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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 11907 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0) | |
3 | elnn0uz 12286 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (ℤ≥‘0)) | |
4 | 2, 3 | sylib 220 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ≥‘0)) |
5 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0)) |
6 | fzisfzounsn 13152 | . . . . . 6 ⊢ (𝑀 ∈ (ℤ≥‘0) → (0...𝑀) = ((0..^𝑀) ∪ {𝑀})) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (0...𝑀) = ((0..^𝑀) ∪ {𝑀})) |
8 | 7 | eleq2d 2900 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) ↔ 𝑖 ∈ ((0..^𝑀) ∪ {𝑀}))) |
9 | elun 4127 | . . . . 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 4585 | . . . . . 6 ⊢ (𝑖 ∈ {𝑀} ↔ 𝑖 = 𝑀) | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑖 ∈ {𝑀} ↔ 𝑖 = 𝑀)) |
13 | 12 | orbi2d 912 | . . . 4 ⊢ (𝜑 → ((𝑖 ∈ (0..^𝑀) ∨ 𝑖 ∈ {𝑀}) ↔ (𝑖 ∈ (0..^𝑀) ∨ 𝑖 = 𝑀))) |
14 | 8, 10, 13 | 3bitrd 307 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) ↔ (𝑖 ∈ (0..^𝑀) ∨ 𝑖 = 𝑀))) |
15 | 1 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑀 ∈ ℕ) |
16 | iccpartgtprec.p | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) | |
17 | 16 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑃 ∈ (RePart‘𝑀)) |
18 | fzossfz 13059 | . . . . . . . . . 10 ⊢ (0..^𝑀) ⊆ (0...𝑀) | |
19 | 18 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → (0..^𝑀) ⊆ (0...𝑀)) |
20 | 19 | sselda 3969 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀)) |
21 | 15, 17, 20 | iccpartxr 43586 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) ∈ ℝ*) |
22 | nn0fz0 13008 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (0...𝑀)) | |
23 | 2, 22 | sylib 220 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀)) |
24 | 1, 23 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀)) |
25 | 1, 16, 24 | iccpartxr 43586 | . . . . . . . 8 ⊢ (𝜑 → (𝑃‘𝑀) ∈ ℝ*) |
26 | 25 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑀) ∈ ℝ*) |
27 | 1, 16 | iccpartltu 43592 | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀)) |
28 | fveq2 6672 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖)) | |
29 | 28 | breq1d 5078 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → ((𝑃‘𝑘) < (𝑃‘𝑀) ↔ (𝑃‘𝑖) < (𝑃‘𝑀))) |
30 | 29 | rspccv 3622 | . . . . . . . . 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 409 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀)) |
33 | 21, 26, 32 | xrltled 12546 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) ≤ (𝑃‘𝑀)) |
34 | 33 | expcom 416 | . . . . 5 ⊢ (𝑖 ∈ (0..^𝑀) → (𝜑 → (𝑃‘𝑖) ≤ (𝑃‘𝑀))) |
35 | fveq2 6672 | . . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝑃‘𝑖) = (𝑃‘𝑀)) | |
36 | 35 | adantr 483 | . . . . . . 7 ⊢ ((𝑖 = 𝑀 ∧ 𝜑) → (𝑃‘𝑖) = (𝑃‘𝑀)) |
37 | 25 | xrleidd 12548 | . . . . . . . 8 ⊢ (𝜑 → (𝑃‘𝑀) ≤ (𝑃‘𝑀)) |
38 | 37 | adantl 484 | . . . . . . 7 ⊢ ((𝑖 = 𝑀 ∧ 𝜑) → (𝑃‘𝑀) ≤ (𝑃‘𝑀)) |
39 | 36, 38 | eqbrtrd 5090 | . . . . . 6 ⊢ ((𝑖 = 𝑀 ∧ 𝜑) → (𝑃‘𝑖) ≤ (𝑃‘𝑀)) |
40 | 39 | ex 415 | . . . . 5 ⊢ (𝑖 = 𝑀 → (𝜑 → (𝑃‘𝑖) ≤ (𝑃‘𝑀))) |
41 | 34, 40 | jaoi 853 | . . . 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 242 | . 2 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃‘𝑖) ≤ (𝑃‘𝑀))) |
44 | 43 | ralrimiv 3183 | 1 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃‘𝑖) ≤ (𝑃‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∪ cun 3936 ⊆ wss 3938 {csn 4569 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 0cc0 10539 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 ℕcn 11640 ℕ0cn0 11900 ℤ≥cuz 12246 ...cfz 12895 ..^cfzo 13036 RePartciccp 43580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-iccp 43581 |
This theorem is referenced by: iccpartrn 43597 iccpartiun 43601 iccpartdisj 43604 |
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