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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 47403 | . . . . . 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 8889 | . . 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 2108 ∀wral 3061 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 0cc0 11155 1c1 11156 + caddc 11158 ℝ*cxr 11294 < clt 11295 ℕcn 12266 ...cfz 13547 ..^cfzo 13694 RePartciccp 47400 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-iccp 47401 | 
| This theorem is referenced by: iccpartipre 47408 iccpartiltu 47409 iccpartigtl 47410 iccpartlt 47411 iccpartleu 47415 iccpartgel 47416 iccpartrn 47417 iccelpart 47420 iccpartiun 47421 icceuelpartlem 47422 icceuelpart 47423 iccpartdisj 47424 iccpartnel 47425 bgoldbtbndlem2 47793 | 
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