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Mirrors > Home > MPE Home > Th. List > quseccl | Structured version Visualization version GIF version |
Description: Closure of the quotient map for 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) |
quseccl.b | ⊢ 𝐵 = (Base‘𝐻) |
Ref | Expression |
---|---|
quseccl | ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7302 | . . . 4 ⊢ (𝐺 ~QG 𝑆) ∈ V | |
2 | 1 | ecelqsi 8537 | . . 3 ⊢ (𝑋 ∈ 𝑉 → [𝑋](𝐺 ~QG 𝑆) ∈ (𝑉 / (𝐺 ~QG 𝑆))) |
3 | 2 | adantl 482 | . 2 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ (𝑉 / (𝐺 ~QG 𝑆))) |
4 | qusgrp.h | . . . . 5 ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) | |
5 | 4 | a1i 11 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))) |
6 | qusadd.v | . . . . 5 ⊢ 𝑉 = (Base‘𝐺) | |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → 𝑉 = (Base‘𝐺)) |
8 | 1 | a1i 11 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝐺 ~QG 𝑆) ∈ V) |
9 | nsgsubg 18776 | . . . . . 6 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
10 | subgrcl 18750 | . . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp) |
12 | 11 | adantr 481 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ Grp) |
13 | 5, 7, 8, 12 | qusbas 17246 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝑉 / (𝐺 ~QG 𝑆)) = (Base‘𝐻)) |
14 | quseccl.b | . . 3 ⊢ 𝐵 = (Base‘𝐻) | |
15 | 13, 14 | eqtr4di 2798 | . 2 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝑉 / (𝐺 ~QG 𝑆)) = 𝐵) |
16 | 3, 15 | eleqtrd 2843 | 1 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ‘cfv 6431 (class class class)co 7269 [cec 8471 / cqs 8472 Basecbs 16902 /s cqus 17206 Grpcgrp 18567 SubGrpcsubg 18739 NrmSGrpcnsg 18740 ~QG cqg 18741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7702 df-1st 7818 df-2nd 7819 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-1o 8282 df-er 8473 df-ec 8475 df-qs 8479 df-en 8709 df-dom 8710 df-sdom 8711 df-fin 8712 df-sup 9171 df-inf 9172 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-nn 11966 df-2 12028 df-3 12029 df-4 12030 df-5 12031 df-6 12032 df-7 12033 df-8 12034 df-9 12035 df-n0 12226 df-z 12312 df-dec 12429 df-uz 12574 df-fz 13231 df-struct 16838 df-slot 16873 df-ndx 16885 df-base 16903 df-plusg 16965 df-mulr 16966 df-sca 16968 df-vsca 16969 df-ip 16970 df-tset 16971 df-ple 16972 df-ds 16974 df-imas 17209 df-qus 17210 df-subg 18742 df-nsg 18743 |
This theorem is referenced by: qus0 18804 qusinv 18805 qussub 18806 qusghm 18861 qustgplem 23262 |
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