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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 28145 | . 2 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ) | |
2 | 1 | ffvelcdmda 7084 | 1 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐴‘𝐼) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ‘cfv 6541 (class class class)co 7406 ℝcr 11106 1c1 11108 ...cfz 13481 𝔼cee 28136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-map 8819 df-ee 28139 |
This theorem is referenced by: fveecn 28150 eqeelen 28152 brbtwn2 28153 colinearalglem4 28157 colinearalg 28158 eleesub 28159 eleesubd 28160 axcgrid 28164 axsegconlem1 28165 axsegconlem2 28166 axsegconlem3 28167 axsegconlem8 28172 axsegconlem9 28173 axsegconlem10 28174 ax5seglem3a 28178 ax5seg 28186 axpasch 28189 axeuclidlem 28210 axcontlem2 28213 |
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