![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 19113 | . . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
6 | eqid 2728 | . . . 4 ⊢ (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆) | |
7 | 3, 6 | eqger 19133 | . . 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 19086 | . . 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 19137 | . 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 18898 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉) → (𝑝 + 𝑞) ∈ 𝑉) |
14 | 13 | 3expb 1118 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 + 𝑞) ∈ 𝑉) |
15 | 10, 14 | sylan 579 | . 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 17535 | 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 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 Er wer 8722 [cec 8723 Basecbs 17180 +gcplusg 17233 /s cqus 17487 Grpcgrp 18890 SubGrpcsubg 19075 NrmSGrpcnsg 19076 ~QG cqg 19077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-ec 8727 df-qs 8731 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-0g 17423 df-imas 17490 df-qus 17491 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-subg 19078 df-nsg 19079 df-eqg 19080 |
This theorem is referenced by: qus0 19144 qusinv 19145 qussub 19146 ecqusaddd 19147 qusghm 19209 ghmquskerlem3 19237 qusabl 19820 nsgmgclem 33134 nsgqusf1olem1 33136 ghmqusnsg 33144 opprqusplusg 33213 |
Copyright terms: Public domain | W3C validator |