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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 12938 | . . . 4 β’ β€ β β | |
2 | 1z 12590 | . . . 4 β’ 1 β β€ | |
3 | 1, 2 | sselii 3972 | . . 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 33455 | . . 3 β’ (((π β DivRing β§ (chrβπ ) = 0) β§ 1 β β) β ((βHomβπ )β1) = ((πΏβ(numerβ1)) / (πΏβ(denomβ1)))) |
8 | 3, 7 | mpan2 688 | . 2 β’ ((π β DivRing β§ (chrβπ ) = 0) β ((βHomβπ )β1) = ((πΏβ(numerβ1)) / (πΏβ(denomβ1)))) |
9 | gcd1 16468 | . . . . . . . . . 10 β’ (1 β β€ β (1 gcd 1) = 1) | |
10 | 2, 9 | ax-mp 5 | . . . . . . . . 9 β’ (1 gcd 1) = 1 |
11 | 1div1e1 11902 | . . . . . . . . . 10 β’ (1 / 1) = 1 | |
12 | 11 | eqcomi 2733 | . . . . . . . . 9 β’ 1 = (1 / 1) |
13 | 10, 12 | pm3.2i 470 | . . . . . . . 8 β’ ((1 gcd 1) = 1 β§ 1 = (1 / 1)) |
14 | 1nn 12221 | . . . . . . . . 9 β’ 1 β β | |
15 | qnumdenbi 16681 | . . . . . . . . 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 1457 | . . . . . . . 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 6885 | . . . . 5 β’ (πΏβ(numerβ1)) = (πΏβ1) |
20 | 17 | simpri 485 | . . . . . 6 β’ (denomβ1) = 1 |
21 | 20 | fveq2i 6885 | . . . . 5 β’ (πΏβ(denomβ1)) = (πΏβ1) |
22 | 19, 21 | oveq12i 7414 | . . . 4 β’ ((πΏβ(numerβ1)) / (πΏβ(denomβ1))) = ((πΏβ1) / (πΏβ1)) |
23 | drngring 20586 | . . . . . 6 β’ (π β DivRing β π β Ring) | |
24 | eqid 2724 | . . . . . . . 8 β’ (1rβπ ) = (1rβπ ) | |
25 | 6, 24 | zrh1 21369 | . . . . . . 7 β’ (π β Ring β (πΏβ1) = (1rβπ )) |
26 | 25, 25 | oveq12d 7420 | . . . . . 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 20157 | . . . . . 6 β’ (π β Ring β (1rβπ ) β π΅) |
29 | 4, 5, 24 | dvr1 20301 | . . . . . 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 2764 | . . . 4 β’ (π β DivRing β ((πΏβ1) / (πΏβ1)) = (1rβπ )) |
32 | 22, 31 | eqtrid 2776 | . . 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 2764 | 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 1533 β wcel 2098 βcfv 6534 (class class class)co 7402 0cc0 11107 1c1 11108 / cdiv 11869 βcn 12210 β€cz 12556 βcq 12930 gcd cgcd 16434 numercnumer 16670 denomcdenom 16671 Basecbs 17145 1rcur 20078 Ringcrg 20130 /rcdvr 20294 DivRingcdr 20579 β€RHomczrh 21356 chrcchr 21358 βHomcqqh 33444 |
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-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-q 12931 df-rp 12973 df-fz 13483 df-fl 13755 df-mod 13833 df-seq 13965 df-exp 14026 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-dvds 16197 df-gcd 16435 df-numer 16672 df-denom 16673 df-gz 16864 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-starv 17213 df-tset 17217 df-ple 17218 df-ds 17220 df-unif 17221 df-0g 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18988 df-subg 19042 df-ghm 19131 df-od 19440 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-cring 20133 df-oppr 20228 df-dvdsr 20251 df-unit 20252 df-invr 20282 df-dvr 20295 df-rhm 20366 df-subrng 20438 df-subrg 20463 df-drng 20581 df-cnfld 21231 df-zring 21304 df-zrh 21360 df-chr 21362 df-qqh 33445 |
This theorem is referenced by: qqhrhm 33461 |
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