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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 20144 | . . . . 5 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
2 | zssre 11976 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
3 | resubdrg 20297 | . . . . . . 7 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
4 | 3 | simpli 487 | . . . . . 6 ⊢ ℝ ∈ (SubRing‘ℂfld) |
5 | df-refld 20294 | . . . . . . 7 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
6 | 5 | subsubrg 19554 | . . . . . 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 710 | . . . 4 ⊢ ℤ ∈ (SubRing‘ℝfld) |
9 | subrgsubg 19534 | . . . 4 ⊢ (ℤ ∈ (SubRing‘ℝfld) → ℤ ∈ (SubGrp‘ℝfld)) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ℤ ∈ (SubGrp‘ℝfld) |
11 | simpl 486 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ) | |
12 | 11 | recnd 10658 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ) |
13 | simpr 488 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
14 | 13 | recnd 10658 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ) |
15 | 12, 14 | addcomd 10831 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
16 | 15 | eleq1d 2874 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 + 𝑦) ∈ ℤ ↔ (𝑦 + 𝑥) ∈ ℤ)) |
17 | 16 | rgen2 3168 | . . 3 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ((𝑥 + 𝑦) ∈ ℤ ↔ (𝑦 + 𝑥) ∈ ℤ) |
18 | rebase 20295 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
19 | replusg 20299 | . . . 4 ⊢ + = (+g‘ℝfld) | |
20 | 18, 19 | isnsg 18299 | . . 3 ⊢ (ℤ ∈ (NrmSGrp‘ℝfld) ↔ (ℤ ∈ (SubGrp‘ℝfld) ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ((𝑥 + 𝑦) ∈ ℤ ↔ (𝑦 + 𝑥) ∈ ℤ))) |
21 | 10, 17, 20 | mpbir2an 710 | . 2 ⊢ ℤ ∈ (NrmSGrp‘ℝfld) |
22 | rzgrp.r | . . 3 ⊢ 𝑅 = (ℝfld /s (ℝfld ~QG ℤ)) | |
23 | 22 | qusgrp 18327 | . 2 ⊢ (ℤ ∈ (NrmSGrp‘ℝfld) → 𝑅 ∈ Grp) |
24 | 21, 23 | ax-mp 5 | 1 ⊢ 𝑅 ∈ Grp |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 + caddc 10529 ℤcz 11969 /s cqus 16770 Grpcgrp 18095 SubGrpcsubg 18265 NrmSGrpcnsg 18266 ~QG cqg 18267 DivRingcdr 19495 SubRingcsubrg 19524 ℂfldccnfld 20091 ℝfldcrefld 20293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-ec 8274 df-qs 8278 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-imas 16773 df-qus 16774 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-subg 18268 df-nsg 18269 df-eqg 18270 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-subrg 19526 df-cnfld 20092 df-refld 20294 |
This theorem is referenced by: (None) |
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