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Mirrors > Home > MPE Home > Th. List > ecqusaddcl | Structured version Visualization version GIF version |
Description: Closure of the addition in a quotient group. (Contributed by AV, 24-Feb-2025.) |
Ref | Expression |
---|---|
ecqusaddd.i | ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) |
ecqusaddd.b | ⊢ 𝐵 = (Base‘𝑅) |
ecqusaddd.g | ⊢ ∼ = (𝑅 ~QG 𝐼) |
ecqusaddd.q | ⊢ 𝑄 = (𝑅 /s ∼ ) |
Ref | Expression |
---|---|
ecqusaddcl | ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ ) ∈ (Base‘𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecqusaddd.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) | |
2 | ecqusaddd.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | ecqusaddd.g | . . 3 ⊢ ∼ = (𝑅 ~QG 𝐼) | |
4 | ecqusaddd.q | . . 3 ⊢ 𝑄 = (𝑅 /s ∼ ) | |
5 | 1, 2, 3, 4 | ecqusaddd 19149 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴(+g‘𝑅)𝐶)] ∼ = ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ )) |
6 | 1 | elfvexd 6930 | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) |
7 | nsgsubg 19115 | . . . . . . 7 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅)) | |
8 | subgrcl 19088 | . . . . . . 7 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → 𝑅 ∈ Grp) | |
9 | 1, 7, 8 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
10 | 9 | anim1i 613 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (𝑅 ∈ Grp ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵))) |
11 | 3anass 1092 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) ↔ (𝑅 ∈ Grp ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵))) | |
12 | 10, 11 | sylibr 233 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (𝑅 ∈ Grp ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) |
13 | eqid 2725 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
14 | 2, 13 | grpcl 18900 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐴(+g‘𝑅)𝐶) ∈ 𝐵) |
15 | 12, 14 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (𝐴(+g‘𝑅)𝐶) ∈ 𝐵) |
16 | eqid 2725 | . . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
17 | 3, 4, 2, 16 | quseccl0 19142 | . . 3 ⊢ ((𝑅 ∈ V ∧ (𝐴(+g‘𝑅)𝐶) ∈ 𝐵) → [(𝐴(+g‘𝑅)𝐶)] ∼ ∈ (Base‘𝑄)) |
18 | 6, 15, 17 | syl2an2r 683 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴(+g‘𝑅)𝐶)] ∼ ∈ (Base‘𝑄)) |
19 | 5, 18 | eqeltrrd 2826 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ ) ∈ (Base‘𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ‘cfv 6542 (class class class)co 7415 [cec 8719 Basecbs 17177 +gcplusg 17230 /s cqus 17484 Grpcgrp 18892 SubGrpcsubg 19077 NrmSGrpcnsg 19078 ~QG cqg 19079 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-ec 8723 df-qs 8727 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-inf 9464 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-0g 17420 df-imas 17487 df-qus 17488 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-minusg 18896 df-subg 19080 df-nsg 19081 df-eqg 19082 |
This theorem is referenced by: rngqiprngghm 21191 |
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