| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > qus0 | Structured version Visualization version GIF version | ||
| Description: Value of the group identity operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) |
| Ref | Expression |
|---|---|
| qusgrp.h | ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) |
| qus0.p | ⊢ 0 = (0g‘𝐺) |
| Ref | Expression |
|---|---|
| qus0 | ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [ 0 ](𝐺 ~QG 𝑆) = (0g‘𝐻)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nsgsubg 19176 | . . . . . . 7 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
| 2 | subgrcl 19149 | . . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
| 3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp) |
| 4 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 5 | qus0.p | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
| 6 | 4, 5 | grpidcl 18983 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺)) |
| 7 | 3, 6 | syl 17 | . . . . 5 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 0 ∈ (Base‘𝐺)) |
| 8 | qusgrp.h | . . . . . 6 ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) | |
| 9 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 10 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
| 11 | 8, 4, 9, 10 | qusadd 19206 | . . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 0 ∈ (Base‘𝐺) ∧ 0 ∈ (Base‘𝐺)) → ([ 0 ](𝐺 ~QG 𝑆)(+g‘𝐻)[ 0 ](𝐺 ~QG 𝑆)) = [( 0 (+g‘𝐺) 0 )](𝐺 ~QG 𝑆)) |
| 12 | 7, 7, 11 | mpd3an23 1465 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ([ 0 ](𝐺 ~QG 𝑆)(+g‘𝐻)[ 0 ](𝐺 ~QG 𝑆)) = [( 0 (+g‘𝐺) 0 )](𝐺 ~QG 𝑆)) |
| 13 | 4, 9, 5 | grplid 18985 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
| 14 | 3, 7, 13 | syl2anc 584 | . . . . 5 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
| 15 | 14 | eceq1d 8785 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [( 0 (+g‘𝐺) 0 )](𝐺 ~QG 𝑆) = [ 0 ](𝐺 ~QG 𝑆)) |
| 16 | 12, 15 | eqtrd 2777 | . . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ([ 0 ](𝐺 ~QG 𝑆)(+g‘𝐻)[ 0 ](𝐺 ~QG 𝑆)) = [ 0 ](𝐺 ~QG 𝑆)) |
| 17 | 8 | qusgrp 19204 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp) |
| 18 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
| 19 | 8, 4, 18 | quseccl 19205 | . . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 0 ∈ (Base‘𝐺)) → [ 0 ](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
| 20 | 7, 19 | mpdan 687 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [ 0 ](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
| 21 | eqid 2737 | . . . . 5 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
| 22 | 18, 10, 21 | grpid 18993 | . . . 4 ⊢ ((𝐻 ∈ Grp ∧ [ 0 ](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) → (([ 0 ](𝐺 ~QG 𝑆)(+g‘𝐻)[ 0 ](𝐺 ~QG 𝑆)) = [ 0 ](𝐺 ~QG 𝑆) ↔ (0g‘𝐻) = [ 0 ](𝐺 ~QG 𝑆))) |
| 23 | 17, 20, 22 | syl2anc 584 | . . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (([ 0 ](𝐺 ~QG 𝑆)(+g‘𝐻)[ 0 ](𝐺 ~QG 𝑆)) = [ 0 ](𝐺 ~QG 𝑆) ↔ (0g‘𝐻) = [ 0 ](𝐺 ~QG 𝑆))) |
| 24 | 16, 23 | mpbid 232 | . 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (0g‘𝐻) = [ 0 ](𝐺 ~QG 𝑆)) |
| 25 | 24 | eqcomd 2743 | 1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [ 0 ](𝐺 ~QG 𝑆) = (0g‘𝐻)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 [cec 8743 Basecbs 17247 +gcplusg 17297 0gc0g 17484 /s cqus 17550 Grpcgrp 18951 SubGrpcsubg 19138 NrmSGrpcnsg 19139 ~QG cqg 19140 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-ec 8747 df-qs 8751 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-imas 17553 df-qus 17554 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-nsg 19142 df-eqg 19143 |
| This theorem is referenced by: qusinv 19208 ghmqusker 19305 rngqiprngimf1lem 21304 rngqiprngimf1 21310 qustgphaus 24131 qusker 33377 qus0g 33435 nsgqus0 33438 qsidomlem1 33480 qsidomlem2 33481 qsnzr 33483 qsdrngi 33523 |
| Copyright terms: Public domain | W3C validator |