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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 12638 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
5 | fzval3 13770 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (1...𝑀) = (1..^(𝑀 + 1))) | |
6 | 5 | eleq2d 2825 | . . . . . . 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 232 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (1..^(𝑀 + 1))) |
9 | 1 | nncnd 12280 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
10 | pncan1 11685 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℂ → ((𝑀 + 1) − 1) = 𝑀) | |
11 | 9, 10 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀) |
12 | 11 | eqcomd 2741 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 = ((𝑀 + 1) − 1)) |
13 | 12 | oveq2d 7447 | . . . . . . 7 ⊢ (𝜑 → (0..^𝑀) = (0..^((𝑀 + 1) − 1))) |
14 | 13 | eleq2d 2825 | . . . . . 6 ⊢ (𝜑 → ((𝐼 − 1) ∈ (0..^𝑀) ↔ (𝐼 − 1) ∈ (0..^((𝑀 + 1) − 1)))) |
15 | 3 | elfzelzd 13562 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
16 | 4 | peano2zd 12723 | . . . . . . 7 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ) |
17 | elfzom1b 13802 | . . . . . . 7 ⊢ ((𝐼 ∈ ℤ ∧ (𝑀 + 1) ∈ ℤ) → (𝐼 ∈ (1..^(𝑀 + 1)) ↔ (𝐼 − 1) ∈ (0..^((𝑀 + 1) − 1)))) | |
18 | 15, 16, 17 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐼 ∈ (1..^(𝑀 + 1)) ↔ (𝐼 − 1) ∈ (0..^((𝑀 + 1) − 1)))) |
19 | 14, 18 | bitr4d 282 | . . . . 5 ⊢ (𝜑 → ((𝐼 − 1) ∈ (0..^𝑀) ↔ 𝐼 ∈ (1..^(𝑀 + 1)))) |
20 | 8, 19 | mpbird 257 | . . . 4 ⊢ (𝜑 → (𝐼 − 1) ∈ (0..^𝑀)) |
21 | iccpartimp 47342 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ (𝐼 − 1) ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘(𝐼 − 1)) < (𝑃‘((𝐼 − 1) + 1)))) | |
22 | 1, 2, 20, 21 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘(𝐼 − 1)) < (𝑃‘((𝐼 − 1) + 1)))) |
23 | 22 | simprd 495 | . 2 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘((𝐼 − 1) + 1))) |
24 | 15 | zcnd 12721 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
25 | npcan1 11686 | . . . . 5 ⊢ (𝐼 ∈ ℂ → ((𝐼 − 1) + 1) = 𝐼) | |
26 | 24, 25 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐼 − 1) + 1) = 𝐼) |
27 | 26 | eqcomd 2741 | . . 3 ⊢ (𝜑 → 𝐼 = ((𝐼 − 1) + 1)) |
28 | 27 | fveq2d 6911 | . 2 ⊢ (𝜑 → (𝑃‘𝐼) = (𝑃‘((𝐼 − 1) + 1))) |
29 | 23, 28 | breqtrrd 5176 | 1 ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 ℝ*cxr 11292 < clt 11293 − cmin 11490 ℕcn 12264 ℤcz 12611 ...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-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: iccpartipre 47346 iccpartiltu 47347 |
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