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Theorem ghmgrp 18161
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)
8 grpmnd 18048 . . . 4 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
97, 8syl 17 . . 3 (𝜑𝐺 ∈ Mnd)
101, 2, 3, 4, 5, 6, 9mhmmnd 18159 . 2 (𝜑𝐻 ∈ Mnd)
11 fof 6583 . . . . . . . 8 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
126, 11syl 17 . . . . . . 7 (𝜑𝐹:𝑋𝑌)
1312ad3antrrr 726 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐹:𝑋𝑌)
147ad3antrrr 726 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐺 ∈ Grp)
15 simplr 765 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝑖𝑋)
16 eqid 2818 . . . . . . . 8 (invg𝐺) = (invg𝐺)
172, 16grpinvcl 18089 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → ((invg𝐺)‘𝑖) ∈ 𝑋)
1814, 15, 17syl2anc 584 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((invg𝐺)‘𝑖) ∈ 𝑋)
1913, 18ffvelrnd 6844 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘((invg𝐺)‘𝑖)) ∈ 𝑌)
2013adant1r 1169 . . . . . . . 8 (((𝜑𝑖𝑋) ∧ 𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
217, 17sylan 580 . . . . . . . 8 ((𝜑𝑖𝑋) → ((invg𝐺)‘𝑖) ∈ 𝑋)
22 simpr 485 . . . . . . . 8 ((𝜑𝑖𝑋) → 𝑖𝑋)
2320, 21, 22mhmlem 18157 . . . . . . 7 ((𝜑𝑖𝑋) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)))
2423ad4ant13 747 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)))
25 eqid 2818 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
262, 4, 25, 16grplinv 18090 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → (((invg𝐺)‘𝑖) + 𝑖) = (0g𝐺))
2726fveq2d 6667 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = (𝐹‘(0g𝐺)))
2814, 15, 27syl2anc 584 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = (𝐹‘(0g𝐺)))
291, 2, 3, 4, 5, 6, 9, 25mhmid 18158 . . . . . . . 8 (𝜑 → (𝐹‘(0g𝐺)) = (0g𝐻))
3029ad3antrrr 726 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(0g𝐺)) = (0g𝐻))
3128, 30eqtrd 2853 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = (0g𝐻))
32 simpr 485 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹𝑖) = 𝑎)
3332oveq2d 7161 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) 𝑎))
3424, 31, 333eqtr3rd 2862 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻))
35 oveq1 7152 . . . . . . 7 (𝑓 = (𝐹‘((invg𝐺)‘𝑖)) → (𝑓 𝑎) = ((𝐹‘((invg𝐺)‘𝑖)) 𝑎))
3635eqeq1d 2820 . . . . . 6 (𝑓 = (𝐹‘((invg𝐺)‘𝑖)) → ((𝑓 𝑎) = (0g𝐻) ↔ ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻)))
3736rspcev 3620 . . . . 5 (((𝐹‘((invg𝐺)‘𝑖)) ∈ 𝑌 ∧ ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻)) → ∃𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
3819, 34, 37syl2anc 584 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ∃𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
39 foelrni 6720 . . . . 5 ((𝐹:𝑋onto𝑌𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
406, 39sylan 580 . . . 4 ((𝜑𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
4138, 40r19.29a 3286 . . 3 ((𝜑𝑎𝑌) → ∃𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
4241ralrimiva 3179 . 2 (𝜑 → ∀𝑎𝑌𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
43 eqid 2818 . . 3 (0g𝐻) = (0g𝐻)
443, 5, 43isgrp 18047 . 2 (𝐻 ∈ Grp ↔ (𝐻 ∈ Mnd ∧ ∀𝑎𝑌𝑓𝑌 (𝑓 𝑎) = (0g𝐻)))
4510, 42, 44sylanbrc 583 1 (𝜑𝐻 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wf 6344  ontowfo 6346  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701  Mndcmnd 17899  Grpcgrp 18041  invgcminusg 18042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-riota 7103  df-ov 7148  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045
This theorem is referenced by:  ghmfghm  18880  ghmabl  18882
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