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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpartf | Structured version Visualization version GIF version |
Description: The range of the partition is between its starting point and its ending point. Corresponds to fourierdlem15 41949 in GS's mathbox. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 14-Jul-2020.) |
Ref | Expression |
---|---|
iccpartgtprec.m | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
iccpartgtprec.p | ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) |
Ref | Expression |
---|---|
iccpartf | ⊢ (𝜑 → 𝑃:(0...𝑀)⟶((𝑃‘0)[,](𝑃‘𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccpartgtprec.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
2 | iccpartgtprec.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) | |
3 | iccpart 43058 | . . . 4 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))) | |
4 | elmapfn 8279 | . . . . 5 ⊢ (𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) → 𝑃 Fn (0...𝑀)) | |
5 | 4 | adantr 481 | . . . 4 ⊢ ((𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 Fn (0...𝑀)) |
6 | 3, 5 | syl6bi 254 | . . 3 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) → 𝑃 Fn (0...𝑀))) |
7 | 1, 2, 6 | sylc 65 | . 2 ⊢ (𝜑 → 𝑃 Fn (0...𝑀)) |
8 | 1, 2 | iccpartrn 43072 | . 2 ⊢ (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃‘𝑀))) |
9 | df-f 6229 | . 2 ⊢ (𝑃:(0...𝑀)⟶((𝑃‘0)[,](𝑃‘𝑀)) ↔ (𝑃 Fn (0...𝑀) ∧ ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃‘𝑀)))) | |
10 | 7, 8, 9 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝑃:(0...𝑀)⟶((𝑃‘0)[,](𝑃‘𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2081 ∀wral 3105 ⊆ wss 3859 class class class wbr 4962 ran crn 5444 Fn wfn 6220 ⟶wf 6221 ‘cfv 6225 (class class class)co 7016 ↑𝑚 cmap 8256 0cc0 10383 1c1 10384 + caddc 10386 ℝ*cxr 10520 < clt 10521 ℕcn 11486 [,]cicc 12591 ...cfz 12742 ..^cfzo 12883 RePartciccp 43055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-n0 11746 df-z 11830 df-uz 12094 df-icc 12595 df-fz 12743 df-fzo 12884 df-iccp 43056 |
This theorem is referenced by: iccpartel 43074 |
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