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Mathbox for Thierry Arnoux |
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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 12889 | . . . 4 β’ β€ β β | |
2 | 1z 12541 | . . . 4 β’ 1 β β€ | |
3 | 1, 2 | sselii 3945 | . . 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 32628 | . . 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 16416 | . . . . . . . . . 10 β’ (1 β β€ β (1 gcd 1) = 1) | |
10 | 2, 9 | ax-mp 5 | . . . . . . . . 9 β’ (1 gcd 1) = 1 |
11 | 1div1e1 11853 | . . . . . . . . . 10 β’ (1 / 1) = 1 | |
12 | 11 | eqcomi 2742 | . . . . . . . . 9 β’ 1 = (1 / 1) |
13 | 10, 12 | pm3.2i 472 | . . . . . . . 8 β’ ((1 gcd 1) = 1 β§ 1 = (1 / 1)) |
14 | 1nn 12172 | . . . . . . . . 9 β’ 1 β β | |
15 | qnumdenbi 16627 | . . . . . . . . 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 1462 | . . . . . . . 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 485 | . . . . . 6 β’ (numerβ1) = 1 |
19 | 18 | fveq2i 6849 | . . . . 5 β’ (πΏβ(numerβ1)) = (πΏβ1) |
20 | 17 | simpri 487 | . . . . . 6 β’ (denomβ1) = 1 |
21 | 20 | fveq2i 6849 | . . . . 5 β’ (πΏβ(denomβ1)) = (πΏβ1) |
22 | 19, 21 | oveq12i 7373 | . . . 4 β’ ((πΏβ(numerβ1)) / (πΏβ(denomβ1))) = ((πΏβ1) / (πΏβ1)) |
23 | drngring 20226 | . . . . . 6 β’ (π β DivRing β π β Ring) | |
24 | eqid 2733 | . . . . . . . 8 β’ (1rβπ ) = (1rβπ ) | |
25 | 6, 24 | zrh1 20936 | . . . . . . 7 β’ (π β Ring β (πΏβ1) = (1rβπ )) |
26 | 25, 25 | oveq12d 7379 | . . . . . 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 19997 | . . . . . 6 β’ (π β Ring β (1rβπ ) β π΅) |
29 | 4, 5, 24 | dvr1 20126 | . . . . . 6 β’ ((π β Ring β§ (1rβπ ) β π΅) β ((1rβπ ) / (1rβπ )) = (1rβπ )) |
30 | 23, 28, 29 | syl2anc2 586 | . . . . 5 β’ (π β DivRing β ((1rβπ ) / (1rβπ )) = (1rβπ )) |
31 | 27, 30 | eqtrd 2773 | . . . 4 β’ (π β DivRing β ((πΏβ1) / (πΏβ1)) = (1rβπ )) |
32 | 22, 31 | eqtrid 2785 | . . 3 β’ (π β DivRing β ((πΏβ(numerβ1)) / (πΏβ(denomβ1))) = (1rβπ )) |
33 | 32 | adantr 482 | . 2 β’ ((π β DivRing β§ (chrβπ ) = 0) β ((πΏβ(numerβ1)) / (πΏβ(denomβ1))) = (1rβπ )) |
34 | 8, 33 | eqtrd 2773 | 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 397 = wceq 1542 β wcel 2107 βcfv 6500 (class class class)co 7361 0cc0 11059 1c1 11060 / cdiv 11820 βcn 12161 β€cz 12507 βcq 12881 gcd cgcd 16382 numercnumer 16616 denomcdenom 16617 Basecbs 17091 1rcur 19921 Ringcrg 19972 /rcdvr 20119 DivRingcdr 20219 β€RHomczrh 20923 chrcchr 20925 βHomcqqh 32617 |
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-rep 5246 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 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 |
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-tp 4595 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-1st 7925 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-inf 9387 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-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-fz 13434 df-fl 13706 df-mod 13784 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-dvds 16145 df-gcd 16383 df-numer 16618 df-denom 16619 df-gz 16810 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-grp 18759 df-minusg 18760 df-sbg 18761 df-mulg 18881 df-subg 18933 df-ghm 19014 df-od 19318 df-cmn 19572 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-rnghom 20156 df-drng 20221 df-subrg 20262 df-cnfld 20820 df-zring 20893 df-zrh 20927 df-chr 20929 df-qqh 32618 |
This theorem is referenced by: qqhrhm 32634 |
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