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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 19118 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴(+g‘𝑅)𝐶)] ∼ = ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ )) |
6 | 1 | elfvexd 6924 | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) |
7 | nsgsubg 19085 | . . . . . . 7 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅)) | |
8 | subgrcl 19058 | . . . . . . 7 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → 𝑅 ∈ Grp) | |
9 | 1, 7, 8 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
10 | 9 | anim1i 614 | . . . . 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 2726 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
14 | 2, 13 | grpcl 18871 | . . . 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 2726 | . . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
17 | 3, 4, 2, 16 | quseccl0 19111 | . . 3 ⊢ ((𝑅 ∈ V ∧ (𝐴(+g‘𝑅)𝐶) ∈ 𝐵) → [(𝐴(+g‘𝑅)𝐶)] ∼ ∈ (Base‘𝑄)) |
18 | 6, 15, 17 | syl2an2r 682 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴(+g‘𝑅)𝐶)] ∼ ∈ (Base‘𝑄)) |
19 | 5, 18 | eqeltrrd 2828 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ ) ∈ (Base‘𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ‘cfv 6537 (class class class)co 7405 [cec 8703 Basecbs 17153 +gcplusg 17206 /s cqus 17460 Grpcgrp 18863 SubGrpcsubg 19047 NrmSGrpcnsg 19048 ~QG cqg 19049 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-ec 8707 df-qs 8711 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-0g 17396 df-imas 17463 df-qus 17464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-subg 19050 df-nsg 19051 df-eqg 19052 |
This theorem is referenced by: rngqiprngghm 21152 |
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