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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpartleu | Structured version Visualization version GIF version |
Description: If there is a partition, then all intermediate points and the lower and the upper bound are less than or equal to the upper bound. (Contributed by AV, 14-Jul-2020.) |
Ref | Expression |
---|---|
iccpartgtprec.m | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
iccpartgtprec.p | ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) |
Ref | Expression |
---|---|
iccpartleu | ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃‘𝑖) ≤ (𝑃‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccpartgtprec.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
2 | nnnn0 11892 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0) | |
3 | elnn0uz 12271 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (ℤ≥‘0)) | |
4 | 2, 3 | sylib 221 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ≥‘0)) |
5 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0)) |
6 | fzisfzounsn 13144 | . . . . . 6 ⊢ (𝑀 ∈ (ℤ≥‘0) → (0...𝑀) = ((0..^𝑀) ∪ {𝑀})) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (0...𝑀) = ((0..^𝑀) ∪ {𝑀})) |
8 | 7 | eleq2d 2875 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) ↔ 𝑖 ∈ ((0..^𝑀) ∪ {𝑀}))) |
9 | elun 4076 | . . . . 5 ⊢ (𝑖 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑖 ∈ (0..^𝑀) ∨ 𝑖 ∈ {𝑀})) | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑖 ∈ (0..^𝑀) ∨ 𝑖 ∈ {𝑀}))) |
11 | velsn 4541 | . . . . . 6 ⊢ (𝑖 ∈ {𝑀} ↔ 𝑖 = 𝑀) | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑖 ∈ {𝑀} ↔ 𝑖 = 𝑀)) |
13 | 12 | orbi2d 913 | . . . 4 ⊢ (𝜑 → ((𝑖 ∈ (0..^𝑀) ∨ 𝑖 ∈ {𝑀}) ↔ (𝑖 ∈ (0..^𝑀) ∨ 𝑖 = 𝑀))) |
14 | 8, 10, 13 | 3bitrd 308 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) ↔ (𝑖 ∈ (0..^𝑀) ∨ 𝑖 = 𝑀))) |
15 | 1 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑀 ∈ ℕ) |
16 | iccpartgtprec.p | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) | |
17 | 16 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑃 ∈ (RePart‘𝑀)) |
18 | fzossfz 13051 | . . . . . . . . . 10 ⊢ (0..^𝑀) ⊆ (0...𝑀) | |
19 | 18 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → (0..^𝑀) ⊆ (0...𝑀)) |
20 | 19 | sselda 3915 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀)) |
21 | 15, 17, 20 | iccpartxr 43936 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) ∈ ℝ*) |
22 | nn0fz0 13000 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (0...𝑀)) | |
23 | 2, 22 | sylib 221 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀)) |
24 | 1, 23 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀)) |
25 | 1, 16, 24 | iccpartxr 43936 | . . . . . . . 8 ⊢ (𝜑 → (𝑃‘𝑀) ∈ ℝ*) |
26 | 25 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑀) ∈ ℝ*) |
27 | 1, 16 | iccpartltu 43942 | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀)) |
28 | fveq2 6645 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖)) | |
29 | 28 | breq1d 5040 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → ((𝑃‘𝑘) < (𝑃‘𝑀) ↔ (𝑃‘𝑖) < (𝑃‘𝑀))) |
30 | 29 | rspccv 3568 | . . . . . . . . 9 ⊢ (∀𝑘 ∈ (0..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀) → (𝑖 ∈ (0..^𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
31 | 27, 30 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
32 | 31 | imp 410 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀)) |
33 | 21, 26, 32 | xrltled 12531 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) ≤ (𝑃‘𝑀)) |
34 | 33 | expcom 417 | . . . . 5 ⊢ (𝑖 ∈ (0..^𝑀) → (𝜑 → (𝑃‘𝑖) ≤ (𝑃‘𝑀))) |
35 | fveq2 6645 | . . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝑃‘𝑖) = (𝑃‘𝑀)) | |
36 | 35 | adantr 484 | . . . . . . 7 ⊢ ((𝑖 = 𝑀 ∧ 𝜑) → (𝑃‘𝑖) = (𝑃‘𝑀)) |
37 | 25 | xrleidd 12533 | . . . . . . . 8 ⊢ (𝜑 → (𝑃‘𝑀) ≤ (𝑃‘𝑀)) |
38 | 37 | adantl 485 | . . . . . . 7 ⊢ ((𝑖 = 𝑀 ∧ 𝜑) → (𝑃‘𝑀) ≤ (𝑃‘𝑀)) |
39 | 36, 38 | eqbrtrd 5052 | . . . . . 6 ⊢ ((𝑖 = 𝑀 ∧ 𝜑) → (𝑃‘𝑖) ≤ (𝑃‘𝑀)) |
40 | 39 | ex 416 | . . . . 5 ⊢ (𝑖 = 𝑀 → (𝜑 → (𝑃‘𝑖) ≤ (𝑃‘𝑀))) |
41 | 34, 40 | jaoi 854 | . . . 4 ⊢ ((𝑖 ∈ (0..^𝑀) ∨ 𝑖 = 𝑀) → (𝜑 → (𝑃‘𝑖) ≤ (𝑃‘𝑀))) |
42 | 41 | com12 32 | . . 3 ⊢ (𝜑 → ((𝑖 ∈ (0..^𝑀) ∨ 𝑖 = 𝑀) → (𝑃‘𝑖) ≤ (𝑃‘𝑀))) |
43 | 14, 42 | sylbid 243 | . 2 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃‘𝑖) ≤ (𝑃‘𝑀))) |
44 | 43 | ralrimiv 3148 | 1 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃‘𝑖) ≤ (𝑃‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∪ cun 3879 ⊆ wss 3881 {csn 4525 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 0cc0 10526 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 ℕcn 11625 ℕ0cn0 11885 ℤ≥cuz 12231 ...cfz 12885 ..^cfzo 13028 RePartciccp 43930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-iccp 43931 |
This theorem is referenced by: iccpartrn 43947 iccpartiun 43951 iccpartdisj 43954 |
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