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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpartltu | Structured version Visualization version GIF version |
Description: If there is a partition, then all intermediate points and the lower bound are strictly less than the upper bound. (Contributed by AV, 14-Jul-2020.) |
Ref | Expression |
---|---|
iccpartgtprec.m | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
iccpartgtprec.p | ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) |
Ref | Expression |
---|---|
iccpartltu | ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccpartgtprec.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
2 | 0zd 11596 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 0 ∈ ℤ) | |
3 | nnz 11606 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
4 | nngt0 11255 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀) | |
5 | 2, 3, 4 | 3jca 1122 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀)) |
6 | 1, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀)) |
7 | fzopred 41855 | . . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀) → (0..^𝑀) = ({0} ∪ ((0 + 1)..^𝑀))) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → (0..^𝑀) = ({0} ∪ ((0 + 1)..^𝑀))) |
9 | 0p1e1 11338 | . . . . . . . 8 ⊢ (0 + 1) = 1 | |
10 | 9 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (0 + 1) = 1) |
11 | 10 | oveq1d 6811 | . . . . . 6 ⊢ (𝜑 → ((0 + 1)..^𝑀) = (1..^𝑀)) |
12 | 11 | uneq2d 3918 | . . . . 5 ⊢ (𝜑 → ({0} ∪ ((0 + 1)..^𝑀)) = ({0} ∪ (1..^𝑀))) |
13 | 8, 12 | eqtrd 2805 | . . . 4 ⊢ (𝜑 → (0..^𝑀) = ({0} ∪ (1..^𝑀))) |
14 | 13 | eleq2d 2836 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) ↔ 𝑖 ∈ ({0} ∪ (1..^𝑀)))) |
15 | elun 3904 | . . . 4 ⊢ (𝑖 ∈ ({0} ∪ (1..^𝑀)) ↔ (𝑖 ∈ {0} ∨ 𝑖 ∈ (1..^𝑀))) | |
16 | elsni 4334 | . . . . . . 7 ⊢ (𝑖 ∈ {0} → 𝑖 = 0) | |
17 | fveq2 6333 | . . . . . . . . . 10 ⊢ (𝑖 = 0 → (𝑃‘𝑖) = (𝑃‘0)) | |
18 | 17 | adantr 466 | . . . . . . . . 9 ⊢ ((𝑖 = 0 ∧ 𝜑) → (𝑃‘𝑖) = (𝑃‘0)) |
19 | iccpartgtprec.p | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) | |
20 | 1, 19 | iccpartlt 41883 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀)) |
21 | 20 | adantl 467 | . . . . . . . . 9 ⊢ ((𝑖 = 0 ∧ 𝜑) → (𝑃‘0) < (𝑃‘𝑀)) |
22 | 18, 21 | eqbrtrd 4809 | . . . . . . . 8 ⊢ ((𝑖 = 0 ∧ 𝜑) → (𝑃‘𝑖) < (𝑃‘𝑀)) |
23 | 22 | ex 397 | . . . . . . 7 ⊢ (𝑖 = 0 → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
24 | 16, 23 | syl 17 | . . . . . 6 ⊢ (𝑖 ∈ {0} → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
25 | fveq2 6333 | . . . . . . . . 9 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖)) | |
26 | 25 | breq1d 4797 | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → ((𝑃‘𝑘) < (𝑃‘𝑀) ↔ (𝑃‘𝑖) < (𝑃‘𝑀))) |
27 | 26 | rspccv 3457 | . . . . . . 7 ⊢ (∀𝑘 ∈ (1..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀) → (𝑖 ∈ (1..^𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
28 | 1, 19 | iccpartiltu 41881 | . . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀)) |
29 | 27, 28 | syl11 33 | . . . . . 6 ⊢ (𝑖 ∈ (1..^𝑀) → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
30 | 24, 29 | jaoi 846 | . . . . 5 ⊢ ((𝑖 ∈ {0} ∨ 𝑖 ∈ (1..^𝑀)) → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
31 | 30 | com12 32 | . . . 4 ⊢ (𝜑 → ((𝑖 ∈ {0} ∨ 𝑖 ∈ (1..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
32 | 15, 31 | syl5bi 232 | . . 3 ⊢ (𝜑 → (𝑖 ∈ ({0} ∪ (1..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
33 | 14, 32 | sylbid 230 | . 2 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
34 | 33 | ralrimiv 3114 | 1 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∨ wo 836 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∀wral 3061 ∪ cun 3721 {csn 4317 class class class wbr 4787 ‘cfv 6030 (class class class)co 6796 0cc0 10142 1c1 10143 + caddc 10145 < clt 10280 ℕcn 11226 ℤcz 11584 ..^cfzo 12673 RePartciccp 41872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-er 7900 df-map 8015 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-nn 11227 df-2 11285 df-n0 11500 df-z 11585 df-uz 11894 df-fz 12534 df-fzo 12674 df-iccp 41873 |
This theorem is referenced by: iccpartleu 41887 |
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