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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 47408 | . . . . . 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 8890 | . . 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 7104 | 1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2107 ∀wral 3060 class class class wbr 5142 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 0cc0 11156 1c1 11157 + caddc 11159 ℝ*cxr 11295 < clt 11296 ℕcn 12267 ...cfz 13548 ..^cfzo 13695 RePartciccp 47405 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-map 8869 df-iccp 47406 | 
| This theorem is referenced by: iccpartipre 47413 iccpartiltu 47414 iccpartigtl 47415 iccpartlt 47416 iccpartleu 47420 iccpartgel 47421 iccpartrn 47422 iccelpart 47425 iccpartiun 47426 icceuelpartlem 47427 icceuelpart 47428 iccpartdisj 47429 iccpartnel 47430 bgoldbtbndlem2 47798 | 
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