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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 12170 | . . . 4 ⊢ ℤ ⊆ ℚ | |
2 | 0z 11804 | . . . 4 ⊢ 0 ∈ ℤ | |
3 | 1, 2 | sselii 3855 | . . 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 30874 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ 0 ∈ ℚ) → ((ℚHom‘𝑅)‘0) = ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0)))) |
8 | 3, 7 | mpan2 678 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0)))) |
9 | 1z 11825 | . . . . . . . . . . 11 ⊢ 1 ∈ ℤ | |
10 | gcd0id 15727 | . . . . . . . . . . 11 ⊢ (1 ∈ ℤ → (0 gcd 1) = (abs‘1)) | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . . 10 ⊢ (0 gcd 1) = (abs‘1) |
12 | abs1 14518 | . . . . . . . . . 10 ⊢ (abs‘1) = 1 | |
13 | 11, 12 | eqtri 2802 | . . . . . . . . 9 ⊢ (0 gcd 1) = 1 |
14 | 0cn 10431 | . . . . . . . . . . 11 ⊢ 0 ∈ ℂ | |
15 | 14 | div1i 11169 | . . . . . . . . . 10 ⊢ (0 / 1) = 0 |
16 | 15 | eqcomi 2787 | . . . . . . . . 9 ⊢ 0 = (0 / 1) |
17 | 13, 16 | pm3.2i 463 | . . . . . . . 8 ⊢ ((0 gcd 1) = 1 ∧ 0 = (0 / 1)) |
18 | 1nn 11452 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
19 | qnumdenbi 15940 | . . . . . . . . 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 1440 | . . . . . . . 8 ⊢ (((0 gcd 1) = 1 ∧ 0 = (0 / 1)) ↔ ((numer‘0) = 0 ∧ (denom‘0) = 1)) |
21 | 17, 20 | mpbi 222 | . . . . . . 7 ⊢ ((numer‘0) = 0 ∧ (denom‘0) = 1) |
22 | 21 | simpli 476 | . . . . . 6 ⊢ (numer‘0) = 0 |
23 | 22 | fveq2i 6502 | . . . . 5 ⊢ (𝐿‘(numer‘0)) = (𝐿‘0) |
24 | 21 | simpri 478 | . . . . . 6 ⊢ (denom‘0) = 1 |
25 | 24 | fveq2i 6502 | . . . . 5 ⊢ (𝐿‘(denom‘0)) = (𝐿‘1) |
26 | 23, 25 | oveq12i 6988 | . . . 4 ⊢ ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = ((𝐿‘0) / (𝐿‘1)) |
27 | drngring 19232 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
28 | eqid 2778 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
29 | 6, 28 | zrh0 20363 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐿‘0) = (0g‘𝑅)) |
30 | eqid 2778 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
31 | 6, 30 | zrh1 20362 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐿‘1) = (1r‘𝑅)) |
32 | 29, 31 | oveq12d 6994 | . . . . . 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 19233 | . . . . . . 7 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp) | |
35 | 4, 28 | grpidcl 17919 | . . . . . . 7 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) ∈ 𝐵) |
36 | 34, 35 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → (0g‘𝑅) ∈ 𝐵) |
37 | 4, 5, 30 | dvr1 19162 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (0g‘𝑅) ∈ 𝐵) → ((0g‘𝑅) / (1r‘𝑅)) = (0g‘𝑅)) |
38 | 27, 36, 37 | syl2anc 576 | . . . . 5 ⊢ (𝑅 ∈ DivRing → ((0g‘𝑅) / (1r‘𝑅)) = (0g‘𝑅)) |
39 | 33, 38 | eqtrd 2814 | . . . 4 ⊢ (𝑅 ∈ DivRing → ((𝐿‘0) / (𝐿‘1)) = (0g‘𝑅)) |
40 | 26, 39 | syl5eq 2826 | . . 3 ⊢ (𝑅 ∈ DivRing → ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = (0g‘𝑅)) |
41 | 40 | adantr 473 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = (0g‘𝑅)) |
42 | 8, 41 | eqtrd 2814 | 1 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = (0g‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ‘cfv 6188 (class class class)co 6976 0cc0 10335 1c1 10336 / cdiv 11098 ℕcn 11439 ℤcz 11793 ℚcq 12162 abscabs 14454 gcd cgcd 15703 numercnumer 15929 denomcdenom 15930 Basecbs 16339 0gc0g 16569 Grpcgrp 17891 1rcur 18974 Ringcrg 19020 /rcdvr 19155 DivRingcdr 19225 ℤRHomczrh 20349 chrcchr 20351 ℚHomcqqh 30863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 ax-addf 10414 ax-mulf 10415 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-tpos 7695 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-inf 8702 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-q 12163 df-rp 12205 df-fz 12709 df-fl 12977 df-mod 13053 df-seq 13185 df-exp 13245 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-dvds 15468 df-gcd 15704 df-numer 15931 df-denom 15932 df-gz 16122 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-starv 16436 df-tset 16440 df-ple 16441 df-ds 16443 df-unif 16444 df-0g 16571 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-mhm 17803 df-grp 17894 df-minusg 17895 df-sbg 17896 df-mulg 18012 df-subg 18060 df-ghm 18127 df-od 18418 df-cmn 18668 df-mgp 18963 df-ur 18975 df-ring 19022 df-cring 19023 df-oppr 19096 df-dvdsr 19114 df-unit 19115 df-invr 19145 df-dvr 19156 df-rnghom 19190 df-drng 19227 df-subrg 19256 df-cnfld 20248 df-zring 20320 df-zrh 20353 df-chr 20355 df-qqh 30864 |
This theorem is referenced by: qqhcn 30882 rrh0 30906 |
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