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Mirrors > Home > MPE Home > Th. List > fveere | Structured version Visualization version GIF version |
Description: The function value of a point is a real. (Contributed by Scott Fenton, 10-Jun-2013.) |
Ref | Expression |
---|---|
fveere | ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐴‘𝐼) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleei 25777 | . 2 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ) | |
2 | 1 | ffvelrnda 6359 | 1 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐴‘𝐼) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 1c1 9937 ...cfz 12326 𝔼cee 25768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-ee 25771 |
This theorem is referenced by: fveecn 25782 eqeelen 25784 brbtwn2 25785 colinearalglem4 25789 colinearalg 25790 eleesub 25791 eleesubd 25792 axcgrid 25796 axsegconlem1 25797 axsegconlem2 25798 axsegconlem3 25799 axsegconlem8 25804 axsegconlem9 25805 axsegconlem10 25806 ax5seglem3a 25810 ax5seg 25818 axpasch 25821 axeuclidlem 25842 axcontlem2 25845 |
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