Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rzgrp | Structured version Visualization version GIF version |
Description: The quotient group R/Z is a group. (Contributed by Thierry Arnoux, 26-Jan-2020.) |
Ref | Expression |
---|---|
rzgrp.r | ⊢ 𝑅 = (ℝfld /s (ℝfld ~QG ℤ)) |
Ref | Expression |
---|---|
rzgrp | ⊢ 𝑅 ∈ Grp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zsubrg 19618 | . . . . 5 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
2 | zssre 11261 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
3 | resubdrg 19773 | . . . . . . 7 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
4 | 3 | simpli 473 | . . . . . 6 ⊢ ℝ ∈ (SubRing‘ℂfld) |
5 | df-refld 19770 | . . . . . . 7 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
6 | 5 | subsubrg 18629 | . . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂfld) → (ℤ ∈ (SubRing‘ℝfld) ↔ (ℤ ∈ (SubRing‘ℂfld) ∧ ℤ ⊆ ℝ))) |
7 | 4, 6 | ax-mp 5 | . . . . 5 ⊢ (ℤ ∈ (SubRing‘ℝfld) ↔ (ℤ ∈ (SubRing‘ℂfld) ∧ ℤ ⊆ ℝ)) |
8 | 1, 2, 7 | mpbir2an 957 | . . . 4 ⊢ ℤ ∈ (SubRing‘ℝfld) |
9 | subrgsubg 18609 | . . . 4 ⊢ (ℤ ∈ (SubRing‘ℝfld) → ℤ ∈ (SubGrp‘ℝfld)) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ℤ ∈ (SubGrp‘ℝfld) |
11 | simpl 472 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ) | |
12 | 11 | recnd 9947 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ) |
13 | simpr 476 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
14 | 13 | recnd 9947 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ) |
15 | 12, 14 | addcomd 10117 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
16 | 15 | eleq1d 2672 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 + 𝑦) ∈ ℤ ↔ (𝑦 + 𝑥) ∈ ℤ)) |
17 | 16 | rgen2a 2960 | . . 3 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ((𝑥 + 𝑦) ∈ ℤ ↔ (𝑦 + 𝑥) ∈ ℤ) |
18 | rebase 19771 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
19 | replusg 19775 | . . . 4 ⊢ + = (+g‘ℝfld) | |
20 | 18, 19 | isnsg 17446 | . . 3 ⊢ (ℤ ∈ (NrmSGrp‘ℝfld) ↔ (ℤ ∈ (SubGrp‘ℝfld) ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ((𝑥 + 𝑦) ∈ ℤ ↔ (𝑦 + 𝑥) ∈ ℤ))) |
21 | 10, 17, 20 | mpbir2an 957 | . 2 ⊢ ℤ ∈ (NrmSGrp‘ℝfld) |
22 | rzgrp.r | . . 3 ⊢ 𝑅 = (ℝfld /s (ℝfld ~QG ℤ)) | |
23 | 22 | qusgrp 17472 | . 2 ⊢ (ℤ ∈ (NrmSGrp‘ℝfld) → 𝑅 ∈ Grp) |
24 | 21, 23 | ax-mp 5 | 1 ⊢ 𝑅 ∈ Grp |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 + caddc 9818 ℤcz 11254 /s cqus 15988 Grpcgrp 17245 SubGrpcsubg 17411 NrmSGrpcnsg 17412 ~QG cqg 17413 DivRingcdr 18570 SubRingcsubrg 18599 ℂfldccnfld 19567 ℝfldcrefld 19769 |
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-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-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-ec 7631 df-qs 7635 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-fz 12198 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-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-subg 17414 df-nsg 17415 df-eqg 17416 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-drng 18572 df-subrg 18601 df-cnfld 19568 df-refld 19770 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |