![]() |
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 17977 | . . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
6 | eqid 2825 | . . . 4 ⊢ (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆) | |
7 | 3, 6 | eqger 17995 | . . 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 17950 | . . 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 17999 | . 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 17784 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉) → (𝑝 + 𝑞) ∈ 𝑉) |
14 | 13 | 3expb 1155 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 + 𝑞) ∈ 𝑉) |
15 | 10, 14 | sylan 577 | . 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 16566 | 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 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 (class class class)co 6905 Er wer 8006 [cec 8007 Basecbs 16222 +gcplusg 16305 /s cqus 16518 Grpcgrp 17776 SubGrpcsubg 17939 NrmSGrpcnsg 17940 ~QG cqg 17941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-ec 8011 df-qs 8015 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-ip 16323 df-tset 16324 df-ple 16325 df-ds 16327 df-0g 16455 df-imas 16521 df-qus 16522 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-minusg 17780 df-subg 17942 df-nsg 17943 df-eqg 17944 |
This theorem is referenced by: qus0 18003 qusinv 18004 qussub 18005 qusghm 18048 qusabl 18621 |
Copyright terms: Public domain | W3C validator |