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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 11626 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0) | |
3 | elnn0uz 12007 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (ℤ≥‘0)) | |
4 | 2, 3 | sylib 210 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ≥‘0)) |
5 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0)) |
6 | fzisfzounsn 12875 | . . . . . 6 ⊢ (𝑀 ∈ (ℤ≥‘0) → (0...𝑀) = ((0..^𝑀) ∪ {𝑀})) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (0...𝑀) = ((0..^𝑀) ∪ {𝑀})) |
8 | 7 | eleq2d 2892 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) ↔ 𝑖 ∈ ((0..^𝑀) ∪ {𝑀}))) |
9 | elun 3980 | . . . . 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 4413 | . . . . . 6 ⊢ (𝑖 ∈ {𝑀} ↔ 𝑖 = 𝑀) | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑖 ∈ {𝑀} ↔ 𝑖 = 𝑀)) |
13 | 12 | orbi2d 946 | . . . 4 ⊢ (𝜑 → ((𝑖 ∈ (0..^𝑀) ∨ 𝑖 ∈ {𝑀}) ↔ (𝑖 ∈ (0..^𝑀) ∨ 𝑖 = 𝑀))) |
14 | 8, 10, 13 | 3bitrd 297 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) ↔ (𝑖 ∈ (0..^𝑀) ∨ 𝑖 = 𝑀))) |
15 | 1 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑀 ∈ ℕ) |
16 | iccpartgtprec.p | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) | |
17 | 16 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑃 ∈ (RePart‘𝑀)) |
18 | fzossfz 12783 | . . . . . . . . . 10 ⊢ (0..^𝑀) ⊆ (0...𝑀) | |
19 | 18 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → (0..^𝑀) ⊆ (0...𝑀)) |
20 | 19 | sselda 3827 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀)) |
21 | 15, 17, 20 | iccpartxr 42243 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) ∈ ℝ*) |
22 | nn0fz0 12732 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (0...𝑀)) | |
23 | 2, 22 | sylib 210 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀)) |
24 | 1, 23 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀)) |
25 | 1, 16, 24 | iccpartxr 42243 | . . . . . . . 8 ⊢ (𝜑 → (𝑃‘𝑀) ∈ ℝ*) |
26 | 25 | adantr 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑀) ∈ ℝ*) |
27 | 1, 16 | iccpartltu 42249 | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀)) |
28 | fveq2 6433 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖)) | |
29 | 28 | breq1d 4883 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → ((𝑃‘𝑘) < (𝑃‘𝑀) ↔ (𝑃‘𝑖) < (𝑃‘𝑀))) |
30 | 29 | rspccv 3523 | . . . . . . . . 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 397 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀)) |
33 | 21, 26, 32 | xrltled 12269 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) ≤ (𝑃‘𝑀)) |
34 | 33 | expcom 404 | . . . . 5 ⊢ (𝑖 ∈ (0..^𝑀) → (𝜑 → (𝑃‘𝑖) ≤ (𝑃‘𝑀))) |
35 | fveq2 6433 | . . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝑃‘𝑖) = (𝑃‘𝑀)) | |
36 | 35 | adantr 474 | . . . . . . 7 ⊢ ((𝑖 = 𝑀 ∧ 𝜑) → (𝑃‘𝑖) = (𝑃‘𝑀)) |
37 | 25 | xrleidd 12271 | . . . . . . . 8 ⊢ (𝜑 → (𝑃‘𝑀) ≤ (𝑃‘𝑀)) |
38 | 37 | adantl 475 | . . . . . . 7 ⊢ ((𝑖 = 𝑀 ∧ 𝜑) → (𝑃‘𝑀) ≤ (𝑃‘𝑀)) |
39 | 36, 38 | eqbrtrd 4895 | . . . . . 6 ⊢ ((𝑖 = 𝑀 ∧ 𝜑) → (𝑃‘𝑖) ≤ (𝑃‘𝑀)) |
40 | 39 | ex 403 | . . . . 5 ⊢ (𝑖 = 𝑀 → (𝜑 → (𝑃‘𝑖) ≤ (𝑃‘𝑀))) |
41 | 34, 40 | jaoi 890 | . . . 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 232 | . 2 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃‘𝑖) ≤ (𝑃‘𝑀))) |
44 | 43 | ralrimiv 3174 | 1 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃‘𝑖) ≤ (𝑃‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 880 = wceq 1658 ∈ wcel 2166 ∀wral 3117 ∪ cun 3796 ⊆ wss 3798 {csn 4397 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 0cc0 10252 ℝ*cxr 10390 < clt 10391 ≤ cle 10392 ℕcn 11350 ℕ0cn0 11618 ℤ≥cuz 11968 ...cfz 12619 ..^cfzo 12760 RePartciccp 42237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-iccp 42238 |
This theorem is referenced by: iccpartrn 42254 iccpartiun 42258 iccpartdisj 42261 |
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