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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 12080 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
5 | fzval3 13100 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (1...𝑀) = (1..^(𝑀 + 1))) | |
6 | 5 | eleq2d 2898 | . . . . . . 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 234 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (1..^(𝑀 + 1))) |
9 | 1 | nncnd 11648 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
10 | pncan1 11058 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℂ → ((𝑀 + 1) − 1) = 𝑀) | |
11 | 9, 10 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀) |
12 | 11 | eqcomd 2827 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 = ((𝑀 + 1) − 1)) |
13 | 12 | oveq2d 7166 | . . . . . . 7 ⊢ (𝜑 → (0..^𝑀) = (0..^((𝑀 + 1) − 1))) |
14 | 13 | eleq2d 2898 | . . . . . 6 ⊢ (𝜑 → ((𝐼 − 1) ∈ (0..^𝑀) ↔ (𝐼 − 1) ∈ (0..^((𝑀 + 1) − 1)))) |
15 | elfzelz 12902 | . . . . . . . 8 ⊢ (𝐼 ∈ (1...𝑀) → 𝐼 ∈ ℤ) | |
16 | 3, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
17 | 4 | peano2zd 12084 | . . . . . . 7 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ) |
18 | elfzom1b 13130 | . . . . . . 7 ⊢ ((𝐼 ∈ ℤ ∧ (𝑀 + 1) ∈ ℤ) → (𝐼 ∈ (1..^(𝑀 + 1)) ↔ (𝐼 − 1) ∈ (0..^((𝑀 + 1) − 1)))) | |
19 | 16, 17, 18 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝐼 ∈ (1..^(𝑀 + 1)) ↔ (𝐼 − 1) ∈ (0..^((𝑀 + 1) − 1)))) |
20 | 14, 19 | bitr4d 284 | . . . . 5 ⊢ (𝜑 → ((𝐼 − 1) ∈ (0..^𝑀) ↔ 𝐼 ∈ (1..^(𝑀 + 1)))) |
21 | 8, 20 | mpbird 259 | . . . 4 ⊢ (𝜑 → (𝐼 − 1) ∈ (0..^𝑀)) |
22 | iccpartimp 43571 | . . . 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 1367 | . . 3 ⊢ (𝜑 → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘(𝐼 − 1)) < (𝑃‘((𝐼 − 1) + 1)))) |
24 | 23 | simprd 498 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘((𝐼 − 1) + 1))) |
25 | 16 | zcnd 12082 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
26 | npcan1 11059 | . . . . 5 ⊢ (𝐼 ∈ ℂ → ((𝐼 − 1) + 1) = 𝐼) | |
27 | 25, 26 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐼 − 1) + 1) = 𝐼) |
28 | 27 | eqcomd 2827 | . . 3 ⊢ (𝜑 → 𝐼 = ((𝐼 − 1) + 1)) |
29 | 28 | fveq2d 6668 | . 2 ⊢ (𝜑 → (𝑃‘𝐼) = (𝑃‘((𝐼 − 1) + 1))) |
30 | 24, 29 | breqtrrd 5086 | 1 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 ℂcc 10529 0cc0 10531 1c1 10532 + caddc 10534 ℝ*cxr 10668 < clt 10669 − cmin 10864 ℕcn 11632 ℤcz 11975 ...cfz 12886 ..^cfzo 13027 RePartciccp 43567 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-iccp 43568 |
This theorem is referenced by: iccpartipre 43575 iccpartiltu 43576 |
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