Theorem eqglact 17645

Description: A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
eqger.x	⊢ 𝑋 = (Base‘𝐺)
eqger.r	⊢ ∼ = (𝐺 ~QG 𝑌)
eqglact.3	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
eqglact	⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))

Distinct variable groups:   𝑥, +   𝑥, ∼   𝑥,𝐺   𝑥,𝑋   𝑥,𝐴   𝑥,𝑌

Proof of Theorem eqglact

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqger.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
2		eqid 2622	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
3		eqglact.3	. . . . . . 7 ⊢ + = (+g‘𝐺)
4		eqger.r	. . . . . . 7 ⊢ ∼ = (𝐺 ~QG 𝑌)
5	1, 2, 3, 4	eqgval 17643	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
6		3anass 1042	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
7	5, 6	syl6bb 276	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝐴 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌))))
8	7	baibd 948	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
9	8	3impa 1259	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
10	9	abbidv 2741	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∣ 𝐴 ∼ 𝑥} = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
11		dfec2 7745	. . 3 ⊢ (𝐴 ∈ 𝑋 → [𝐴] ∼ = {𝑥 ∣ 𝐴 ∼ 𝑥})
12	11	3ad2ant3 1084	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = {𝑥 ∣ 𝐴 ∼ 𝑥})
13		eqid 2622	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥))) = (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))
14	13, 1, 3, 2	grplactcnv 17518	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴))))
15	14	simprd 479	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)))
16	13, 1	grplactfval 17516	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
17	16	adantl 482	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
18	17	cnveqd 5298	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
19	1, 2	grpinvcl 17467	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
20	13, 1	grplactfval 17516	. . . . . . . 8 ⊢ (((invg‘𝐺)‘𝐴) ∈ 𝑋 → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
21	19, 20	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
22	15, 18, 21	3eqtr3d 2664	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
23	22	cnveqd 5298	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = ◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
24	23	3adant2 1080	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = ◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
25	24	imaeq1d 5465	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌))
26		imacnvcnv 5599	. . 3 ⊢ (◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌)
27		eqid 2622	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥))
28	27	mptpreima 5628	. . . 4 ⊢ (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∈ 𝑋 ∣ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌}
29		df-rab 2921	. . . 4 ⊢ {𝑥 ∈ 𝑋 ∣ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌} = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
30	28, 29	eqtri 2644	. . 3 ⊢ (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
31	25, 26, 30	3eqtr3g 2679	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
32	10, 12, 31	3eqtr4d 2666	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608  {crab 2916   ⊆ wss 3574   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113   “ cima 5117  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  [cec 7740  Basecbs 15857  +_gcplusg 15941  Grpcgrp 17422  inv_gcminusg 17423   ~_QG cqg 17590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-ec 7744  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-eqg 17593

This theorem is referenced by:  eqgen  17647  cldsubg  21914  tgpconncompeqg  21915  snclseqg  21919