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Mirrors > Home > MPE Home > Th. List > abvcl | Structured version Visualization version GIF version |
Description: An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.) |
Ref | Expression |
---|---|
abvf.a | ⊢ 𝐴 = (AbsVal‘𝑅) |
abvf.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
abvcl | ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abvf.a | . . 3 ⊢ 𝐴 = (AbsVal‘𝑅) | |
2 | abvf.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | 1, 2 | abvf 19288 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝐵⟶ℝ) |
4 | 3 | ffvelrnda 6723 | 1 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ‘cfv 6232 ℝcr 10389 Basecbs 16316 AbsValcabv 19281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-addrcl 10451 ax-rnegex 10461 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-po 5369 df-so 5370 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-ov 7026 df-oprab 7027 df-mpo 7028 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-ico 12598 df-abv 19282 |
This theorem is referenced by: abvgt0 19293 abv1z 19297 abvneg 19299 abvrec 19301 abvdiv 19302 abvdom 19303 abvcxp 25877 qabvle 25887 qabvexp 25888 ostth1 25895 ostth2lem2 25896 ostth2lem3 25897 ostth2lem4 25898 ostth2 25899 ostth3 25900 |
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