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Metamath Proof Explorer

Theorem ghmgrp

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)

Ref Expression
Hypotheses ghmgrp.f ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) )
ghmgrp.x ⊢ 𝑋 = ( Base ‘ 𝐺 )
ghmgrp.y ⊢ 𝑌 = ( Base ‘ 𝐻 )
ghmgrp.p ⊢ + = ( +g ‘ 𝐺 )
ghmgrp.q ⊢ ⨣ = ( +g ‘ 𝐻 )
ghmgrp.1 ⊢ ( 𝜑 → 𝐹 : 𝑋 –onto→ 𝑌 )
ghmgrp.3 ⊢ ( 𝜑 → 𝐺 ∈ Grp )
Assertion ghmgrp ( 𝜑 → 𝐻 ∈ Grp )

Proof

Step Hyp Ref Expression
1 ghmgrp.f ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) )
2 ghmgrp.x ⊢ 𝑋 = ( Base ‘ 𝐺 )
3 ghmgrp.y ⊢ 𝑌 = ( Base ‘ 𝐻 )
4 ghmgrp.p ⊢ + = ( +g ‘ 𝐺 )
5 ghmgrp.q ⊢ ⨣ = ( +g ‘ 𝐻 )
6 ghmgrp.1 ⊢ ( 𝜑 → 𝐹 : 𝑋 –onto→ 𝑌 )
7 ghmgrp.3 ⊢ ( 𝜑 → 𝐺 ∈ Grp )
8 7 grpmndd ⊢ ( 𝜑 → 𝐺 ∈ Mnd )
9 1 2 3 4 5 6 8 mhmmnd ⊢ ( 𝜑 → 𝐻 ∈ Mnd )
10 fof ⊢ ( 𝐹 : 𝑋 –onto→ 𝑌 → 𝐹 : 𝑋 ⟶ 𝑌 )
11 6 10 syl ⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ 𝑌 )
12 11 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → 𝐹 : 𝑋 ⟶ 𝑌 )
13 7 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → 𝐺 ∈ Grp )
14 simplr ⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → 𝑖 ∈ 𝑋 )
15 eqid ⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
16 2 15 grpinvcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑖 ∈ 𝑋 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ∈ 𝑋 )
17 13 14 16 syl2anc ⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ∈ 𝑋 )
18 12 17 ffvelcdmd ⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ) ∈ 𝑌 )
19 1 3adant1r ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) )
20 7 16 sylan ⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ∈ 𝑋 )
21 simpr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → 𝑖 ∈ 𝑋 )
22 19 20 21 mhmlem ⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐹 ‘ ( ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) + 𝑖 ) ) = ( ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ) ⨣ ( 𝐹 ‘ 𝑖 ) ) )
23 22 ad4ant13 ⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → ( 𝐹 ‘ ( ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) + 𝑖 ) ) = ( ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ) ⨣ ( 𝐹 ‘ 𝑖 ) ) )
24 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
25 2 4 24 15 grplinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑖 ∈ 𝑋 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) + 𝑖 ) = ( 0g ‘ 𝐺 ) )
26 25 fveq2d ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑖 ∈ 𝑋 ) → ( 𝐹 ‘ ( ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) + 𝑖 ) ) = ( 𝐹 ‘ ( 0g ‘ 𝐺 ) ) )
27 13 14 26 syl2anc ⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → ( 𝐹 ‘ ( ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) + 𝑖 ) ) = ( 𝐹 ‘ ( 0g ‘ 𝐺 ) ) )
28 1 2 3 4 5 6 8 24 mhmid ⊢ ( 𝜑 → ( 𝐹 ‘ ( 0g ‘ 𝐺 ) ) = ( 0g ‘ 𝐻 ) )
29 28 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → ( 𝐹 ‘ ( 0g ‘ 𝐺 ) ) = ( 0g ‘ 𝐻 ) )
30 27 29 eqtrd ⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → ( 𝐹 ‘ ( ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) + 𝑖 ) ) = ( 0g ‘ 𝐻 ) )
31 simpr ⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → ( 𝐹 ‘ 𝑖 ) = 𝑎 )
32 31 oveq2d ⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → ( ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ) ⨣ ( 𝐹 ‘ 𝑖 ) ) = ( ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ) ⨣ 𝑎 ) )
33 23 30 32 3eqtr3rd ⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → ( ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ) ⨣ 𝑎 ) = ( 0g ‘ 𝐻 ) )
34 oveq1 ⊢ ( 𝑓 = ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ) → ( 𝑓 ⨣ 𝑎 ) = ( ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ) ⨣ 𝑎 ) )
35 34 eqeq1d ⊢ ( 𝑓 = ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ) → ( ( 𝑓 ⨣ 𝑎 ) = ( 0g ‘ 𝐻 ) ↔ ( ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ) ⨣ 𝑎 ) = ( 0g ‘ 𝐻 ) ) )
36 35 rspcev ⊢ ( ( ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ) ∈ 𝑌 ∧ ( ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑖 ) ) ⨣ 𝑎 ) = ( 0g ‘ 𝐻 ) ) → ∃ 𝑓 ∈ 𝑌 ( 𝑓 ⨣ 𝑎 ) = ( 0g ‘ 𝐻 ) )
37 18 33 36 syl2anc ⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → ∃ 𝑓 ∈ 𝑌 ( 𝑓 ⨣ 𝑎 ) = ( 0g ‘ 𝐻 ) )
38 foelcdmi ⊢ ( ( 𝐹 : 𝑋 –onto→ 𝑌 ∧ 𝑎 ∈ 𝑌 ) → ∃ 𝑖 ∈ 𝑋 ( 𝐹 ‘ 𝑖 ) = 𝑎 )
39 6 38 sylan ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) → ∃ 𝑖 ∈ 𝑋 ( 𝐹 ‘ 𝑖 ) = 𝑎 )
40 37 39 r19.29a ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑌 ) → ∃ 𝑓 ∈ 𝑌 ( 𝑓 ⨣ 𝑎 ) = ( 0g ‘ 𝐻 ) )
41 40 ralrimiva ⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑌 ∃ 𝑓 ∈ 𝑌 ( 𝑓 ⨣ 𝑎 ) = ( 0g ‘ 𝐻 ) )
42 eqid ⊢ ( 0g ‘ 𝐻 ) = ( 0g ‘ 𝐻 )
43 3 5 42 isgrp ⊢ ( 𝐻 ∈ Grp ↔ ( 𝐻 ∈ Mnd ∧ ∀ 𝑎 ∈ 𝑌 ∃ 𝑓 ∈ 𝑌 ( 𝑓 ⨣ 𝑎 ) = ( 0g ‘ 𝐻 ) ) )
44 9 41 43 sylanbrc ⊢ ( 𝜑 → 𝐻 ∈ Grp )