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Mirrors > Home > MPE Home > Th. List > abv1z | Structured version Visualization version GIF version |
Description: The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.) |
Ref | Expression |
---|---|
abv0.a | β’ π΄ = (AbsValβπ ) |
abv1.p | β’ 1 = (1rβπ ) |
abv1z.z | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
abv1z | β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ 1 ) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abv0.a | . . . . . . . 8 β’ π΄ = (AbsValβπ ) | |
2 | 1 | abvrcl 20323 | . . . . . . 7 β’ (πΉ β π΄ β π β Ring) |
3 | eqid 2733 | . . . . . . . 8 β’ (Baseβπ ) = (Baseβπ ) | |
4 | abv1.p | . . . . . . . 8 β’ 1 = (1rβπ ) | |
5 | 3, 4 | ringidcl 19997 | . . . . . . 7 β’ (π β Ring β 1 β (Baseβπ )) |
6 | 2, 5 | syl 17 | . . . . . 6 β’ (πΉ β π΄ β 1 β (Baseβπ )) |
7 | 1, 3 | abvcl 20326 | . . . . . 6 β’ ((πΉ β π΄ β§ 1 β (Baseβπ )) β (πΉβ 1 ) β β) |
8 | 6, 7 | mpdan 686 | . . . . 5 β’ (πΉ β π΄ β (πΉβ 1 ) β β) |
9 | 8 | adantr 482 | . . . 4 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ 1 ) β β) |
10 | 9 | recnd 11191 | . . 3 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ 1 ) β β) |
11 | simpl 484 | . . . 4 β’ ((πΉ β π΄ β§ 1 β 0 ) β πΉ β π΄) | |
12 | 6 | adantr 482 | . . . 4 β’ ((πΉ β π΄ β§ 1 β 0 ) β 1 β (Baseβπ )) |
13 | simpr 486 | . . . 4 β’ ((πΉ β π΄ β§ 1 β 0 ) β 1 β 0 ) | |
14 | abv1z.z | . . . . 5 β’ 0 = (0gβπ ) | |
15 | 1, 3, 14 | abvne0 20329 | . . . 4 β’ ((πΉ β π΄ β§ 1 β (Baseβπ ) β§ 1 β 0 ) β (πΉβ 1 ) β 0) |
16 | 11, 12, 13, 15 | syl3anc 1372 | . . 3 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ 1 ) β 0) |
17 | 10, 10, 16 | divcan3d 11944 | . 2 β’ ((πΉ β π΄ β§ 1 β 0 ) β (((πΉβ 1 ) Β· (πΉβ 1 )) / (πΉβ 1 )) = (πΉβ 1 )) |
18 | eqid 2733 | . . . . . . . 8 β’ (.rβπ ) = (.rβπ ) | |
19 | 3, 18, 4 | ringlidm 20000 | . . . . . . 7 β’ ((π β Ring β§ 1 β (Baseβπ )) β ( 1 (.rβπ ) 1 ) = 1 ) |
20 | 2, 12, 19 | syl2an2r 684 | . . . . . 6 β’ ((πΉ β π΄ β§ 1 β 0 ) β ( 1 (.rβπ ) 1 ) = 1 ) |
21 | 20 | fveq2d 6850 | . . . . 5 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ( 1 (.rβπ ) 1 )) = (πΉβ 1 )) |
22 | 1, 3, 18 | abvmul 20331 | . . . . . 6 β’ ((πΉ β π΄ β§ 1 β (Baseβπ ) β§ 1 β (Baseβπ )) β (πΉβ( 1 (.rβπ ) 1 )) = ((πΉβ 1 ) Β· (πΉβ 1 ))) |
23 | 11, 12, 12, 22 | syl3anc 1372 | . . . . 5 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ( 1 (.rβπ ) 1 )) = ((πΉβ 1 ) Β· (πΉβ 1 ))) |
24 | 21, 23 | eqtr3d 2775 | . . . 4 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ 1 ) = ((πΉβ 1 ) Β· (πΉβ 1 ))) |
25 | 24 | oveq1d 7376 | . . 3 β’ ((πΉ β π΄ β§ 1 β 0 ) β ((πΉβ 1 ) / (πΉβ 1 )) = (((πΉβ 1 ) Β· (πΉβ 1 )) / (πΉβ 1 ))) |
26 | 10, 16 | dividd 11937 | . . 3 β’ ((πΉ β π΄ β§ 1 β 0 ) β ((πΉβ 1 ) / (πΉβ 1 )) = 1) |
27 | 25, 26 | eqtr3d 2775 | . 2 β’ ((πΉ β π΄ β§ 1 β 0 ) β (((πΉβ 1 ) Β· (πΉβ 1 )) / (πΉβ 1 )) = 1) |
28 | 17, 27 | eqtr3d 2775 | 1 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ 1 ) = 1) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 βcfv 6500 (class class class)co 7361 βcr 11058 0cc0 11059 1c1 11060 Β· cmul 11064 / cdiv 11820 Basecbs 17091 .rcmulr 17142 0gc0g 17329 1rcur 19921 Ringcrg 19972 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-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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-rmo 3352 df-reu 3353 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-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 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-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 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-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 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-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-ico 13279 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mgp 19905 df-ur 19922 df-ring 19974 df-abv 20319 |
This theorem is referenced by: abv1 20335 abvneg 20336 nm1 24054 qabvle 26996 qabvexp 26997 ostthlem2 26999 ostth3 27009 ostth 27010 |
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