Theorem imasgrp 13365

Description: The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)

Hypotheses

Ref	Expression
imasgrp.u	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
imasgrp.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasgrp.p	⊢ (𝜑 → + = (+g‘𝑅))
imasgrp.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasgrp.e	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
imasgrp.r	⊢ (𝜑 → 𝑅 ∈ Grp)
imasgrp.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
imasgrp	⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈)))

Distinct variable groups:   𝑞,𝑝,𝐵   𝑎,𝑏,𝑝,𝑞,𝜑   𝑅,𝑝,𝑞   𝐹,𝑎,𝑏,𝑝,𝑞   + ,𝑝,𝑞   𝑈,𝑎,𝑏,𝑝,𝑞   𝑉,𝑎,𝑏,𝑝,𝑞   0 ,𝑝,𝑞

Allowed substitution hints:   𝐵(𝑎,𝑏)   + (𝑎,𝑏)   𝑅(𝑎,𝑏)   0 (𝑎,𝑏)

Proof of Theorem imasgrp

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasgrp.u	. 2 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
2		imasgrp.v	. 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasgrp.p	. 2 ⊢ (𝜑 → + = (+g‘𝑅))
4		imasgrp.f	. 2 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
5		imasgrp.e	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
6		imasgrp.r	. 2 ⊢ (𝜑 → 𝑅 ∈ Grp)
7	6	3ad2ant1 1020	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑅 ∈ Grp)
8		simp2 1000	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉)
9	2	3ad2ant1 1020	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑉 = (Base‘𝑅))
10	8, 9	eleqtrd 2283	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
11		simp3 1001	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
12	11, 9	eleqtrd 2283	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ (Base‘𝑅))
13		eqid 2204	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
14		eqid 2204	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
15	13, 14	grpcl 13258	. . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
16	7, 10, 12, 15	syl3anc 1249	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
17	3	3ad2ant1 1020	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → + = (+g‘𝑅))
18	17	oveqd 5951	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) = (𝑥(+g‘𝑅)𝑦))
19	16, 18, 9	3eltr4d 2288	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
20	6	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑅 ∈ Grp)
21	10	3adant3r3 1216	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅))
22	12	3adant3r3 1216	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅))
23		simpr3 1007	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
24	2	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
25	23, 24	eleqtrd 2283	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ (Base‘𝑅))
26	13, 14	grpass 13259	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥(+g‘𝑅)𝑦)(+g‘𝑅)𝑧) = (𝑥(+g‘𝑅)(𝑦(+g‘𝑅)𝑧)))
27	20, 21, 22, 25, 26	syl13anc 1251	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥(+g‘𝑅)𝑦)(+g‘𝑅)𝑧) = (𝑥(+g‘𝑅)(𝑦(+g‘𝑅)𝑧)))
28	3	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → + = (+g‘𝑅))
29	18	3adant3r3 1216	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + 𝑦) = (𝑥(+g‘𝑅)𝑦))
30		eqidd 2205	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 = 𝑧)
31	28, 29, 30	oveq123d 5955	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) = ((𝑥(+g‘𝑅)𝑦)(+g‘𝑅)𝑧))
32		eqidd 2205	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 = 𝑥)
33	28	oveqd 5951	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) = (𝑦(+g‘𝑅)𝑧))
34	28, 32, 33	oveq123d 5955	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g‘𝑅)(𝑦(+g‘𝑅)𝑧)))
35	27, 31, 34	3eqtr4d 2247	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
36	35	fveq2d 5574	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
37		imasgrp.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
38	13, 37	grpidcl 13279	. . . 4 ⊢ (𝑅 ∈ Grp → 0 ∈ (Base‘𝑅))
39	6, 38	syl 14	. . 3 ⊢ (𝜑 → 0 ∈ (Base‘𝑅))
40	39, 2	eleqtrrd 2284	. 2 ⊢ (𝜑 → 0 ∈ 𝑉)
41	3	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → + = (+g‘𝑅))
42	41	oveqd 5951	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) = ( 0 (+g‘𝑅)𝑥))
43	2	eleq2d 2274	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅)))
44	43	biimpa 296	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
45	13, 14, 37	grplid 13281	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ( 0 (+g‘𝑅)𝑥) = 𝑥)
46	6, 44, 45	syl2an2r 595	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 (+g‘𝑅)𝑥) = 𝑥)
47	42, 46	eqtrd 2237	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) = 𝑥)
48	47	fveq2d 5574	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥))
49		eqid 2204	. . . . 5 ⊢ (invg‘𝑅) = (invg‘𝑅)
50	13, 49	grpinvcl 13298	. . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((invg‘𝑅)‘𝑥) ∈ (Base‘𝑅))
51	6, 44, 50	syl2an2r 595	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((invg‘𝑅)‘𝑥) ∈ (Base‘𝑅))
52	2	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑉 = (Base‘𝑅))
53	51, 52	eleqtrrd 2284	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((invg‘𝑅)‘𝑥) ∈ 𝑉)
54	41	oveqd 5951	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (((invg‘𝑅)‘𝑥) + 𝑥) = (((invg‘𝑅)‘𝑥)(+g‘𝑅)𝑥))
55	13, 14, 37, 49	grplinv 13300	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (((invg‘𝑅)‘𝑥)(+g‘𝑅)𝑥) = 0 )
56	6, 44, 55	syl2an2r 595	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (((invg‘𝑅)‘𝑥)(+g‘𝑅)𝑥) = 0 )
57	54, 56	eqtrd 2237	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (((invg‘𝑅)‘𝑥) + 𝑥) = 0 )
58	57	fveq2d 5574	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(((invg‘𝑅)‘𝑥) + 𝑥)) = (𝐹‘ 0 ))
59	1, 2, 3, 4, 5, 6, 19, 36, 40, 48, 53, 58	imasgrp2 13364	1 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1372   ∈ wcel 2175  –onto→wfo 5266  ‘cfv 5268  (class class class)co 5934  Basecbs 12751  +_gcplusg 12828  0_gc0g 13006   “_s cimas 13049  Grpcgrp 13250  inv_gcminusg 13251

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-addcom 8007  ax-addass 8009  ax-i2m1 8012  ax-0lt1 8013  ax-0id 8015  ax-rnegex 8016  ax-pre-ltirr 8019  ax-pre-lttrn 8021  ax-pre-ltadd 8023

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-tp 3640  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-pnf 8091  df-mnf 8092  df-ltxr 8094  df-inn 9019  df-2 9077  df-3 9078  df-ndx 12754  df-slot 12755  df-base 12757  df-plusg 12841  df-mulr 12842  df-0g 13008  df-iimas 13052  df-mgm 13106  df-sgrp 13152  df-mnd 13167  df-grp 13253  df-minusg 13254

This theorem is referenced by:  imasgrpf1  13366  imasabl  13590  imasring  13744