Theorem ghmgrp 19085

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 18965	. . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd)
9	1, 2, 3, 4, 5, 6, 8	mhmmnd 19083	. 2 ⊢ (𝜑 → 𝐻 ∈ Mnd)
10		fof 6819	. . . . . . . 8 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
11	6, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
12	11	ad3antrrr 730	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝐹:𝑋⟶𝑌)
13	7	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝐺 ∈ Grp)
14		simplr 768	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝑖 ∈ 𝑋)
15		eqid 2736	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
16	2, 15	grpinvcl 19006	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑖 ∈ 𝑋) → ((invg‘𝐺)‘𝑖) ∈ 𝑋)
17	13, 14, 16	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((invg‘𝐺)‘𝑖) ∈ 𝑋)
18	12, 17	ffvelcdmd 7104	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘((invg‘𝐺)‘𝑖)) ∈ 𝑌)
19	1	3adant1r 1177	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
20	7, 16	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((invg‘𝐺)‘𝑖) ∈ 𝑋)
21		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
22	19, 20, 21	mhmlem 19081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ (𝐹‘𝑖)))
23	22	ad4ant13 751	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ (𝐹‘𝑖)))
24		eqid 2736	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
25	2, 4, 24, 15	grplinv 19008	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑖 ∈ 𝑋) → (((invg‘𝐺)‘𝑖) + 𝑖) = (0g‘𝐺))
26	25	fveq2d 6909	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑖 ∈ 𝑋) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = (𝐹‘(0g‘𝐺)))
27	13, 14, 26	syl2anc 584	. . . . . . 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 19082	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(0g‘𝐺)) = (0g‘𝐻))
29	28	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(0g‘𝐺)) = (0g‘𝐻))
30	27, 29	eqtrd 2776	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = (0g‘𝐻))
31		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘𝑖) = 𝑎)
32	31	oveq2d 7448	. . . . . 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 2785	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎) = (0g‘𝐻))
34		oveq1 7439	. . . . . . 7 ⊢ (𝑓 = (𝐹‘((invg‘𝐺)‘𝑖)) → (𝑓 ⨣ 𝑎) = ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎))
35	34	eqeq1d 2738	. . . . . 6 ⊢ (𝑓 = (𝐹‘((invg‘𝐺)‘𝑖)) → ((𝑓 ⨣ 𝑎) = (0g‘𝐻) ↔ ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎) = (0g‘𝐻)))
36	35	rspcev 3621	. . . . 5 ⊢ (((𝐹‘((invg‘𝐺)‘𝑖)) ∈ 𝑌 ∧ ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎) = (0g‘𝐻)) → ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻))
37	18, 33, 36	syl2anc 584	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻))
38		foelcdmi 6969	. . . . 5 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
39	6, 38	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
40	37, 39	r19.29a 3161	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻))
41	40	ralrimiva 3145	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑌 ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻))
42		eqid 2736	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
43	3, 5, 42	isgrp 18958	. 2 ⊢ (𝐻 ∈ Grp ↔ (𝐻 ∈ Mnd ∧ ∀𝑎 ∈ 𝑌 ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻)))
44	9, 41, 43	sylanbrc 583	1 ⊢ (𝜑 → 𝐻 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069  ⟶wf 6556  –onto→wfo 6558  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  +_gcplusg 17298  0_gc0g 17485  Mndcmnd 18748  Grpcgrp 18952  inv_gcminusg 18953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-riota 7389  df-ov 7435  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-minusg 18956

This theorem is referenced by:  ghmfghm  19849  ghmabl  19851