Theorem frgpup1 19296

Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)

Hypotheses

Ref	Expression
frgpup.b	⊢ 𝐵 = (Base‘𝐻)
frgpup.n	⊢ 𝑁 = (invg‘𝐻)
frgpup.t	⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
frgpup.h	⊢ (𝜑 → 𝐻 ∈ Grp)
frgpup.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
frgpup.a	⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
frgpup.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
frgpup.r	⊢ ∼ = ( ~FG ‘𝐼)
frgpup.g	⊢ 𝐺 = (freeGrp‘𝐼)
frgpup.x	⊢ 𝑋 = (Base‘𝐺)
frgpup.e	⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))⟩)

Assertion

Ref	Expression
frgpup1	⊢ (𝜑 → 𝐸 ∈ (𝐺 GrpHom 𝐻))

Distinct variable groups:   𝑦,𝑔,𝑧   𝑔,𝐻   𝑦,𝐹,𝑧   𝑦,𝑁,𝑧   𝐵,𝑔,𝑦,𝑧   𝑇,𝑔   ∼ ,𝑔   𝜑,𝑔,𝑦,𝑧   𝑦,𝐼,𝑧   𝑔,𝑊

Allowed substitution hints:   ∼ (𝑦,𝑧)   𝑇(𝑦,𝑧)   𝐸(𝑦,𝑧,𝑔)   𝐹(𝑔)   𝐺(𝑦,𝑧,𝑔)   𝐻(𝑦,𝑧)   𝐼(𝑔)   𝑁(𝑔)   𝑉(𝑦,𝑧,𝑔)   𝑊(𝑦,𝑧)   𝑋(𝑦,𝑧,𝑔)

Proof of Theorem frgpup1

Dummy variables 𝑎 𝑢 𝑐 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgpup.x	. 2 ⊢ 𝑋 = (Base‘𝐺)
2		frgpup.b	. 2 ⊢ 𝐵 = (Base‘𝐻)
3		eqid 2738	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2738	. 2 ⊢ (+g‘𝐻) = (+g‘𝐻)
5		frgpup.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
6		frgpup.g	. . . 4 ⊢ 𝐺 = (freeGrp‘𝐼)
7	6	frgpgrp 19283	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐺 ∈ Grp)
8	5, 7	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
9		frgpup.h	. 2 ⊢ (𝜑 → 𝐻 ∈ Grp)
10		frgpup.n	. . 3 ⊢ 𝑁 = (invg‘𝐻)
11		frgpup.t	. . 3 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
12		frgpup.a	. . 3 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
13		frgpup.w	. . 3 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
14		frgpup.r	. . 3 ⊢ ∼ = ( ~FG ‘𝐼)
15		frgpup.e	. . 3 ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))⟩)
16	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupf 19294	. 2 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵)
17		eqid 2738	. . . . . . . . . . 11 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o))
18	6, 17, 14	frgpval 19279	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
19	5, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
20		2on 8275	. . . . . . . . . . . . 13 ⊢ 2o ∈ On
21		xpexg 7578	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
22	5, 20, 21	sylancl 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 × 2o) ∈ V)
23		wrdexg 14155	. . . . . . . . . . . 12 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
24		fvi 6826	. . . . . . . . . . . 12 ⊢ (Word (𝐼 × 2o) ∈ V → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o))
25	22, 23, 24	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o))
26	13, 25	eqtrid 2790	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o))
27		eqid 2738	. . . . . . . . . . . 12 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o)))
28	17, 27	frmdbas 18406	. . . . . . . . . . 11 ⊢ ((𝐼 × 2o) ∈ V → (Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 × 2o))
29	22, 28	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 × 2o))
30	26, 29	eqtr4d 2781	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o))))
31	14	fvexi 6770	. . . . . . . . . 10 ⊢ ∼ ∈ V
32	31	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ∼ ∈ V)
33		fvexd 6771	. . . . . . . . 9 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2o)) ∈ V)
34	19, 30, 32, 33	qusbas 17173	. . . . . . . 8 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺))
35	1, 34	eqtr4id 2798	. . . . . . 7 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ ))
36		eqimss 3973	. . . . . . 7 ⊢ (𝑋 = (𝑊 / ∼ ) → 𝑋 ⊆ (𝑊 / ∼ ))
37	35, 36	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ (𝑊 / ∼ ))
38	37	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑋) → 𝑋 ⊆ (𝑊 / ∼ ))
39	38	sselda 3917	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑐 ∈ 𝑋) → 𝑐 ∈ (𝑊 / ∼ ))
40		eqid 2738	. . . . 5 ⊢ (𝑊 / ∼ ) = (𝑊 / ∼ )
41		oveq2 7263	. . . . . . 7 ⊢ ([𝑢] ∼ = 𝑐 → (𝑎(+g‘𝐺)[𝑢] ∼ ) = (𝑎(+g‘𝐺)𝑐))
42	41	fveq2d 6760	. . . . . 6 ⊢ ([𝑢] ∼ = 𝑐 → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = (𝐸‘(𝑎(+g‘𝐺)𝑐)))
43		fveq2 6756	. . . . . . 7 ⊢ ([𝑢] ∼ = 𝑐 → (𝐸‘[𝑢] ∼ ) = (𝐸‘𝑐))
44	43	oveq2d 7271	. . . . . 6 ⊢ ([𝑢] ∼ = 𝑐 → ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘𝑐)))
45	42, 44	eqeq12d 2754	. . . . 5 ⊢ ([𝑢] ∼ = 𝑐 → ((𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )) ↔ (𝐸‘(𝑎(+g‘𝐺)𝑐)) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘𝑐))))
46	37	sselda 3917	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (𝑊 / ∼ ))
47	46	adantlr 711	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (𝑊 / ∼ ))
48		fvoveq1 7278	. . . . . . . . 9 ⊢ ([𝑡] ∼ = 𝑎 → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )))
49		fveq2 6756	. . . . . . . . . 10 ⊢ ([𝑡] ∼ = 𝑎 → (𝐸‘[𝑡] ∼ ) = (𝐸‘𝑎))
50	49	oveq1d 7270	. . . . . . . . 9 ⊢ ([𝑡] ∼ = 𝑎 → ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
51	48, 50	eqeq12d 2754	. . . . . . . 8 ⊢ ([𝑡] ∼ = 𝑎 → ((𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )) ↔ (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ ))))
52		fviss 6827	. . . . . . . . . . . . . . . 16 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
53	13, 52	eqsstri 3951	. . . . . . . . . . . . . . 15 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
54	53	sseli 3913	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑊 → 𝑡 ∈ Word (𝐼 × 2o))
55	53	sseli 3913	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ 𝑊 → 𝑢 ∈ Word (𝐼 × 2o))
56		ccatcl 14205	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ Word (𝐼 × 2o) ∧ 𝑢 ∈ Word (𝐼 × 2o)) → (𝑡 ++ 𝑢) ∈ Word (𝐼 × 2o))
57	54, 55, 56	syl2an 595	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → (𝑡 ++ 𝑢) ∈ Word (𝐼 × 2o))
58	13	efgrcl 19236	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
59	58	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
60	59	simprd 495	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → 𝑊 = Word (𝐼 × 2o))
61	57, 60	eleqtrrd 2842	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → (𝑡 ++ 𝑢) ∈ 𝑊)
62	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupval 19295	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ++ 𝑢) ∈ 𝑊) → (𝐸‘[(𝑡 ++ 𝑢)] ∼ ) = (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))))
63	61, 62	sylan2 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘[(𝑡 ++ 𝑢)] ∼ ) = (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))))
64	54	ad2antrl 724	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝑡 ∈ Word (𝐼 × 2o))
65	55	ad2antll 725	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝑢 ∈ Word (𝐼 × 2o))
66	2, 10, 11, 9, 5, 12	frgpuptf 19291	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵)
67	66	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝑇:(𝐼 × 2o)⟶𝐵)
68		ccatco 14476	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ Word (𝐼 × 2o) ∧ 𝑢 ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ (𝑡 ++ 𝑢)) = ((𝑇 ∘ 𝑡) ++ (𝑇 ∘ 𝑢)))
69	64, 65, 67, 68	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝑇 ∘ (𝑡 ++ 𝑢)) = ((𝑇 ∘ 𝑡) ++ (𝑇 ∘ 𝑢)))
70	69	oveq2d 7271	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))) = (𝐻 Σg ((𝑇 ∘ 𝑡) ++ (𝑇 ∘ 𝑢))))
71	9	grpmndd 18504	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ∈ Mnd)
72	71	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝐻 ∈ Mnd)
73		wrdco 14472	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ 𝑡) ∈ Word 𝐵)
74	54, 66, 73	syl2anr 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑇 ∘ 𝑡) ∈ Word 𝐵)
75	74	adantrr 713	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝑇 ∘ 𝑡) ∈ Word 𝐵)
76		wrdco 14472	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ 𝑢) ∈ Word 𝐵)
77	65, 67, 76	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝑇 ∘ 𝑢) ∈ Word 𝐵)
78	2, 4	gsumccat 18395	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ 𝑡) ∈ Word 𝐵 ∧ (𝑇 ∘ 𝑢) ∈ Word 𝐵) → (𝐻 Σg ((𝑇 ∘ 𝑡) ++ (𝑇 ∘ 𝑢))) = ((𝐻 Σg (𝑇 ∘ 𝑡))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ 𝑢))))
79	72, 75, 77, 78	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐻 Σg ((𝑇 ∘ 𝑡) ++ (𝑇 ∘ 𝑢))) = ((𝐻 Σg (𝑇 ∘ 𝑡))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ 𝑢))))
80	63, 70, 79	3eqtrd 2782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘[(𝑡 ++ 𝑢)] ∼ ) = ((𝐻 Σg (𝑇 ∘ 𝑡))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ 𝑢))))
81	13, 6, 14, 3	frgpadd 19284	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → ([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ ) = [(𝑡 ++ 𝑢)] ∼ )
82	81	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → ([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ ) = [(𝑡 ++ 𝑢)] ∼ )
83	82	fveq2d 6760	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = (𝐸‘[(𝑡 ++ 𝑢)] ∼ ))
84	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupval 19295	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐸‘[𝑡] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑡)))
85	84	adantrr 713	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘[𝑡] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑡)))
86	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupval 19295	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑊) → (𝐸‘[𝑢] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑢)))
87	86	adantrl 712	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘[𝑢] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑢)))
88	85, 87	oveq12d 7273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )) = ((𝐻 Σg (𝑇 ∘ 𝑡))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ 𝑢))))
89	80, 83, 88	3eqtr4d 2788	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
90	89	anass1rs 651	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑡 ∈ 𝑊) → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
91	40, 51, 90	ectocld 8531	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑎 ∈ (𝑊 / ∼ )) → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
92	47, 91	syldan 590	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑎 ∈ 𝑋) → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
93	92	an32s 648	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑢 ∈ 𝑊) → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
94	40, 45, 93	ectocld 8531	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑐 ∈ (𝑊 / ∼ )) → (𝐸‘(𝑎(+g‘𝐺)𝑐)) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘𝑐)))
95	39, 94	syldan 590	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑐 ∈ 𝑋) → (𝐸‘(𝑎(+g‘𝐺)𝑐)) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘𝑐)))
96	95	anasss 466	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐸‘(𝑎(+g‘𝐺)𝑐)) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘𝑐)))
97	1, 2, 3, 4, 8, 9, 16, 96	isghmd 18758	1 ⊢ (𝜑 → 𝐸 ∈ (𝐺 GrpHom 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ifcif 4456  ⟨cop 4564   ↦ cmpt 5153   I cid 5479   × cxp 5578  ran crn 5581   ∘ ccom 5584  Oncon0 6251  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  2_oc2o 8261  [cec 8454   / cqs 8455  Word cword 14145   ++ cconcat 14201  Basecbs 16840  +_gcplusg 16888   Σ_g cgsu 17068   /_s cqus 17133  Mndcmnd 18300  freeMndcfrmd 18401  Grpcgrp 18492  inv_gcminusg 18493   GrpHom cghm 18746   ~_FG cefg 19227  freeGrpcfrgp 19228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-ec 8458  df-qs 8462  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-xnn0 12236  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-word 14146  df-lsw 14194  df-concat 14202  df-s1 14229  df-substr 14282  df-pfx 14312  df-splice 14391  df-reverse 14400  df-s2 14489  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-0g 17069  df-gsum 17070  df-imas 17136  df-qus 17137  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-frmd 18403  df-grp 18495  df-minusg 18496  df-ghm 18747  df-efg 19230  df-frgp 19231

This theorem is referenced by:  frgpup3lem  19298  frgpup3  19299