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 20600 | . . . . 5 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
2 | zssre 11991 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
3 | resubdrg 20754 | . . . . . . 7 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
4 | 3 | simpli 486 | . . . . . 6 ⊢ ℝ ∈ (SubRing‘ℂfld) |
5 | df-refld 20751 | . . . . . . 7 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
6 | 5 | subsubrg 19563 | . . . . . 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 709 | . . . 4 ⊢ ℤ ∈ (SubRing‘ℝfld) |
9 | subrgsubg 19543 | . . . 4 ⊢ (ℤ ∈ (SubRing‘ℝfld) → ℤ ∈ (SubGrp‘ℝfld)) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ℤ ∈ (SubGrp‘ℝfld) |
11 | simpl 485 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ) | |
12 | 11 | recnd 10671 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ) |
13 | simpr 487 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
14 | 13 | recnd 10671 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ) |
15 | 12, 14 | addcomd 10844 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
16 | 15 | eleq1d 2899 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 + 𝑦) ∈ ℤ ↔ (𝑦 + 𝑥) ∈ ℤ)) |
17 | 16 | rgen2 3205 | . . 3 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ((𝑥 + 𝑦) ∈ ℤ ↔ (𝑦 + 𝑥) ∈ ℤ) |
18 | rebase 20752 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
19 | replusg 20756 | . . . 4 ⊢ + = (+g‘ℝfld) | |
20 | 18, 19 | isnsg 18309 | . . 3 ⊢ (ℤ ∈ (NrmSGrp‘ℝfld) ↔ (ℤ ∈ (SubGrp‘ℝfld) ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ((𝑥 + 𝑦) ∈ ℤ ↔ (𝑦 + 𝑥) ∈ ℤ))) |
21 | 10, 17, 20 | mpbir2an 709 | . 2 ⊢ ℤ ∈ (NrmSGrp‘ℝfld) |
22 | rzgrp.r | . . 3 ⊢ 𝑅 = (ℝfld /s (ℝfld ~QG ℤ)) | |
23 | 22 | qusgrp 18337 | . 2 ⊢ (ℤ ∈ (NrmSGrp‘ℝfld) → 𝑅 ∈ Grp) |
24 | 21, 23 | ax-mp 5 | 1 ⊢ 𝑅 ∈ Grp |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 + caddc 10542 ℤcz 11984 /s cqus 16780 Grpcgrp 18105 SubGrpcsubg 18275 NrmSGrpcnsg 18276 ~QG cqg 18277 DivRingcdr 19504 SubRingcsubrg 19533 ℂfldccnfld 20547 ℝfldcrefld 20750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-ec 8293 df-qs 8297 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-imas 16783 df-qus 16784 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-subg 18278 df-nsg 18279 df-eqg 18280 df-cmn 18910 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-subrg 19535 df-cnfld 20548 df-refld 20751 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |