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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpartgtl | 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 lower bound. (Contributed by AV, 14-Jul-2020.) |
Ref | Expression |
---|---|
iccpartgtprec.m | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
iccpartgtprec.p | ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) |
Ref | Expression |
---|---|
iccpartgtl | ⊢ (𝜑 → ∀𝑖 ∈ (1...𝑀)(𝑃‘0) < (𝑃‘𝑖)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccpartgtprec.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
2 | elnnuz 12848 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ≥‘1)) | |
3 | 1, 2 | sylib 217 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1)) |
4 | fzisfzounsn 13726 | . . . . . 6 ⊢ (𝑀 ∈ (ℤ≥‘1) → (1...𝑀) = ((1..^𝑀) ∪ {𝑀})) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → (1...𝑀) = ((1..^𝑀) ∪ {𝑀})) |
6 | 5 | eleq2d 2818 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ (1...𝑀) ↔ 𝑖 ∈ ((1..^𝑀) ∪ {𝑀}))) |
7 | elun 4144 | . . . . 5 ⊢ (𝑖 ∈ ((1..^𝑀) ∪ {𝑀}) ↔ (𝑖 ∈ (1..^𝑀) ∨ 𝑖 ∈ {𝑀})) | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ ((1..^𝑀) ∪ {𝑀}) ↔ (𝑖 ∈ (1..^𝑀) ∨ 𝑖 ∈ {𝑀}))) |
9 | velsn 4638 | . . . . . 6 ⊢ (𝑖 ∈ {𝑀} ↔ 𝑖 = 𝑀) | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑖 ∈ {𝑀} ↔ 𝑖 = 𝑀)) |
11 | 10 | orbi2d 914 | . . . 4 ⊢ (𝜑 → ((𝑖 ∈ (1..^𝑀) ∨ 𝑖 ∈ {𝑀}) ↔ (𝑖 ∈ (1..^𝑀) ∨ 𝑖 = 𝑀))) |
12 | 6, 8, 11 | 3bitrd 304 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (1...𝑀) ↔ (𝑖 ∈ (1..^𝑀) ∨ 𝑖 = 𝑀))) |
13 | fveq2 6878 | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖)) | |
14 | 13 | breq2d 5153 | . . . . . . 7 ⊢ (𝑘 = 𝑖 → ((𝑃‘0) < (𝑃‘𝑘) ↔ (𝑃‘0) < (𝑃‘𝑖))) |
15 | 14 | rspccv 3606 | . . . . . 6 ⊢ (∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑘) → (𝑖 ∈ (1..^𝑀) → (𝑃‘0) < (𝑃‘𝑖))) |
16 | iccpartgtprec.p | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) | |
17 | 1, 16 | iccpartigtl 45863 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑘)) |
18 | 15, 17 | syl11 33 | . . . . 5 ⊢ (𝑖 ∈ (1..^𝑀) → (𝜑 → (𝑃‘0) < (𝑃‘𝑖))) |
19 | 1, 16 | iccpartlt 45864 | . . . . . . . 8 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀)) |
20 | 19 | adantl 482 | . . . . . . 7 ⊢ ((𝑖 = 𝑀 ∧ 𝜑) → (𝑃‘0) < (𝑃‘𝑀)) |
21 | fveq2 6878 | . . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝑃‘𝑖) = (𝑃‘𝑀)) | |
22 | 21 | adantr 481 | . . . . . . 7 ⊢ ((𝑖 = 𝑀 ∧ 𝜑) → (𝑃‘𝑖) = (𝑃‘𝑀)) |
23 | 20, 22 | breqtrrd 5169 | . . . . . 6 ⊢ ((𝑖 = 𝑀 ∧ 𝜑) → (𝑃‘0) < (𝑃‘𝑖)) |
24 | 23 | ex 413 | . . . . 5 ⊢ (𝑖 = 𝑀 → (𝜑 → (𝑃‘0) < (𝑃‘𝑖))) |
25 | 18, 24 | jaoi 855 | . . . 4 ⊢ ((𝑖 ∈ (1..^𝑀) ∨ 𝑖 = 𝑀) → (𝜑 → (𝑃‘0) < (𝑃‘𝑖))) |
26 | 25 | com12 32 | . . 3 ⊢ (𝜑 → ((𝑖 ∈ (1..^𝑀) ∨ 𝑖 = 𝑀) → (𝑃‘0) < (𝑃‘𝑖))) |
27 | 12, 26 | sylbid 239 | . 2 ⊢ (𝜑 → (𝑖 ∈ (1...𝑀) → (𝑃‘0) < (𝑃‘𝑖))) |
28 | 27 | ralrimiv 3144 | 1 ⊢ (𝜑 → ∀𝑖 ∈ (1...𝑀)(𝑃‘0) < (𝑃‘𝑖)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∀wral 3060 ∪ cun 3942 {csn 4622 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 0cc0 11092 1c1 11093 < clt 11230 ℕcn 12194 ℤ≥cuz 12804 ...cfz 13466 ..^cfzo 13609 RePartciccp 45853 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-fzo 13610 df-iccp 45854 |
This theorem is referenced by: iccpartgel 45869 |
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