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Mirrors > Home > MPE Home > Th. List > eqg0el | Structured version Visualization version GIF version |
Description: Equivalence class of a quotient group for a subgroup. (Contributed by Thierry Arnoux, 15-Jan-2024.) |
Ref | Expression |
---|---|
eqg0el.1 | ⊢ ∼ = (𝐺 ~QG 𝐻) |
Ref | Expression |
---|---|
eqg0el | ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] ∼ = 𝐻 ↔ 𝑋 ∈ 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqg0el.1 | . . . . . 6 ⊢ ∼ = (𝐺 ~QG 𝐻) | |
3 | 1, 2 | eqger 19140 | . . . . 5 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∼ Er (Base‘𝐺)) |
4 | 3 | adantl 480 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ∼ Er (Base‘𝐺)) |
5 | eqid 2728 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
6 | 1, 5 | grpidcl 18929 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺)) |
7 | 6 | adantr 479 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (0g‘𝐺) ∈ (Base‘𝐺)) |
8 | 4, 7 | erth 8781 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ((0g‘𝐺) ∼ 𝑋 ↔ [(0g‘𝐺)] ∼ = [𝑋] ∼ )) |
9 | 1, 2, 5 | eqgid 19142 | . . . . 5 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → [(0g‘𝐺)] ∼ = 𝐻) |
10 | 9 | adantl 480 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → [(0g‘𝐺)] ∼ = 𝐻) |
11 | 10 | eqeq1d 2730 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([(0g‘𝐺)] ∼ = [𝑋] ∼ ↔ 𝐻 = [𝑋] ∼ )) |
12 | eqcom 2735 | . . . 4 ⊢ (𝐻 = [𝑋] ∼ ↔ [𝑋] ∼ = 𝐻) | |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (𝐻 = [𝑋] ∼ ↔ [𝑋] ∼ = 𝐻)) |
14 | 8, 11, 13 | 3bitrrd 305 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] ∼ = 𝐻 ↔ (0g‘𝐺) ∼ 𝑋)) |
15 | errel 8740 | . . . 4 ⊢ ( ∼ Er (Base‘𝐺) → Rel ∼ ) | |
16 | relelec 8777 | . . . 4 ⊢ (Rel ∼ → (𝑋 ∈ [(0g‘𝐺)] ∼ ↔ (0g‘𝐺) ∼ 𝑋)) | |
17 | 3, 15, 16 | 3syl 18 | . . 3 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → (𝑋 ∈ [(0g‘𝐺)] ∼ ↔ (0g‘𝐺) ∼ 𝑋)) |
18 | 17 | adantl 480 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ [(0g‘𝐺)] ∼ ↔ (0g‘𝐺) ∼ 𝑋)) |
19 | 10 | eleq2d 2815 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ [(0g‘𝐺)] ∼ ↔ 𝑋 ∈ 𝐻)) |
20 | 14, 18, 19 | 3bitr2d 306 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] ∼ = 𝐻 ↔ 𝑋 ∈ 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 Rel wrel 5687 ‘cfv 6553 (class class class)co 7426 Er wer 8728 [cec 8729 Basecbs 17187 0gc0g 17428 Grpcgrp 18897 SubGrpcsubg 19082 ~QG cqg 19084 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-ec 8733 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-subg 19085 df-eqg 19087 |
This theorem is referenced by: ghmqusker 19245 qsidomlem1 33193 qsidomlem2 33194 qsdrngi 33231 |
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