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Theorem ghmgrp 12858
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 12766 . . 3 (𝜑𝐺 ∈ Mnd)
91, 2, 3, 4, 5, 6, 8mhmmnd 12856 . 2 (𝜑𝐻 ∈ Mnd)
10 fof 5433 . . . . . . . 8 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
116, 10syl 14 . . . . . . 7 (𝜑𝐹:𝑋𝑌)
1211ad3antrrr 492 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐹:𝑋𝑌)
137ad3antrrr 492 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐺 ∈ Grp)
14 simplr 528 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝑖𝑋)
15 eqid 2177 . . . . . . . 8 (invg𝐺) = (invg𝐺)
162, 15grpinvcl 12798 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → ((invg𝐺)‘𝑖) ∈ 𝑋)
1713, 14, 16syl2anc 411 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((invg𝐺)‘𝑖) ∈ 𝑋)
1812, 17ffvelcdmd 5647 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘((invg𝐺)‘𝑖)) ∈ 𝑌)
1913adant1r 1231 . . . . . . . 8 (((𝜑𝑖𝑋) ∧ 𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
207, 16sylan 283 . . . . . . . 8 ((𝜑𝑖𝑋) → ((invg𝐺)‘𝑖) ∈ 𝑋)
21 simpr 110 . . . . . . . 8 ((𝜑𝑖𝑋) → 𝑖𝑋)
2219, 20, 21mhmlem 12854 . . . . . . 7 ((𝜑𝑖𝑋) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)))
2322ad4ant13 513 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)))
24 eqid 2177 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
252, 4, 24, 15grplinv 12799 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → (((invg𝐺)‘𝑖) + 𝑖) = (0g𝐺))
2625fveq2d 5514 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = (𝐹‘(0g𝐺)))
2713, 14, 26syl2anc 411 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = (𝐹‘(0g𝐺)))
281, 2, 3, 4, 5, 6, 8, 24mhmid 12855 . . . . . . . 8 (𝜑 → (𝐹‘(0g𝐺)) = (0g𝐻))
2928ad3antrrr 492 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(0g𝐺)) = (0g𝐻))
3027, 29eqtrd 2210 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = (0g𝐻))
31 simpr 110 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹𝑖) = 𝑎)
3231oveq2d 5884 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) 𝑎))
3323, 30, 323eqtr3rd 2219 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻))
34 oveq1 5875 . . . . . . 7 (𝑓 = (𝐹‘((invg𝐺)‘𝑖)) → (𝑓 𝑎) = ((𝐹‘((invg𝐺)‘𝑖)) 𝑎))
3534eqeq1d 2186 . . . . . 6 (𝑓 = (𝐹‘((invg𝐺)‘𝑖)) → ((𝑓 𝑎) = (0g𝐻) ↔ ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻)))
3635rspcev 2841 . . . . 5 (((𝐹‘((invg𝐺)‘𝑖)) ∈ 𝑌 ∧ ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻)) → ∃𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
3718, 33, 36syl2anc 411 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ∃𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
38 foelcdmi 5563 . . . . 5 ((𝐹:𝑋onto𝑌𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
396, 38sylan 283 . . . 4 ((𝜑𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
4037, 39r19.29a 2620 . . 3 ((𝜑𝑎𝑌) → ∃𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
4140ralrimiva 2550 . 2 (𝜑 → ∀𝑎𝑌𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
42 eqid 2177 . . 3 (0g𝐻) = (0g𝐻)
433, 5, 42isgrp 12760 . 2 (𝐻 ∈ Grp ↔ (𝐻 ∈ Mnd ∧ ∀𝑎𝑌𝑓𝑌 (𝑓 𝑎) = (0g𝐻)))
449, 41, 43sylanbrc 417 1 (𝜑𝐻 ∈ Grp)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  wral 2455  wrex 2456  wf 5207  ontowfo 5209  cfv 5211  (class class class)co 5868  Basecbs 12432  +gcplusg 12505  0gc0g 12640  Mndcmnd 12696  Grpcgrp 12754  invgcminusg 12755
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757  df-minusg 12758
This theorem is referenced by: (None)
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