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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 2737 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 2 | eqg0el.1 | . . . . . 6 ⊢ ∼ = (𝐺 ~QG 𝐻) | |
| 3 | 1, 2 | eqger 19196 | . . . . 5 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∼ Er (Base‘𝐺)) |
| 4 | 3 | adantl 481 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ∼ Er (Base‘𝐺)) |
| 5 | eqid 2737 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 6 | 1, 5 | grpidcl 18983 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺)) |
| 7 | 6 | adantr 480 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (0g‘𝐺) ∈ (Base‘𝐺)) |
| 8 | 4, 7 | erth 8796 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ((0g‘𝐺) ∼ 𝑋 ↔ [(0g‘𝐺)] ∼ = [𝑋] ∼ )) |
| 9 | 1, 2, 5 | eqgid 19198 | . . . . 5 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → [(0g‘𝐺)] ∼ = 𝐻) |
| 10 | 9 | adantl 481 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → [(0g‘𝐺)] ∼ = 𝐻) |
| 11 | 10 | eqeq1d 2739 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([(0g‘𝐺)] ∼ = [𝑋] ∼ ↔ 𝐻 = [𝑋] ∼ )) |
| 12 | eqcom 2744 | . . . 4 ⊢ (𝐻 = [𝑋] ∼ ↔ [𝑋] ∼ = 𝐻) | |
| 13 | 12 | a1i 11 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (𝐻 = [𝑋] ∼ ↔ [𝑋] ∼ = 𝐻)) |
| 14 | 8, 11, 13 | 3bitrrd 306 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] ∼ = 𝐻 ↔ (0g‘𝐺) ∼ 𝑋)) |
| 15 | errel 8754 | . . . 4 ⊢ ( ∼ Er (Base‘𝐺) → Rel ∼ ) | |
| 16 | relelec 8792 | . . . 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 481 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ [(0g‘𝐺)] ∼ ↔ (0g‘𝐺) ∼ 𝑋)) |
| 19 | 10 | eleq2d 2827 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ [(0g‘𝐺)] ∼ ↔ 𝑋 ∈ 𝐻)) |
| 20 | 14, 18, 19 | 3bitr2d 307 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] ∼ = 𝐻 ↔ 𝑋 ∈ 𝐻)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 Rel wrel 5690 ‘cfv 6561 (class class class)co 7431 Er wer 8742 [cec 8743 Basecbs 17247 0gc0g 17484 Grpcgrp 18951 SubGrpcsubg 19138 ~QG cqg 19140 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-ec 8747 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-eqg 19143 |
| This theorem is referenced by: ghmqusker 19305 qsidomlem1 33480 qsidomlem2 33481 qsdrngi 33523 |
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