MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghmgrp Structured version   Visualization version   GIF version

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 ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
ghmgrp.x 𝑋 = (Base‘𝐺)
ghmgrp.y 𝑌 = (Base‘𝐻)
ghmgrp.p + = (+g𝐺)
ghmgrp.q = (+g𝐻)
ghmgrp.1 (𝜑𝐹:𝑋onto𝑌)
ghmgrp.3 (𝜑𝐺 ∈ Grp)
Assertion
Ref Expression
ghmgrp (𝜑𝐻 ∈ 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.
StepHypRef Expression
1 ghmgrp.f . . 3 ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
2 ghmgrp.x . . 3 𝑋 = (Base‘𝐺)
3 ghmgrp.y . . 3 𝑌 = (Base‘𝐻)
4 ghmgrp.p . . 3 + = (+g𝐺)
5 ghmgrp.q . . 3 = (+g𝐻)
6 ghmgrp.1 . . 3 (𝜑𝐹:𝑋onto𝑌)
7 ghmgrp.3 . . . 4 (𝜑𝐺 ∈ Grp)
87grpmndd 18965 . . 3 (𝜑𝐺 ∈ Mnd)
91, 2, 3, 4, 5, 6, 8mhmmnd 19083 . 2 (𝜑𝐻 ∈ Mnd)
10 fof 6819 . . . . . . . 8 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
116, 10syl 17 . . . . . . 7 (𝜑𝐹:𝑋𝑌)
1211ad3antrrr 730 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐹:𝑋𝑌)
137ad3antrrr 730 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐺 ∈ Grp)
14 simplr 768 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝑖𝑋)
15 eqid 2736 . . . . . . . 8 (invg𝐺) = (invg𝐺)
162, 15grpinvcl 19006 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → ((invg𝐺)‘𝑖) ∈ 𝑋)
1713, 14, 16syl2anc 584 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((invg𝐺)‘𝑖) ∈ 𝑋)
1812, 17ffvelcdmd 7104 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘((invg𝐺)‘𝑖)) ∈ 𝑌)
1913adant1r 1177 . . . . . . . 8 (((𝜑𝑖𝑋) ∧ 𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
207, 16sylan 580 . . . . . . . 8 ((𝜑𝑖𝑋) → ((invg𝐺)‘𝑖) ∈ 𝑋)
21 simpr 484 . . . . . . . 8 ((𝜑𝑖𝑋) → 𝑖𝑋)
2219, 20, 21mhmlem 19081 . . . . . . 7 ((𝜑𝑖𝑋) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)))
2322ad4ant13 751 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)))
24 eqid 2736 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
252, 4, 24, 15grplinv 19008 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → (((invg𝐺)‘𝑖) + 𝑖) = (0g𝐺))
2625fveq2d 6909 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = (𝐹‘(0g𝐺)))
2713, 14, 26syl2anc 584 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = (𝐹‘(0g𝐺)))
281, 2, 3, 4, 5, 6, 8, 24mhmid 19082 . . . . . . . 8 (𝜑 → (𝐹‘(0g𝐺)) = (0g𝐻))
2928ad3antrrr 730 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(0g𝐺)) = (0g𝐻))
3027, 29eqtrd 2776 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = (0g𝐻))
31 simpr 484 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹𝑖) = 𝑎)
3231oveq2d 7448 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) 𝑎))
3323, 30, 323eqtr3rd 2785 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻))
34 oveq1 7439 . . . . . . 7 (𝑓 = (𝐹‘((invg𝐺)‘𝑖)) → (𝑓 𝑎) = ((𝐹‘((invg𝐺)‘𝑖)) 𝑎))
3534eqeq1d 2738 . . . . . 6 (𝑓 = (𝐹‘((invg𝐺)‘𝑖)) → ((𝑓 𝑎) = (0g𝐻) ↔ ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻)))
3635rspcev 3621 . . . . 5 (((𝐹‘((invg𝐺)‘𝑖)) ∈ 𝑌 ∧ ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻)) → ∃𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
3718, 33, 36syl2anc 584 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ∃𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
38 foelcdmi 6969 . . . . 5 ((𝐹:𝑋onto𝑌𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
396, 38sylan 580 . . . 4 ((𝜑𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
4037, 39r19.29a 3161 . . 3 ((𝜑𝑎𝑌) → ∃𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
4140ralrimiva 3145 . 2 (𝜑 → ∀𝑎𝑌𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
42 eqid 2736 . . 3 (0g𝐻) = (0g𝐻)
433, 5, 42isgrp 18958 . 2 (𝐻 ∈ Grp ↔ (𝐻 ∈ Mnd ∧ ∀𝑎𝑌𝑓𝑌 (𝑓 𝑎) = (0g𝐻)))
449, 41, 43sylanbrc 583 1 (𝜑𝐻 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wrex 3069  wf 6556  ontowfo 6558  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  0gc0g 17485  Mndcmnd 18748  Grpcgrp 18952  invgcminusg 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
  Copyright terms: Public domain W3C validator