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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 47341 | . . . . . 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 232 | . . . 4 ⊢ (𝜑 → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) |
6 | 5 | simpld 494 | . . 3 ⊢ (𝜑 → 𝑃 ∈ (ℝ* ↑m (0...𝑀))) |
7 | elmapi 8888 | . . 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 | ffvelcdmd 7105 | 1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 0cc0 11153 1c1 11154 + caddc 11156 ℝ*cxr 11292 < clt 11293 ℕcn 12264 ...cfz 13544 ..^cfzo 13691 RePartciccp 47338 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-iccp 47339 |
This theorem is referenced by: iccpartipre 47346 iccpartiltu 47347 iccpartigtl 47348 iccpartlt 47349 iccpartleu 47353 iccpartgel 47354 iccpartrn 47355 iccelpart 47358 iccpartiun 47359 icceuelpartlem 47360 icceuelpart 47361 iccpartdisj 47362 iccpartnel 47363 bgoldbtbndlem2 47731 |
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