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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 11924 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
5 | fzval3 12944 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (1...𝑀) = (1..^(𝑀 + 1))) | |
6 | 5 | eleq2d 2866 | . . . . . . 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 11491 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
10 | pncan1 10901 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℂ → ((𝑀 + 1) − 1) = 𝑀) | |
11 | 9, 10 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀) |
12 | 11 | eqcomd 2799 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 = ((𝑀 + 1) − 1)) |
13 | 12 | oveq2d 7023 | . . . . . . 7 ⊢ (𝜑 → (0..^𝑀) = (0..^((𝑀 + 1) − 1))) |
14 | 13 | eleq2d 2866 | . . . . . 6 ⊢ (𝜑 → ((𝐼 − 1) ∈ (0..^𝑀) ↔ (𝐼 − 1) ∈ (0..^((𝑀 + 1) − 1)))) |
15 | elfzelz 12747 | . . . . . . . 8 ⊢ (𝐼 ∈ (1...𝑀) → 𝐼 ∈ ℤ) | |
16 | 3, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
17 | 4 | peano2zd 11928 | . . . . . . 7 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ) |
18 | elfzom1b 12974 | . . . . . . 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 43013 | . . . 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 1362 | . . 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 11926 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
26 | npcan1 10902 | . . . . 5 ⊢ (𝐼 ∈ ℂ → ((𝐼 − 1) + 1) = 𝐼) | |
27 | 25, 26 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐼 − 1) + 1) = 𝐼) |
28 | 27 | eqcomd 2799 | . . 3 ⊢ (𝜑 → 𝐼 = ((𝐼 − 1) + 1)) |
29 | 28 | fveq2d 6534 | . 2 ⊢ (𝜑 → (𝑃‘𝐼) = (𝑃‘((𝐼 − 1) + 1))) |
30 | 24, 29 | breqtrrd 4984 | 1 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1520 ∈ wcel 2079 class class class wbr 4956 ‘cfv 6217 (class class class)co 7007 ↑𝑚 cmap 8247 ℂcc 10370 0cc0 10372 1c1 10373 + caddc 10375 ℝ*cxr 10509 < clt 10510 − cmin 10706 ℕcn 11475 ℤcz 11818 ...cfz 12731 ..^cfzo 12872 RePartciccp 43009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-n0 11735 df-z 11819 df-uz 12083 df-fz 12732 df-fzo 12873 df-iccp 43010 |
This theorem is referenced by: iccpartipre 43017 iccpartiltu 43018 |
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