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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 20654 | . . . . . . 7 β’ (πΉ β π΄ β π β Ring) |
3 | eqid 2724 | . . . . . . . 8 β’ (Baseβπ ) = (Baseβπ ) | |
4 | abv1.p | . . . . . . . 8 β’ 1 = (1rβπ ) | |
5 | 3, 4 | ringidcl 20155 | . . . . . . 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 20657 | . . . . . 6 β’ ((πΉ β π΄ β§ 1 β (Baseβπ )) β (πΉβ 1 ) β β) |
8 | 6, 7 | mpdan 684 | . . . . 5 β’ (πΉ β π΄ β (πΉβ 1 ) β β) |
9 | 8 | adantr 480 | . . . 4 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ 1 ) β β) |
10 | 9 | recnd 11239 | . . 3 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ 1 ) β β) |
11 | simpl 482 | . . . 4 β’ ((πΉ β π΄ β§ 1 β 0 ) β πΉ β π΄) | |
12 | 6 | adantr 480 | . . . 4 β’ ((πΉ β π΄ β§ 1 β 0 ) β 1 β (Baseβπ )) |
13 | simpr 484 | . . . 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 20660 | . . . 4 β’ ((πΉ β π΄ β§ 1 β (Baseβπ ) β§ 1 β 0 ) β (πΉβ 1 ) β 0) |
16 | 11, 12, 13, 15 | syl3anc 1368 | . . 3 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ 1 ) β 0) |
17 | 10, 10, 16 | divcan3d 11992 | . 2 β’ ((πΉ β π΄ β§ 1 β 0 ) β (((πΉβ 1 ) Β· (πΉβ 1 )) / (πΉβ 1 )) = (πΉβ 1 )) |
18 | eqid 2724 | . . . . . . . 8 β’ (.rβπ ) = (.rβπ ) | |
19 | 3, 18, 4 | ringlidm 20158 | . . . . . . 7 β’ ((π β Ring β§ 1 β (Baseβπ )) β ( 1 (.rβπ ) 1 ) = 1 ) |
20 | 2, 12, 19 | syl2an2r 682 | . . . . . 6 β’ ((πΉ β π΄ β§ 1 β 0 ) β ( 1 (.rβπ ) 1 ) = 1 ) |
21 | 20 | fveq2d 6885 | . . . . 5 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ( 1 (.rβπ ) 1 )) = (πΉβ 1 )) |
22 | 1, 3, 18 | abvmul 20662 | . . . . . 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 1368 | . . . . 5 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ( 1 (.rβπ ) 1 )) = ((πΉβ 1 ) Β· (πΉβ 1 ))) |
24 | 21, 23 | eqtr3d 2766 | . . . 4 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ 1 ) = ((πΉβ 1 ) Β· (πΉβ 1 ))) |
25 | 24 | oveq1d 7416 | . . 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 11985 | . . 3 β’ ((πΉ β π΄ β§ 1 β 0 ) β ((πΉβ 1 ) / (πΉβ 1 )) = 1) |
27 | 25, 26 | eqtr3d 2766 | . 2 β’ ((πΉ β π΄ β§ 1 β 0 ) β (((πΉβ 1 ) Β· (πΉβ 1 )) / (πΉβ 1 )) = 1) |
28 | 17, 27 | eqtr3d 2766 | 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 395 = wceq 1533 β wcel 2098 β wne 2932 βcfv 6533 (class class class)co 7401 βcr 11105 0cc0 11106 1c1 11107 Β· cmul 11111 / cdiv 11868 Basecbs 17143 .rcmulr 17197 0gc0g 17384 1rcur 20076 Ringcrg 20128 AbsValcabv 20649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-ico 13327 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mgp 20030 df-ur 20077 df-ring 20130 df-abv 20650 |
This theorem is referenced by: abv1 20666 abvneg 20667 nm1 24506 qabvle 27474 qabvexp 27475 ostthlem2 27477 ostth3 27487 ostth 27488 |
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