Theorem eqglact 13777

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 2229	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
3		eqglact.3	. . . . . . 7 ⊢ + = (+g‘𝐺)
4		eqger.r	. . . . . . 7 ⊢ ∼ = (𝐺 ~QG 𝑌)
5	1, 2, 3, 4	eqgval 13775	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
6		3anass 1006	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
7	5, 6	bitrdi 196	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝐴 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌))))
8	7	baibd 928	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
9	8	3impa 1218	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
10	9	abbidv 2347	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∣ 𝐴 ∼ 𝑥} = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
11		dfec2 6691	. . 3 ⊢ (𝐴 ∈ 𝑋 → [𝐴] ∼ = {𝑥 ∣ 𝐴 ∼ 𝑥})
12	11	3ad2ant3 1044	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = {𝑥 ∣ 𝐴 ∼ 𝑥})
13		eqid 2229	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥))) = (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))
14	13, 1, 3, 2	grplactcnv 13650	. . . . . . . 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 114	. . . . . . 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 13649	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
17	16	adantl 277	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
18	17	cnveqd 4898	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
19	1, 2	grpinvcl 13596	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
20	13, 1	grplactfval 13649	. . . . . . . 8 ⊢ (((invg‘𝐺)‘𝐴) ∈ 𝑋 → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
21	19, 20	syl 14	. . . . . . 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 2270	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
23	22	cnveqd 4898	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = ◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
24	23	3adant2 1040	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = ◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
25	24	imaeq1d 5067	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌))
26		imacnvcnv 5193	. . 3 ⊢ (◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌)
27		eqid 2229	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥))
28	27	mptpreima 5222	. . . 4 ⊢ (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∈ 𝑋 ∣ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌}
29		df-rab 2517	. . . 4 ⊢ {𝑥 ∈ 𝑋 ∣ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌} = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
30	28, 29	eqtri 2250	. . 3 ⊢ (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
31	25, 26, 30	3eqtr3g 2285	. 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 2272	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  {cab 2215  {crab 2512   ⊆ wss 3197   class class class wbr 4083   ↦ cmpt 4145  ^◡ccnv 4718   “ cima 4722  –1-1-onto→wf1o 5317  ‘cfv 5318  (class class class)co 6007  [cec 6686  Basecbs 13047  +_gcplusg 13125  Grpcgrp 13548  inv_gcminusg 13549   ~_QG cqg 13721

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-ec 6690  df-inn 9122  df-2 9180  df-ndx 13050  df-slot 13051  df-base 13053  df-plusg 13138  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-grp 13551  df-minusg 13552  df-eqg 13724

This theorem is referenced by:  eqgen  13779