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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 39954 | . . . . . 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 221 | . . . 4 ⊢ (𝜑 → (𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) |
6 | 5 | simpld 474 | . . 3 ⊢ (𝜑 → 𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀))) |
7 | elmapi 7765 | . . 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 6268 | 1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 0cc0 9815 1c1 9816 + caddc 9818 ℝ*cxr 9952 < clt 9953 ℕcn 10897 ...cfz 12197 ..^cfzo 12334 RePartciccp 39951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-iccp 39952 |
This theorem is referenced by: iccpartipre 39959 iccpartiltu 39960 iccpartigtl 39961 iccpartlt 39962 iccpartleu 39966 iccpartgel 39967 iccpartrn 39968 iccelpart 39971 iccpartiun 39972 icceuelpartlem 39973 icceuelpart 39974 iccpartdisj 39975 iccpartnel 39976 bgoldbtbndlem2 40222 |
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