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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpartxr | Structured version Visualization version GIF version |
Description: If there is a partition, then all intermediate points and bounds are extended real numbers. (Contributed by AV, 11-Jul-2020.) |
Ref | Expression |
---|---|
iccpartgtprec.m | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
iccpartgtprec.p | ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) |
iccpartxr.i | ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
Ref | Expression |
---|---|
iccpartxr | ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccpartgtprec.p | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) | |
2 | iccpartgtprec.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
3 | iccpart 43933 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))) |
5 | 1, 4 | mpbid 235 | . . . 4 ⊢ (𝜑 → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) |
6 | 5 | simpld 498 | . . 3 ⊢ (𝜑 → 𝑃 ∈ (ℝ* ↑m (0...𝑀))) |
7 | elmapi 8411 | . . 3 ⊢ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) → 𝑃:(0...𝑀)⟶ℝ*) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → 𝑃:(0...𝑀)⟶ℝ*) |
9 | iccpartxr.i | . 2 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) | |
10 | 8, 9 | ffvelrnd 6829 | 1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ∀wral 3106 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 0cc0 10526 1c1 10527 + caddc 10529 ℝ*cxr 10663 < clt 10664 ℕcn 11625 ...cfz 12885 ..^cfzo 13028 RePartciccp 43930 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-iccp 43931 |
This theorem is referenced by: iccpartipre 43938 iccpartiltu 43939 iccpartigtl 43940 iccpartlt 43941 iccpartleu 43945 iccpartgel 43946 iccpartrn 43947 iccelpart 43950 iccpartiun 43951 icceuelpartlem 43952 icceuelpart 43953 iccpartdisj 43954 iccpartnel 43955 bgoldbtbndlem2 44324 |
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