Theorem frgpup1 19817

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 2740	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2740	. 2 ⊢ (+g‘𝐻) = (+g‘𝐻)
5		frgpup.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
6		frgpup.g	. . . 4 ⊢ 𝐺 = (freeGrp‘𝐼)
7	6	frgpgrp 19804	. . 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 19815	. 2 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵)
17		eqid 2740	. . . . . . . . . . 11 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o))
18	6, 17, 14	frgpval 19800	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
19	5, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
20		2on 8536	. . . . . . . . . . . . 13 ⊢ 2o ∈ On
21		xpexg 7785	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
22	5, 20, 21	sylancl 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 × 2o) ∈ V)
23		wrdexg 14572	. . . . . . . . . . . 12 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
24		fvi 6998	. . . . . . . . . . . 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 2792	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o))
27		eqid 2740	. . . . . . . . . . . 12 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o)))
28	17, 27	frmdbas 18887	. . . . . . . . . . 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 2783	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o))))
31	14	fvexi 6934	. . . . . . . . . 10 ⊢ ∼ ∈ V
32	31	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ∼ ∈ V)
33		fvexd 6935	. . . . . . . . 9 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2o)) ∈ V)
34	19, 30, 32, 33	qusbas 17605	. . . . . . . 8 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺))
35	1, 34	eqtr4id 2799	. . . . . . 7 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ ))
36		eqimss 4067	. . . . . . 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 4008	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑐 ∈ 𝑋) → 𝑐 ∈ (𝑊 / ∼ ))
40		eqid 2740	. . . . 5 ⊢ (𝑊 / ∼ ) = (𝑊 / ∼ )
41		oveq2 7456	. . . . . . 7 ⊢ ([𝑢] ∼ = 𝑐 → (𝑎(+g‘𝐺)[𝑢] ∼ ) = (𝑎(+g‘𝐺)𝑐))
42	41	fveq2d 6924	. . . . . 6 ⊢ ([𝑢] ∼ = 𝑐 → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = (𝐸‘(𝑎(+g‘𝐺)𝑐)))
43		fveq2 6920	. . . . . . 7 ⊢ ([𝑢] ∼ = 𝑐 → (𝐸‘[𝑢] ∼ ) = (𝐸‘𝑐))
44	43	oveq2d 7464	. . . . . 6 ⊢ ([𝑢] ∼ = 𝑐 → ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘𝑐)))
45	42, 44	eqeq12d 2756	. . . . 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 4008	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (𝑊 / ∼ ))
47	46	adantlr 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (𝑊 / ∼ ))
48		fvoveq1 7471	. . . . . . . . 9 ⊢ ([𝑡] ∼ = 𝑎 → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )))
49		fveq2 6920	. . . . . . . . . 10 ⊢ ([𝑡] ∼ = 𝑎 → (𝐸‘[𝑡] ∼ ) = (𝐸‘𝑎))
50	49	oveq1d 7463	. . . . . . . . 9 ⊢ ([𝑡] ∼ = 𝑎 → ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
51	48, 50	eqeq12d 2756	. . . . . . . 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 6999	. . . . . . . . . . . . . . . 16 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
53	13, 52	eqsstri 4043	. . . . . . . . . . . . . . 15 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
54	53	sseli 4004	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑊 → 𝑡 ∈ Word (𝐼 × 2o))
55	53	sseli 4004	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ 𝑊 → 𝑢 ∈ Word (𝐼 × 2o))
56		ccatcl 14622	. . . . . . . . . . . . . 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 19757	. . . . . . . . . . . . . . 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 2847	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → (𝑡 ++ 𝑢) ∈ 𝑊)
62	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupval 19816	. . . . . . . . . . . 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 727	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝑡 ∈ Word (𝐼 × 2o))
65	55	ad2antll 728	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝑢 ∈ Word (𝐼 × 2o))
66	2, 10, 11, 9, 5, 12	frgpuptf 19812	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵)
67	66	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝑇:(𝐼 × 2o)⟶𝐵)
68		ccatco 14884	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝑇 ∘ (𝑡 ++ 𝑢)) = ((𝑇 ∘ 𝑡) ++ (𝑇 ∘ 𝑢)))
70	69	oveq2d 7464	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))) = (𝐻 Σg ((𝑇 ∘ 𝑡) ++ (𝑇 ∘ 𝑢))))
71	9	grpmndd 18986	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ∈ Mnd)
72	71	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝐻 ∈ Mnd)
73		wrdco 14880	. . . . . . . . . . . . . 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 716	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝑇 ∘ 𝑡) ∈ Word 𝐵)
76		wrdco 14880	. . . . . . . . . . . . 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 18876	. . . . . . . . . . . 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 1371	. . . . . . . . . . 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 2784	. . . . . . . . . 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 19805	. . . . . . . . . . . 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 6924	. . . . . . . . . 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 19816	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐸‘[𝑡] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑡)))
85	84	adantrr 716	. . . . . . . . . . 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 19816	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑊) → (𝐸‘[𝑢] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑢)))
87	86	adantrl 715	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘[𝑢] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑢)))
88	85, 87	oveq12d 7466	. . . . . . . . . 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 2790	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
90	89	anass1rs 654	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑡 ∈ 𝑊) → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
91	40, 51, 90	ectocld 8842	. . . . . . 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 651	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑢 ∈ 𝑊) → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
94	40, 45, 93	ectocld 8842	. . . 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 19265	1 ⊢ (𝜑 → 𝐸 ∈ (𝐺 GrpHom 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  ifcif 4548  ⟨cop 4654   ↦ cmpt 5249   I cid 5592   × cxp 5698  ran crn 5701   ∘ ccom 5704  Oncon0 6395  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  2_oc2o 8516  [cec 8761   / cqs 8762  Word cword 14562   ++ cconcat 14618  Basecbs 17258  +_gcplusg 17311   Σ_g cgsu 17500   /_s cqus 17565  Mndcmnd 18772  freeMndcfrmd 18882  Grpcgrp 18973  inv_gcminusg 18974   GrpHom cghm 19252   ~_FG cefg 19748  freeGrpcfrgp 19749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-ec 8765  df-qs 8769  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-xnn0 12626  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-word 14563  df-lsw 14611  df-concat 14619  df-s1 14644  df-substr 14689  df-pfx 14719  df-splice 14798  df-reverse 14807  df-s2 14897  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-0g 17501  df-gsum 17502  df-imas 17568  df-qus 17569  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-frmd 18884  df-grp 18976  df-minusg 18977  df-ghm 19253  df-efg 19751  df-frgp 19752

This theorem is referenced by:  frgpup3lem  19819  frgpup3  19820