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Theorem ghmgrp 17586
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 17476 . . . 4 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
97, 8syl 17 . . 3 (𝜑𝐺 ∈ Mnd)
101, 2, 3, 4, 5, 6, 9mhmmnd 17584 . 2 (𝜑𝐻 ∈ Mnd)
11 fof 6153 . . . . . . . 8 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
126, 11syl 17 . . . . . . 7 (𝜑𝐹:𝑋𝑌)
1312ad3antrrr 766 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐹:𝑋𝑌)
147ad3antrrr 766 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐺 ∈ Grp)
15 simplr 807 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝑖𝑋)
16 eqid 2651 . . . . . . . 8 (invg𝐺) = (invg𝐺)
172, 16grpinvcl 17514 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → ((invg𝐺)‘𝑖) ∈ 𝑋)
1814, 15, 17syl2anc 694 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((invg𝐺)‘𝑖) ∈ 𝑋)
1913, 18ffvelrnd 6400 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘((invg𝐺)‘𝑖)) ∈ 𝑌)
2013adant1r 1359 . . . . . . . . 9 (((𝜑𝑖𝑋) ∧ 𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
217, 17sylan 487 . . . . . . . . 9 ((𝜑𝑖𝑋) → ((invg𝐺)‘𝑖) ∈ 𝑋)
22 simpr 476 . . . . . . . . 9 ((𝜑𝑖𝑋) → 𝑖𝑋)
2320, 21, 22mhmlem 17582 . . . . . . . 8 ((𝜑𝑖𝑋) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)))
2423adantlr 751 . . . . . . 7 (((𝜑𝑎𝑌) ∧ 𝑖𝑋) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)))
2524adantr 480 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)))
26 eqid 2651 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
272, 4, 26, 16grplinv 17515 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → (((invg𝐺)‘𝑖) + 𝑖) = (0g𝐺))
2827fveq2d 6233 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑖𝑋) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = (𝐹‘(0g𝐺)))
2914, 15, 28syl2anc 694 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = (𝐹‘(0g𝐺)))
301, 2, 3, 4, 5, 6, 9, 26mhmid 17583 . . . . . . . 8 (𝜑 → (𝐹‘(0g𝐺)) = (0g𝐻))
3130ad3antrrr 766 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(0g𝐺)) = (0g𝐻))
3229, 31eqtrd 2685 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(((invg𝐺)‘𝑖) + 𝑖)) = (0g𝐻))
33 simpr 476 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹𝑖) = 𝑎)
3433oveq2d 6706 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹‘((invg𝐺)‘𝑖)) (𝐹𝑖)) = ((𝐹‘((invg𝐺)‘𝑖)) 𝑎))
3525, 32, 343eqtr3rd 2694 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻))
36 oveq1 6697 . . . . . . 7 (𝑓 = (𝐹‘((invg𝐺)‘𝑖)) → (𝑓 𝑎) = ((𝐹‘((invg𝐺)‘𝑖)) 𝑎))
3736eqeq1d 2653 . . . . . 6 (𝑓 = (𝐹‘((invg𝐺)‘𝑖)) → ((𝑓 𝑎) = (0g𝐻) ↔ ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻)))
3837rspcev 3340 . . . . 5 (((𝐹‘((invg𝐺)‘𝑖)) ∈ 𝑌 ∧ ((𝐹‘((invg𝐺)‘𝑖)) 𝑎) = (0g𝐻)) → ∃𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
3919, 35, 38syl2anc 694 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ∃𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
40 foelrni 6283 . . . . 5 ((𝐹:𝑋onto𝑌𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
416, 40sylan 487 . . . 4 ((𝜑𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
4239, 41r19.29a 3107 . . 3 ((𝜑𝑎𝑌) → ∃𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
4342ralrimiva 2995 . 2 (𝜑 → ∀𝑎𝑌𝑓𝑌 (𝑓 𝑎) = (0g𝐻))
44 eqid 2651 . . 3 (0g𝐻) = (0g𝐻)
453, 5, 44isgrp 17475 . 2 (𝐻 ∈ Grp ↔ (𝐻 ∈ Mnd ∧ ∀𝑎𝑌𝑓𝑌 (𝑓 𝑎) = (0g𝐻)))
4610, 43, 45sylanbrc 699 1 (𝜑𝐻 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  0gc0g 16147  Mndcmnd 17341  Grpcgrp 17469  invgcminusg 17470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473
This theorem is referenced by:  ghmfghm  18282  ghmabl  18284
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