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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qqh0 | Structured version Visualization version GIF version |
Description: The image of 0 by the ℚHom homomorphism is the ring's zero. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
Ref | Expression |
---|---|
qqhval2.0 | ⊢ 𝐵 = (Base‘𝑅) |
qqhval2.1 | ⊢ / = (/r‘𝑅) |
qqhval2.2 | ⊢ 𝐿 = (ℤRHom‘𝑅) |
Ref | Expression |
---|---|
qqh0 | ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = (0g‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zssq 12835 | . . . 4 ⊢ ℤ ⊆ ℚ | |
2 | 0z 12468 | . . . 4 ⊢ 0 ∈ ℤ | |
3 | 1, 2 | sselii 3939 | . . 3 ⊢ 0 ∈ ℚ |
4 | qqhval2.0 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
5 | qqhval2.1 | . . . 4 ⊢ / = (/r‘𝑅) | |
6 | qqhval2.2 | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
7 | 4, 5, 6 | qqhvval 32392 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ 0 ∈ ℚ) → ((ℚHom‘𝑅)‘0) = ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0)))) |
8 | 3, 7 | mpan2 689 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0)))) |
9 | 1z 12491 | . . . . . . . . . . 11 ⊢ 1 ∈ ℤ | |
10 | gcd0id 16353 | . . . . . . . . . . 11 ⊢ (1 ∈ ℤ → (0 gcd 1) = (abs‘1)) | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . . 10 ⊢ (0 gcd 1) = (abs‘1) |
12 | abs1 15136 | . . . . . . . . . 10 ⊢ (abs‘1) = 1 | |
13 | 11, 12 | eqtri 2764 | . . . . . . . . 9 ⊢ (0 gcd 1) = 1 |
14 | 0cn 11105 | . . . . . . . . . . 11 ⊢ 0 ∈ ℂ | |
15 | 14 | div1i 11841 | . . . . . . . . . 10 ⊢ (0 / 1) = 0 |
16 | 15 | eqcomi 2745 | . . . . . . . . 9 ⊢ 0 = (0 / 1) |
17 | 13, 16 | pm3.2i 471 | . . . . . . . 8 ⊢ ((0 gcd 1) = 1 ∧ 0 = (0 / 1)) |
18 | 1nn 12122 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
19 | qnumdenbi 16573 | . . . . . . . . 9 ⊢ ((0 ∈ ℚ ∧ 0 ∈ ℤ ∧ 1 ∈ ℕ) → (((0 gcd 1) = 1 ∧ 0 = (0 / 1)) ↔ ((numer‘0) = 0 ∧ (denom‘0) = 1))) | |
20 | 3, 2, 18, 19 | mp3an 1461 | . . . . . . . 8 ⊢ (((0 gcd 1) = 1 ∧ 0 = (0 / 1)) ↔ ((numer‘0) = 0 ∧ (denom‘0) = 1)) |
21 | 17, 20 | mpbi 229 | . . . . . . 7 ⊢ ((numer‘0) = 0 ∧ (denom‘0) = 1) |
22 | 21 | simpli 484 | . . . . . 6 ⊢ (numer‘0) = 0 |
23 | 22 | fveq2i 6842 | . . . . 5 ⊢ (𝐿‘(numer‘0)) = (𝐿‘0) |
24 | 21 | simpri 486 | . . . . . 6 ⊢ (denom‘0) = 1 |
25 | 24 | fveq2i 6842 | . . . . 5 ⊢ (𝐿‘(denom‘0)) = (𝐿‘1) |
26 | 23, 25 | oveq12i 7363 | . . . 4 ⊢ ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = ((𝐿‘0) / (𝐿‘1)) |
27 | drngring 20139 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
28 | eqid 2736 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
29 | 6, 28 | zrh0 20861 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐿‘0) = (0g‘𝑅)) |
30 | eqid 2736 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
31 | 6, 30 | zrh1 20860 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐿‘1) = (1r‘𝑅)) |
32 | 29, 31 | oveq12d 7369 | . . . . . 6 ⊢ (𝑅 ∈ Ring → ((𝐿‘0) / (𝐿‘1)) = ((0g‘𝑅) / (1r‘𝑅))) |
33 | 27, 32 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ DivRing → ((𝐿‘0) / (𝐿‘1)) = ((0g‘𝑅) / (1r‘𝑅))) |
34 | drnggrp 20142 | . . . . . . 7 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp) | |
35 | 4, 28 | grpidcl 18732 | . . . . . . 7 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) ∈ 𝐵) |
36 | 34, 35 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → (0g‘𝑅) ∈ 𝐵) |
37 | 4, 5, 30 | dvr1 20065 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (0g‘𝑅) ∈ 𝐵) → ((0g‘𝑅) / (1r‘𝑅)) = (0g‘𝑅)) |
38 | 27, 36, 37 | syl2anc 584 | . . . . 5 ⊢ (𝑅 ∈ DivRing → ((0g‘𝑅) / (1r‘𝑅)) = (0g‘𝑅)) |
39 | 33, 38 | eqtrd 2776 | . . . 4 ⊢ (𝑅 ∈ DivRing → ((𝐿‘0) / (𝐿‘1)) = (0g‘𝑅)) |
40 | 26, 39 | eqtrid 2788 | . . 3 ⊢ (𝑅 ∈ DivRing → ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = (0g‘𝑅)) |
41 | 40 | adantr 481 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = (0g‘𝑅)) |
42 | 8, 41 | eqtrd 2776 | 1 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = (0g‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 0cc0 11009 1c1 11010 / cdiv 11770 ℕcn 12111 ℤcz 12457 ℚcq 12827 abscabs 15073 gcd cgcd 16328 numercnumer 16562 denomcdenom 16563 Basecbs 17037 0gc0g 17275 Grpcgrp 18702 1rcur 19866 Ringcrg 19912 /rcdvr 20058 DivRingcdr 20132 ℤRHomczrh 20847 chrcchr 20849 ℚHomcqqh 32381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-tpos 8149 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-q 12828 df-rp 12870 df-fz 13379 df-fl 13651 df-mod 13729 df-seq 13861 df-exp 13922 df-cj 14938 df-re 14939 df-im 14940 df-sqrt 15074 df-abs 15075 df-dvds 16091 df-gcd 16329 df-numer 16564 df-denom 16565 df-gz 16756 df-struct 16973 df-sets 16990 df-slot 17008 df-ndx 17020 df-base 17038 df-ress 17067 df-plusg 17100 df-mulr 17101 df-starv 17102 df-tset 17106 df-ple 17107 df-ds 17109 df-unif 17110 df-0g 17277 df-mgm 18451 df-sgrp 18500 df-mnd 18511 df-mhm 18555 df-grp 18705 df-minusg 18706 df-sbg 18707 df-mulg 18826 df-subg 18878 df-ghm 18959 df-od 19263 df-cmn 19517 df-mgp 19850 df-ur 19867 df-ring 19914 df-cring 19915 df-oppr 19996 df-dvdsr 20017 df-unit 20018 df-invr 20048 df-dvr 20059 df-rnghom 20093 df-drng 20134 df-subrg 20167 df-cnfld 20744 df-zring 20817 df-zrh 20851 df-chr 20853 df-qqh 32382 |
This theorem is referenced by: qqhcn 32400 rrh0 32424 |
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