Theorem ghmgrp 13621

Description: The image of a group 𝐺 under a group homomorphism 𝐹 is a group. This is a stronger result than that usually found in the literature, since the target of the homomorphism (operator 𝑂 in our model) need not have any of the properties of a group as a prerequisite. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.)

Hypotheses

Ref	Expression
ghmgrp.f	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
ghmgrp.x	⊢ 𝑋 = (Base‘𝐺)
ghmgrp.y	⊢ 𝑌 = (Base‘𝐻)
ghmgrp.p	⊢ + = (+g‘𝐺)
ghmgrp.q	⊢ ⨣ = (+g‘𝐻)
ghmgrp.1	⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
ghmgrp.3	⊢ (𝜑 → 𝐺 ∈ Grp)

Assertion

Ref	Expression
ghmgrp	⊢ (𝜑 → 𝐻 ∈ Grp)

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥, + ,𝑦   𝑥,𝐻,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥, ⨣ ,𝑦   𝜑,𝑥,𝑦

Proof of Theorem ghmgrp

Dummy variables 𝑎 𝑓 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghmgrp.f	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
2		ghmgrp.x	. . 3 ⊢ 𝑋 = (Base‘𝐺)
3		ghmgrp.y	. . 3 ⊢ 𝑌 = (Base‘𝐻)
4		ghmgrp.p	. . 3 ⊢ + = (+g‘𝐺)
5		ghmgrp.q	. . 3 ⊢ ⨣ = (+g‘𝐻)
6		ghmgrp.1	. . 3 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
7		ghmgrp.3	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
8	7	grpmndd 13512	. . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd)
9	1, 2, 3, 4, 5, 6, 8	mhmmnd 13619	. 2 ⊢ (𝜑 → 𝐻 ∈ Mnd)
10		fof 5524	. . . . . . . 8 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
11	6, 10	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
12	11	ad3antrrr 492	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝐹:𝑋⟶𝑌)
13	7	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝐺 ∈ Grp)
14		simplr 528	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝑖 ∈ 𝑋)
15		eqid 2209	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
16	2, 15	grpinvcl 13547	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑖 ∈ 𝑋) → ((invg‘𝐺)‘𝑖) ∈ 𝑋)
17	13, 14, 16	syl2anc 411	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((invg‘𝐺)‘𝑖) ∈ 𝑋)
18	12, 17	ffvelcdmd 5744	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘((invg‘𝐺)‘𝑖)) ∈ 𝑌)
19	1	3adant1r 1236	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
20	7, 16	sylan 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((invg‘𝐺)‘𝑖) ∈ 𝑋)
21		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
22	19, 20, 21	mhmlem 13617	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ (𝐹‘𝑖)))
23	22	ad4ant13 513	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ (𝐹‘𝑖)))
24		eqid 2209	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
25	2, 4, 24, 15	grplinv 13549	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑖 ∈ 𝑋) → (((invg‘𝐺)‘𝑖) + 𝑖) = (0g‘𝐺))
26	25	fveq2d 5607	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑖 ∈ 𝑋) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = (𝐹‘(0g‘𝐺)))
27	13, 14, 26	syl2anc 411	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = (𝐹‘(0g‘𝐺)))
28	1, 2, 3, 4, 5, 6, 8, 24	mhmid 13618	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(0g‘𝐺)) = (0g‘𝐻))
29	28	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(0g‘𝐺)) = (0g‘𝐻))
30	27, 29	eqtrd 2242	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = (0g‘𝐻))
31		simpr 110	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘𝑖) = 𝑎)
32	31	oveq2d 5990	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ (𝐹‘𝑖)) = ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎))
33	23, 30, 32	3eqtr3rd 2251	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎) = (0g‘𝐻))
34		oveq1 5981	. . . . . . 7 ⊢ (𝑓 = (𝐹‘((invg‘𝐺)‘𝑖)) → (𝑓 ⨣ 𝑎) = ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎))
35	34	eqeq1d 2218	. . . . . 6 ⊢ (𝑓 = (𝐹‘((invg‘𝐺)‘𝑖)) → ((𝑓 ⨣ 𝑎) = (0g‘𝐻) ↔ ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎) = (0g‘𝐻)))
36	35	rspcev 2887	. . . . 5 ⊢ (((𝐹‘((invg‘𝐺)‘𝑖)) ∈ 𝑌 ∧ ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎) = (0g‘𝐻)) → ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻))
37	18, 33, 36	syl2anc 411	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻))
38		foelcdmi 5659	. . . . 5 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
39	6, 38	sylan 283	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
40	37, 39	r19.29a 2654	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻))
41	40	ralrimiva 2583	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑌 ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻))
42		eqid 2209	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
43	3, 5, 42	isgrp 13505	. 2 ⊢ (𝐻 ∈ Grp ↔ (𝐻 ∈ Mnd ∧ ∀𝑎 ∈ 𝑌 ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻)))
44	9, 41, 43	sylanbrc 417	1 ⊢ (𝜑 → 𝐻 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 983   = wceq 1375   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489  ⟶wf 5290  –onto→wfo 5292  ‘cfv 5294  (class class class)co 5974  Basecbs 12998  +_gcplusg 13076  0_gc0g 13255  Mndcmnd 13415  Grpcgrp 13499  inv_gcminusg 13500

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-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-cnex 8058  ax-resscn 8059  ax-1re 8061  ax-addrcl 8064

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-inn 9079  df-2 9137  df-ndx 13001  df-slot 13002  df-base 13004  df-plusg 13089  df-0g 13257  df-mgm 13355  df-sgrp 13401  df-mnd 13416  df-grp 13502  df-minusg 13503

This theorem is referenced by:  ghmfghm  13829  ghmabl  13831