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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 12992 | . . . 4 ⊢ ℤ ⊆ ℚ | |
2 | 0z 12621 | . . . 4 ⊢ 0 ∈ ℤ | |
3 | 1, 2 | sselii 3976 | . . 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 33798 | . . 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 12644 | . . . . . . . . . . 11 ⊢ 1 ∈ ℤ | |
10 | gcd0id 16519 | . . . . . . . . . . 11 ⊢ (1 ∈ ℤ → (0 gcd 1) = (abs‘1)) | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . . 10 ⊢ (0 gcd 1) = (abs‘1) |
12 | abs1 15302 | . . . . . . . . . 10 ⊢ (abs‘1) = 1 | |
13 | 11, 12 | eqtri 2754 | . . . . . . . . 9 ⊢ (0 gcd 1) = 1 |
14 | 0cn 11256 | . . . . . . . . . . 11 ⊢ 0 ∈ ℂ | |
15 | 14 | div1i 11993 | . . . . . . . . . 10 ⊢ (0 / 1) = 0 |
16 | 15 | eqcomi 2735 | . . . . . . . . 9 ⊢ 0 = (0 / 1) |
17 | 13, 16 | pm3.2i 469 | . . . . . . . 8 ⊢ ((0 gcd 1) = 1 ∧ 0 = (0 / 1)) |
18 | 1nn 12275 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
19 | qnumdenbi 16746 | . . . . . . . . 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 1458 | . . . . . . . 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 482 | . . . . . 6 ⊢ (numer‘0) = 0 |
23 | 22 | fveq2i 6904 | . . . . 5 ⊢ (𝐿‘(numer‘0)) = (𝐿‘0) |
24 | 21 | simpri 484 | . . . . . 6 ⊢ (denom‘0) = 1 |
25 | 24 | fveq2i 6904 | . . . . 5 ⊢ (𝐿‘(denom‘0)) = (𝐿‘1) |
26 | 23, 25 | oveq12i 7436 | . . . 4 ⊢ ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = ((𝐿‘0) / (𝐿‘1)) |
27 | drngring 20714 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
28 | eqid 2726 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
29 | 6, 28 | zrh0 21503 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐿‘0) = (0g‘𝑅)) |
30 | eqid 2726 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
31 | 6, 30 | zrh1 21502 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐿‘1) = (1r‘𝑅)) |
32 | 29, 31 | oveq12d 7442 | . . . . . 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 20717 | . . . . . . 7 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp) | |
35 | 4, 28 | grpidcl 18960 | . . . . . . 7 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) ∈ 𝐵) |
36 | 34, 35 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → (0g‘𝑅) ∈ 𝐵) |
37 | 4, 5, 30 | dvr1 20389 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (0g‘𝑅) ∈ 𝐵) → ((0g‘𝑅) / (1r‘𝑅)) = (0g‘𝑅)) |
38 | 27, 36, 37 | syl2anc 582 | . . . . 5 ⊢ (𝑅 ∈ DivRing → ((0g‘𝑅) / (1r‘𝑅)) = (0g‘𝑅)) |
39 | 33, 38 | eqtrd 2766 | . . . 4 ⊢ (𝑅 ∈ DivRing → ((𝐿‘0) / (𝐿‘1)) = (0g‘𝑅)) |
40 | 26, 39 | eqtrid 2778 | . . 3 ⊢ (𝑅 ∈ DivRing → ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = (0g‘𝑅)) |
41 | 40 | adantr 479 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = (0g‘𝑅)) |
42 | 8, 41 | eqtrd 2766 | 1 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = (0g‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 0cc0 11158 1c1 11159 / cdiv 11921 ℕcn 12264 ℤcz 12610 ℚcq 12984 abscabs 15239 gcd cgcd 16494 numercnumer 16735 denomcdenom 16736 Basecbs 17213 0gc0g 17454 Grpcgrp 18928 1rcur 20164 Ringcrg 20216 /rcdvr 20382 DivRingcdr 20707 ℤRHomczrh 21489 chrcchr 21491 ℚHomcqqh 33787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 ax-mulf 11238 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-inf 9486 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-fz 13539 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-dvds 16257 df-gcd 16495 df-numer 16737 df-denom 16738 df-gz 16932 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-grp 18931 df-minusg 18932 df-sbg 18933 df-mulg 19062 df-subg 19117 df-ghm 19207 df-od 19526 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-cring 20219 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-dvr 20383 df-rhm 20454 df-subrng 20528 df-subrg 20553 df-drng 20709 df-cnfld 21344 df-zring 21437 df-zrh 21493 df-chr 21495 df-qqh 33788 |
This theorem is referenced by: qqhcn 33806 rrh0 33830 |
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