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| Mirrors > Home > ILE Home > Th. List > ecqusaddcl | 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 13516 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴(+g‘𝑅)𝐶)] ∼ = ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ )) |
| 6 | nsgsubg 13483 | . . . . 5 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅)) | |
| 7 | subgrcl 13457 | . . . . 5 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → 𝑅 ∈ Grp) | |
| 8 | 1, 6, 7 | 3syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) |
| 9 | 8 | adantr 276 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → 𝑅 ∈ Grp) |
| 10 | 8 | anim1i 340 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (𝑅 ∈ Grp ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵))) |
| 11 | 3anass 984 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) ↔ (𝑅 ∈ Grp ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵))) | |
| 12 | 10, 11 | sylibr 134 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (𝑅 ∈ Grp ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) |
| 13 | eqid 2204 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
| 14 | 2, 13 | grpcl 13282 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐴(+g‘𝑅)𝐶) ∈ 𝐵) |
| 15 | 12, 14 | syl 14 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (𝐴(+g‘𝑅)𝐶) ∈ 𝐵) |
| 16 | 1 | adantr 276 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → 𝐼 ∈ (NrmSGrp‘𝑅)) |
| 17 | eqid 2204 | . . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
| 18 | 3, 4, 2, 17 | quseccl0g 13509 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝐴(+g‘𝑅)𝐶) ∈ 𝐵 ∧ 𝐼 ∈ (NrmSGrp‘𝑅)) → [(𝐴(+g‘𝑅)𝐶)] ∼ ∈ (Base‘𝑄)) |
| 19 | 9, 15, 16, 18 | syl3anc 1249 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴(+g‘𝑅)𝐶)] ∼ ∈ (Base‘𝑄)) |
| 20 | 5, 19 | eqeltrrd 2282 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ ) ∈ (Base‘𝑄)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1372 ∈ wcel 2175 ‘cfv 5270 (class class class)co 5943 [cec 6617 Basecbs 12774 +gcplusg 12851 /s cqus 13074 Grpcgrp 13274 SubGrpcsubg 13445 NrmSGrpcnsg 13446 ~QG cqg 13447 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-addcom 8024 ax-addass 8026 ax-i2m1 8029 ax-0lt1 8030 ax-0id 8032 ax-rnegex 8033 ax-pre-ltirr 8036 ax-pre-lttrn 8038 ax-pre-ltadd 8040 |
| This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-pw 3617 df-sn 3638 df-pr 3639 df-tp 3640 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-er 6619 df-ec 6621 df-qs 6625 df-pnf 8108 df-mnf 8109 df-ltxr 8111 df-inn 9036 df-2 9094 df-3 9095 df-ndx 12777 df-slot 12778 df-base 12780 df-sets 12781 df-iress 12782 df-plusg 12864 df-mulr 12865 df-0g 13032 df-iimas 13076 df-qus 13077 df-mgm 13130 df-sgrp 13176 df-mnd 13191 df-grp 13277 df-minusg 13278 df-subg 13448 df-nsg 13449 df-eqg 13450 |
| This theorem is referenced by: (None) |
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