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Mirrors > Home > MPE Home > Th. List > qusadd | Structured version Visualization version GIF version |
Description: Value of the group operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) |
Ref | Expression |
---|---|
qusgrp.h | ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) |
qusadd.v | ⊢ 𝑉 = (Base‘𝐺) |
qusadd.p | ⊢ + = (+g‘𝐺) |
qusadd.a | ⊢ ✚ = (+g‘𝐻) |
Ref | Expression |
---|---|
qusadd | ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋](𝐺 ~QG 𝑆) ✚ [𝑌](𝐺 ~QG 𝑆)) = [(𝑋 + 𝑌)](𝐺 ~QG 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusgrp.h | . . 3 ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))) |
3 | qusadd.v | . . 3 ⊢ 𝑉 = (Base‘𝐺) | |
4 | 3 | a1i 11 | . 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑉 = (Base‘𝐺)) |
5 | nsgsubg 19147 | . . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
6 | eqid 2725 | . . . 4 ⊢ (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆) | |
7 | 3, 6 | eqger 19167 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑆) Er 𝑉) |
8 | 5, 7 | syl 17 | . 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (𝐺 ~QG 𝑆) Er 𝑉) |
9 | subgrcl 19120 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
10 | 5, 9 | syl 17 | . 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp) |
11 | qusadd.p | . . 3 ⊢ + = (+g‘𝐺) | |
12 | 3, 6, 11 | eqgcpbl 19171 | . 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ((𝑎(𝐺 ~QG 𝑆)𝑝 ∧ 𝑏(𝐺 ~QG 𝑆)𝑞) → (𝑎 + 𝑏)(𝐺 ~QG 𝑆)(𝑝 + 𝑞))) |
13 | 3, 11 | grpcl 18931 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉) → (𝑝 + 𝑞) ∈ 𝑉) |
14 | 13 | 3expb 1117 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 + 𝑞) ∈ 𝑉) |
15 | 10, 14 | sylan 578 | . 2 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 + 𝑞) ∈ 𝑉) |
16 | qusadd.a | . 2 ⊢ ✚ = (+g‘𝐻) | |
17 | 2, 4, 8, 10, 12, 15, 11, 16 | qusaddval 17563 | 1 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋](𝐺 ~QG 𝑆) ✚ [𝑌](𝐺 ~QG 𝑆)) = [(𝑋 + 𝑌)](𝐺 ~QG 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7423 Er wer 8730 [cec 8731 Basecbs 17208 +gcplusg 17261 /s cqus 17515 Grpcgrp 18923 SubGrpcsubg 19109 NrmSGrpcnsg 19110 ~QG cqg 19111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-ec 8735 df-qs 8739 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9481 df-inf 9482 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-dec 12725 df-uz 12870 df-fz 13534 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-0g 17451 df-imas 17518 df-qus 17519 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-grp 18926 df-minusg 18927 df-subg 19112 df-nsg 19113 df-eqg 19114 |
This theorem is referenced by: qus0 19178 qusinv 19179 qussub 19180 ecqusaddd 19181 qusghm 19244 ghmqusnsg 19271 ghmquskerlem3 19275 qusabl 19858 nsgmgclem 33263 nsgqusf1olem1 33265 opprqusplusg 33341 |
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