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 11671 | . . . 4 ⊢ ℤ ⊆ ℚ | |
2 | 0z 11265 | . . . 4 ⊢ 0 ∈ ℤ | |
3 | 1, 2 | sselii 3565 | . . 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 29355 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ 0 ∈ ℚ) → ((ℚHom‘𝑅)‘0) = ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0)))) |
8 | 3, 7 | mpan2 703 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0)))) |
9 | 1z 11284 | . . . . . . . . . . 11 ⊢ 1 ∈ ℤ | |
10 | gcd0id 15078 | . . . . . . . . . . 11 ⊢ (1 ∈ ℤ → (0 gcd 1) = (abs‘1)) | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . . 10 ⊢ (0 gcd 1) = (abs‘1) |
12 | abs1 13885 | . . . . . . . . . 10 ⊢ (abs‘1) = 1 | |
13 | 11, 12 | eqtri 2632 | . . . . . . . . 9 ⊢ (0 gcd 1) = 1 |
14 | 0cn 9911 | . . . . . . . . . . 11 ⊢ 0 ∈ ℂ | |
15 | 14 | div1i 10632 | . . . . . . . . . 10 ⊢ (0 / 1) = 0 |
16 | 15 | eqcomi 2619 | . . . . . . . . 9 ⊢ 0 = (0 / 1) |
17 | 13, 16 | pm3.2i 470 | . . . . . . . 8 ⊢ ((0 gcd 1) = 1 ∧ 0 = (0 / 1)) |
18 | 1nn 10908 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
19 | qnumdenbi 15290 | . . . . . . . . 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 1416 | . . . . . . . 8 ⊢ (((0 gcd 1) = 1 ∧ 0 = (0 / 1)) ↔ ((numer‘0) = 0 ∧ (denom‘0) = 1)) |
21 | 17, 20 | mpbi 219 | . . . . . . 7 ⊢ ((numer‘0) = 0 ∧ (denom‘0) = 1) |
22 | 21 | simpli 473 | . . . . . 6 ⊢ (numer‘0) = 0 |
23 | 22 | fveq2i 6106 | . . . . 5 ⊢ (𝐿‘(numer‘0)) = (𝐿‘0) |
24 | 21 | simpri 477 | . . . . . 6 ⊢ (denom‘0) = 1 |
25 | 24 | fveq2i 6106 | . . . . 5 ⊢ (𝐿‘(denom‘0)) = (𝐿‘1) |
26 | 23, 25 | oveq12i 6561 | . . . 4 ⊢ ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = ((𝐿‘0) / (𝐿‘1)) |
27 | drngring 18577 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
28 | eqid 2610 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
29 | 6, 28 | zrh0 19681 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐿‘0) = (0g‘𝑅)) |
30 | eqid 2610 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
31 | 6, 30 | zrh1 19680 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐿‘1) = (1r‘𝑅)) |
32 | 29, 31 | oveq12d 6567 | . . . . . 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 18578 | . . . . . . 7 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp) | |
35 | 4, 28 | grpidcl 17273 | . . . . . . 7 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) ∈ 𝐵) |
36 | 34, 35 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → (0g‘𝑅) ∈ 𝐵) |
37 | 4, 5, 30 | dvr1 18512 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (0g‘𝑅) ∈ 𝐵) → ((0g‘𝑅) / (1r‘𝑅)) = (0g‘𝑅)) |
38 | 27, 36, 37 | syl2anc 691 | . . . . 5 ⊢ (𝑅 ∈ DivRing → ((0g‘𝑅) / (1r‘𝑅)) = (0g‘𝑅)) |
39 | 33, 38 | eqtrd 2644 | . . . 4 ⊢ (𝑅 ∈ DivRing → ((𝐿‘0) / (𝐿‘1)) = (0g‘𝑅)) |
40 | 26, 39 | syl5eq 2656 | . . 3 ⊢ (𝑅 ∈ DivRing → ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = (0g‘𝑅)) |
41 | 40 | adantr 480 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((𝐿‘(numer‘0)) / (𝐿‘(denom‘0))) = (0g‘𝑅)) |
42 | 8, 41 | eqtrd 2644 | 1 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = (0g‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 / cdiv 10563 ℕcn 10897 ℤcz 11254 ℚcq 11664 abscabs 13822 gcd cgcd 15054 numercnumer 15279 denomcdenom 15280 Basecbs 15695 0gc0g 15923 Grpcgrp 17245 1rcur 18324 Ringcrg 18370 /rcdvr 18505 DivRingcdr 18570 ℤRHomczrh 19667 chrcchr 19669 ℚHomcqqh 29344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-numer 15281 df-denom 15282 df-gz 15472 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-od 17771 df-cmn 18018 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-rnghom 18538 df-drng 18572 df-subrg 18601 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-chr 19673 df-qqh 29345 |
This theorem is referenced by: qqhcn 29363 rrh0 29387 |
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