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Mirrors > Home > MPE Home > Th. List > abvf | 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 | β’ π΄ = (AbsValβπ ) |
abvf.b | β’ π΅ = (Baseβπ ) |
Ref | Expression |
---|---|
abvf | β’ (πΉ β π΄ β πΉ:π΅βΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abvf.a | . . 3 β’ π΄ = (AbsValβπ ) | |
2 | abvf.b | . . 3 β’ π΅ = (Baseβπ ) | |
3 | 1, 2 | abvfge0 20324 | . 2 β’ (πΉ β π΄ β πΉ:π΅βΆ(0[,)+β)) |
4 | rge0ssre 13382 | . 2 β’ (0[,)+β) β β | |
5 | fss 6689 | . 2 β’ ((πΉ:π΅βΆ(0[,)+β) β§ (0[,)+β) β β) β πΉ:π΅βΆβ) | |
6 | 3, 4, 5 | sylancl 587 | 1 β’ (πΉ β π΄ β πΉ:π΅βΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3914 βΆwf 6496 βcfv 6500 (class class class)co 7361 βcr 11058 0cc0 11059 +βcpnf 11194 [,)cico 13275 Basecbs 17091 AbsValcabv 20318 |
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 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-addrcl 11120 ax-rnegex 11130 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-ico 13279 df-abv 20319 |
This theorem is referenced by: abvcl 20326 abvres 20341 abvmet 23954 tngnrg 24061 ostthlem1 26998 |
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