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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 20694 | . . . . . . 7 β’ (πΉ β π΄ β π β Ring) |
3 | eqid 2727 | . . . . . . . 8 β’ (Baseβπ ) = (Baseβπ ) | |
4 | abv1.p | . . . . . . . 8 β’ 1 = (1rβπ ) | |
5 | 3, 4 | ringidcl 20195 | . . . . . . 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 20697 | . . . . . 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 480 | . . . 4 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ 1 ) β β) |
10 | 9 | recnd 11266 | . . 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 20700 | . . . 4 β’ ((πΉ β π΄ β§ 1 β (Baseβπ ) β§ 1 β 0 ) β (πΉβ 1 ) β 0) |
16 | 11, 12, 13, 15 | syl3anc 1369 | . . 3 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ 1 ) β 0) |
17 | 10, 10, 16 | divcan3d 12019 | . 2 β’ ((πΉ β π΄ β§ 1 β 0 ) β (((πΉβ 1 ) Β· (πΉβ 1 )) / (πΉβ 1 )) = (πΉβ 1 )) |
18 | eqid 2727 | . . . . . . . 8 β’ (.rβπ ) = (.rβπ ) | |
19 | 3, 18, 4 | ringlidm 20198 | . . . . . . 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 6895 | . . . . 5 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ( 1 (.rβπ ) 1 )) = (πΉβ 1 )) |
22 | 1, 3, 18 | abvmul 20702 | . . . . . 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 1369 | . . . . 5 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ( 1 (.rβπ ) 1 )) = ((πΉβ 1 ) Β· (πΉβ 1 ))) |
24 | 21, 23 | eqtr3d 2769 | . . . 4 β’ ((πΉ β π΄ β§ 1 β 0 ) β (πΉβ 1 ) = ((πΉβ 1 ) Β· (πΉβ 1 ))) |
25 | 24 | oveq1d 7429 | . . 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 12012 | . . 3 β’ ((πΉ β π΄ β§ 1 β 0 ) β ((πΉβ 1 ) / (πΉβ 1 )) = 1) |
27 | 25, 26 | eqtr3d 2769 | . 2 β’ ((πΉ β π΄ β§ 1 β 0 ) β (((πΉβ 1 ) Β· (πΉβ 1 )) / (πΉβ 1 )) = 1) |
28 | 17, 27 | eqtr3d 2769 | 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 1534 β wcel 2099 β wne 2935 βcfv 6542 (class class class)co 7414 βcr 11131 0cc0 11132 1c1 11133 Β· cmul 11137 / cdiv 11895 Basecbs 17173 .rcmulr 17227 0gc0g 17414 1rcur 20114 Ringcrg 20166 AbsValcabv 20689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-ico 13356 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-plusg 17239 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mgp 20068 df-ur 20115 df-ring 20168 df-abv 20690 |
This theorem is referenced by: abv1 20706 abvneg 20707 nm1 24577 qabvle 27551 qabvexp 27552 ostthlem2 27554 ostth3 27564 ostth 27565 |
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