 | 25, 26 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐼 − 1) + 1) = 𝐼) |
28 | 27 | eqcomd 2824 | . . 3 ⊢ (𝜑 → 𝐼 = ((𝐼 − 1) + 1)) |
29 | 28 | fveq2d 6667 | . 2 ⊢ (𝜑 → (𝑃‘𝐼) = (𝑃‘((𝐼 − 1) + 1))) |
30 | 24, 29 | breqtrrd 5085 | 1 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 ℂcc 10523 0cc0 10525 1c1 10526 + caddc 10528 ℝ*cxr 10662 < clt 10663 − cmin 10858 ℕcn 11626 ℤcz 11969 ...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-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: iccpartipre 43458 iccpartiltu 43459 |
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