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Theorem ghmgrp 18699
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 18589 . . 3 (𝜑𝐺 ∈ Mnd)
91, 2, 3, 4, 5, 6, 8mhmmnd 18697 . 2 (𝜑𝐻 ∈ Mnd)
10 fof 6688 . . . . . . . 8 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
116, 10syl 17 . . . . . . 7 (𝜑𝐹:𝑋𝑌)
1211ad3antrrr 727 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐹:𝑋𝑌)
137ad3antrrr 727 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐺 ∈ Grp)
14 simplr 766 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝑖𝑋)
15 eqid 2738 . . . . . . . 8 (invg𝐺) = (invg𝐺)
162, 15grpinvcl 18627 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → ((invg𝐺)‘𝑖) ∈ 𝑋)
1713, 14, 16syl2anc 584 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((invg𝐺)‘𝑖) ∈ 𝑋)
1812, 17ffvelrnd 6962 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘((invg𝐺)‘𝑖)) ∈ 𝑌)
1913adant1r 1176 . . . . . . . 8 (((𝜑𝑖𝑋) ∧ 𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
207, 16sylan 580 . . . . . . . 8 ((𝜑𝑖𝑋) → ((invg𝐺)‘𝑖) ∈ 𝑋)
21 simpr 485 . . . . . . . 8 ((𝜑𝑖𝑋) → 𝑖𝑋)
2219, 20, 21mhmlem 18695 . . . . . . 7 ((𝜑𝑖𝑋) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)))
2322ad4ant13 748 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)))
24 eqid 2738 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
252, 4, 24, 15grplinv 18628 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → (((invg𝐺)‘𝑖) + 𝑖) = (0g𝐺))
2625fveq2d 6778 . . . . . . . 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 18696 . . . . . . . 8 (𝜑 → (𝐹‘(0g𝐺)) = (0g𝐻))
2928ad3antrrr 727 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(0g𝐺)) = (0g𝐻))
3027, 29eqtrd 2778 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = (0g𝐻))
31 simpr 485 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹𝑖) = 𝑎)
3231oveq2d 7291 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) 𝑎))
3323, 30, 323eqtr3rd 2787 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻))
34 oveq1 7282 . . . . . . 7 (𝑓 = (𝐹‘((invg𝐺)‘𝑖)) → (𝑓 𝑎) = ((𝐹‘((invg𝐺)‘𝑖)) 𝑎))
3534eqeq1d 2740 . . . . . 6 (𝑓 = (𝐹‘((invg𝐺)‘𝑖)) → ((𝑓 𝑎) = (0g𝐻) ↔ ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻)))
3635rspcev 3561 . . . . 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 foelrni 6831 . . . . 5 ((𝐹:𝑋onto𝑌𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
396, 38sylan 580 . . . 4 ((𝜑𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
4037, 39r19.29a 3218 . . 3 ((𝜑𝑎𝑌) → ∃𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
4140ralrimiva 3103 . 2 (𝜑 → ∀𝑎𝑌𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
42 eqid 2738 . . 3 (0g𝐻) = (0g𝐻)
433, 5, 42isgrp 18583 . 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 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wrex 3065  wf 6429  ontowfo 6431  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  0gc0g 17150  Mndcmnd 18385  Grpcgrp 18577  invgcminusg 18578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-riota 7232  df-ov 7278  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581
This theorem is referenced by:  ghmfghm  19432  ghmabl  19434
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