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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 28927 | . 2 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ) | |
2 | 1 | ffvelcdmda 7104 | 1 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐴‘𝐼) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 1c1 11154 ...cfz 13544 𝔼cee 28918 |
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 ax-cnex 11209 ax-resscn 11210 |
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-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-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-map 8867 df-ee 28921 |
This theorem is referenced by: fveecn 28932 eqeelen 28934 brbtwn2 28935 colinearalglem4 28939 colinearalg 28940 eleesub 28941 eleesubd 28942 axcgrid 28946 axsegconlem1 28947 axsegconlem2 28948 axsegconlem3 28949 axsegconlem8 28954 axsegconlem9 28955 axsegconlem10 28956 ax5seglem3a 28960 ax5seg 28968 axpasch 28971 axeuclidlem 28992 axcontlem2 28995 |
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