![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > qqh1 | Structured version Visualization version GIF version |
Description: The image of 1 by the βHom homomorphism is the ring unity. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
Ref | Expression |
---|---|
qqhval2.0 | β’ π΅ = (Baseβπ ) |
qqhval2.1 | β’ / = (/rβπ ) |
qqhval2.2 | β’ πΏ = (β€RHomβπ ) |
Ref | Expression |
---|---|
qqh1 | β’ ((π β DivRing β§ (chrβπ ) = 0) β ((βHomβπ )β1) = (1rβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zssq 12965 | . . . 4 β’ β€ β β | |
2 | 1z 12617 | . . . 4 β’ 1 β β€ | |
3 | 1, 2 | sselii 3976 | . . 3 β’ 1 β β |
4 | qqhval2.0 | . . . 4 β’ π΅ = (Baseβπ ) | |
5 | qqhval2.1 | . . . 4 β’ / = (/rβπ ) | |
6 | qqhval2.2 | . . . 4 β’ πΏ = (β€RHomβπ ) | |
7 | 4, 5, 6 | qqhvval 33579 | . . 3 β’ (((π β DivRing β§ (chrβπ ) = 0) β§ 1 β β) β ((βHomβπ )β1) = ((πΏβ(numerβ1)) / (πΏβ(denomβ1)))) |
8 | 3, 7 | mpan2 690 | . 2 β’ ((π β DivRing β§ (chrβπ ) = 0) β ((βHomβπ )β1) = ((πΏβ(numerβ1)) / (πΏβ(denomβ1)))) |
9 | gcd1 16497 | . . . . . . . . . 10 β’ (1 β β€ β (1 gcd 1) = 1) | |
10 | 2, 9 | ax-mp 5 | . . . . . . . . 9 β’ (1 gcd 1) = 1 |
11 | 1div1e1 11929 | . . . . . . . . . 10 β’ (1 / 1) = 1 | |
12 | 11 | eqcomi 2737 | . . . . . . . . 9 β’ 1 = (1 / 1) |
13 | 10, 12 | pm3.2i 470 | . . . . . . . 8 β’ ((1 gcd 1) = 1 β§ 1 = (1 / 1)) |
14 | 1nn 12248 | . . . . . . . . 9 β’ 1 β β | |
15 | qnumdenbi 16710 | . . . . . . . . 9 β’ ((1 β β β§ 1 β β€ β§ 1 β β) β (((1 gcd 1) = 1 β§ 1 = (1 / 1)) β ((numerβ1) = 1 β§ (denomβ1) = 1))) | |
16 | 3, 2, 14, 15 | mp3an 1458 | . . . . . . . 8 β’ (((1 gcd 1) = 1 β§ 1 = (1 / 1)) β ((numerβ1) = 1 β§ (denomβ1) = 1)) |
17 | 13, 16 | mpbi 229 | . . . . . . 7 β’ ((numerβ1) = 1 β§ (denomβ1) = 1) |
18 | 17 | simpli 483 | . . . . . 6 β’ (numerβ1) = 1 |
19 | 18 | fveq2i 6895 | . . . . 5 β’ (πΏβ(numerβ1)) = (πΏβ1) |
20 | 17 | simpri 485 | . . . . . 6 β’ (denomβ1) = 1 |
21 | 20 | fveq2i 6895 | . . . . 5 β’ (πΏβ(denomβ1)) = (πΏβ1) |
22 | 19, 21 | oveq12i 7427 | . . . 4 β’ ((πΏβ(numerβ1)) / (πΏβ(denomβ1))) = ((πΏβ1) / (πΏβ1)) |
23 | drngring 20625 | . . . . . 6 β’ (π β DivRing β π β Ring) | |
24 | eqid 2728 | . . . . . . . 8 β’ (1rβπ ) = (1rβπ ) | |
25 | 6, 24 | zrh1 21432 | . . . . . . 7 β’ (π β Ring β (πΏβ1) = (1rβπ )) |
26 | 25, 25 | oveq12d 7433 | . . . . . 6 β’ (π β Ring β ((πΏβ1) / (πΏβ1)) = ((1rβπ ) / (1rβπ ))) |
27 | 23, 26 | syl 17 | . . . . 5 β’ (π β DivRing β ((πΏβ1) / (πΏβ1)) = ((1rβπ ) / (1rβπ ))) |
28 | 4, 24 | ringidcl 20196 | . . . . . 6 β’ (π β Ring β (1rβπ ) β π΅) |
29 | 4, 5, 24 | dvr1 20340 | . . . . . 6 β’ ((π β Ring β§ (1rβπ ) β π΅) β ((1rβπ ) / (1rβπ )) = (1rβπ )) |
30 | 23, 28, 29 | syl2anc2 584 | . . . . 5 β’ (π β DivRing β ((1rβπ ) / (1rβπ )) = (1rβπ )) |
31 | 27, 30 | eqtrd 2768 | . . . 4 β’ (π β DivRing β ((πΏβ1) / (πΏβ1)) = (1rβπ )) |
32 | 22, 31 | eqtrid 2780 | . . 3 β’ (π β DivRing β ((πΏβ(numerβ1)) / (πΏβ(denomβ1))) = (1rβπ )) |
33 | 32 | adantr 480 | . 2 β’ ((π β DivRing β§ (chrβπ ) = 0) β ((πΏβ(numerβ1)) / (πΏβ(denomβ1))) = (1rβπ )) |
34 | 8, 33 | eqtrd 2768 | 1 β’ ((π β DivRing β§ (chrβπ ) = 0) β ((βHomβπ )β1) = (1rβπ )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 βcfv 6543 (class class class)co 7415 0cc0 11133 1c1 11134 / cdiv 11896 βcn 12237 β€cz 12583 βcq 12957 gcd cgcd 16463 numercnumer 16699 denomcdenom 16700 Basecbs 17174 1rcur 20115 Ringcrg 20167 /rcdvr 20333 DivRingcdr 20618 β€RHomczrh 21419 chrcchr 21421 βHomcqqh 33568 |
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 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212 ax-mulf 11213 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-tpos 8226 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-fz 13512 df-fl 13784 df-mod 13862 df-seq 13994 df-exp 14054 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-dvds 16226 df-gcd 16464 df-numer 16701 df-denom 16702 df-gz 16893 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-grp 18887 df-minusg 18888 df-sbg 18889 df-mulg 19018 df-subg 19072 df-ghm 19162 df-od 19477 df-cmn 19731 df-abl 19732 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-cring 20170 df-oppr 20267 df-dvdsr 20290 df-unit 20291 df-invr 20321 df-dvr 20334 df-rhm 20405 df-subrng 20477 df-subrg 20502 df-drng 20620 df-cnfld 21274 df-zring 21367 df-zrh 21423 df-chr 21425 df-qqh 33569 |
This theorem is referenced by: qqhrhm 33585 |
Copyright terms: Public domain | W3C validator |