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 12349 | . . . 4 ⊢ ℤ ⊆ ℚ | |
2 | 0z 11986 | . . . 4 ⊢ 0 ∈ ℤ | |
3 | 1, 2 | sselii 3963 | . . 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 31219 | . . 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 12006 | . . . . . . . . . . 11 ⊢ 1 ∈ ℤ | |
10 | gcd0id 15861 | . . . . . . . . . . 11 ⊢ (1 ∈ ℤ → (0 gcd 1) = (abs‘1)) | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . . 10 ⊢ (0 gcd 1) = (abs‘1) |
12 | abs1 14651 | . . . . . . . . . 10 ⊢ (abs‘1) = 1 | |
13 | 11, 12 | eqtri 2844 | . . . . . . . . 9 ⊢ (0 gcd 1) = 1 |
14 | 0cn 10627 | . . . . . . . . . . 11 ⊢ 0 ∈ ℂ | |
15 | 14 | div1i 11362 | . . . . . . . . . 10 ⊢ (0 / 1) = 0 |
16 | 15 | eqcomi 2830 | . . . . . . . . 9 ⊢ 0 = (0 / 1) |
17 | 13, 16 | pm3.2i 473 | . . . . . . . 8 ⊢ ((0 gcd 1) = 1 ∧ 0 = (0 / 1)) |
18 | 1nn 11643 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
19 | qnumdenbi 16078 | . . . . . . . . 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 1457 | . . . . . . . 8 ⊢ (((0 gcd 1) = 1 ∧ 0 = (0 / 1)) ↔ ((numer‘0) = 0 ∧ (denom‘0) = 1)) |
21 | 17, 20 | mpbi 232 | . . . . . . 7 ⊢ ((numer‘0) = 0 ∧ (denom‘0) = 1) |
22 | 21 | simpli 486 | . . . . . 6 ⊢ (numer‘0) = 0 |
23 | 22 | fveq2i 6667 | . . . . 5 ⊢ (𝐿‘(numer‘0)) = (𝐿‘0) |
24 | 21 | simpri 488 | . . . . . 6 ⊢ (denom‘0) = 1 |
25 | 24 | fveq2i 6667 | . . . . 5 ⊢ (𝐿‘(denom‘0)) = (𝐿‘1) |
26 | 23, 25 | oveq12i 7162 | . . . 4 ⊢ ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = ((𝐿‘0) / (𝐿‘1)) |
27 | drngring 19503 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
28 | eqid 2821 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
29 | 6, 28 | zrh0 20655 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐿‘0) = (0g‘𝑅)) |
30 | eqid 2821 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
31 | 6, 30 | zrh1 20654 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐿‘1) = (1r‘𝑅)) |
32 | 29, 31 | oveq12d 7168 | . . . . . 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 19504 | . . . . . . 7 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp) | |
35 | 4, 28 | grpidcl 18125 | . . . . . . 7 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) ∈ 𝐵) |
36 | 34, 35 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → (0g‘𝑅) ∈ 𝐵) |
37 | 4, 5, 30 | dvr1 19433 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (0g‘𝑅) ∈ 𝐵) → ((0g‘𝑅) / (1r‘𝑅)) = (0g‘𝑅)) |
38 | 27, 36, 37 | syl2anc 586 | . . . . 5 ⊢ (𝑅 ∈ DivRing → ((0g‘𝑅) / (1r‘𝑅)) = (0g‘𝑅)) |
39 | 33, 38 | eqtrd 2856 | . . . 4 ⊢ (𝑅 ∈ DivRing → ((𝐿‘0) / (𝐿‘1)) = (0g‘𝑅)) |
40 | 26, 39 | syl5eq 2868 | . . 3 ⊢ (𝑅 ∈ DivRing → ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = (0g‘𝑅)) |
41 | 40 | adantr 483 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = (0g‘𝑅)) |
42 | 8, 41 | eqtrd 2856 | 1 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = (0g‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 / cdiv 11291 ℕcn 11632 ℤcz 11975 ℚcq 12342 abscabs 14587 gcd cgcd 15837 numercnumer 16067 denomcdenom 16068 Basecbs 16477 0gc0g 16707 Grpcgrp 18097 1rcur 19245 Ringcrg 19291 /rcdvr 19426 DivRingcdr 19496 ℤRHomczrh 20641 chrcchr 20643 ℚHomcqqh 31208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-fz 12887 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-dvds 15602 df-gcd 15838 df-numer 16069 df-denom 16070 df-gz 16260 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-od 18650 df-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-rnghom 19461 df-drng 19498 df-subrg 19527 df-cnfld 20540 df-zring 20612 df-zrh 20645 df-chr 20647 df-qqh 31209 |
This theorem is referenced by: qqhcn 31227 rrh0 31251 |
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