Theorem frgpup1 18541

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 2825	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2825	. 2 ⊢ (+g‘𝐻) = (+g‘𝐻)
5		frgpup.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
6		frgpup.g	. . . 4 ⊢ 𝐺 = (freeGrp‘𝐼)
7	6	frgpgrp 18528	. . 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 18539	. 2 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵)
17		eqid 2825	. . . . . . . . . . 11 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o))
18	6, 17, 14	frgpval 18524	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
19	5, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
20		2on 7835	. . . . . . . . . . . . 13 ⊢ 2o ∈ On
21		xpexg 7220	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
22	5, 20, 21	sylancl 580	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 × 2o) ∈ V)
23		wrdexg 13584	. . . . . . . . . . . 12 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
24		fvi 6502	. . . . . . . . . . . 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	syl5eq 2873	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o))
27		eqid 2825	. . . . . . . . . . . 12 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o)))
28	17, 27	frmdbas 17743	. . . . . . . . . . 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 2864	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o))))
31	14	fvexi 6447	. . . . . . . . . 10 ⊢ ∼ ∈ V
32	31	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ∼ ∈ V)
33		fvexd 6448	. . . . . . . . 9 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2o)) ∈ V)
34	19, 30, 32, 33	qusbas 16558	. . . . . . . 8 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺))
35	34, 1	syl6reqr 2880	. . . . . . 7 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ ))
36		eqimss 3882	. . . . . . 7 ⊢ (𝑋 = (𝑊 / ∼ ) → 𝑋 ⊆ (𝑊 / ∼ ))
37	35, 36	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ (𝑊 / ∼ ))
38	37	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑋) → 𝑋 ⊆ (𝑊 / ∼ ))
39	38	sselda 3827	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑐 ∈ 𝑋) → 𝑐 ∈ (𝑊 / ∼ ))
40		eqid 2825	. . . . 5 ⊢ (𝑊 / ∼ ) = (𝑊 / ∼ )
41		oveq2 6913	. . . . . . 7 ⊢ ([𝑢] ∼ = 𝑐 → (𝑎(+g‘𝐺)[𝑢] ∼ ) = (𝑎(+g‘𝐺)𝑐))
42	41	fveq2d 6437	. . . . . 6 ⊢ ([𝑢] ∼ = 𝑐 → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = (𝐸‘(𝑎(+g‘𝐺)𝑐)))
43		fveq2 6433	. . . . . . 7 ⊢ ([𝑢] ∼ = 𝑐 → (𝐸‘[𝑢] ∼ ) = (𝐸‘𝑐))
44	43	oveq2d 6921	. . . . . 6 ⊢ ([𝑢] ∼ = 𝑐 → ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘𝑐)))
45	42, 44	eqeq12d 2840	. . . . 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 3827	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (𝑊 / ∼ ))
47	46	adantlr 706	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (𝑊 / ∼ ))
48		fvoveq1 6928	. . . . . . . . 9 ⊢ ([𝑡] ∼ = 𝑎 → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )))
49		fveq2 6433	. . . . . . . . . 10 ⊢ ([𝑡] ∼ = 𝑎 → (𝐸‘[𝑡] ∼ ) = (𝐸‘𝑎))
50	49	oveq1d 6920	. . . . . . . . 9 ⊢ ([𝑡] ∼ = 𝑎 → ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
51	48, 50	eqeq12d 2840	. . . . . . . 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 6503	. . . . . . . . . . . . . . . 16 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
53	13, 52	eqsstri 3860	. . . . . . . . . . . . . . 15 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
54	53	sseli 3823	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑊 → 𝑡 ∈ Word (𝐼 × 2o))
55	53	sseli 3823	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ 𝑊 → 𝑢 ∈ Word (𝐼 × 2o))
56		ccatcl 13634	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ Word (𝐼 × 2o) ∧ 𝑢 ∈ Word (𝐼 × 2o)) → (𝑡 ++ 𝑢) ∈ Word (𝐼 × 2o))
57	54, 55, 56	syl2an 589	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → (𝑡 ++ 𝑢) ∈ Word (𝐼 × 2o))
58	13	efgrcl 18479	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
59	58	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
60	59	simprd 491	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → 𝑊 = Word (𝐼 × 2o))
61	57, 60	eleqtrrd 2909	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → (𝑡 ++ 𝑢) ∈ 𝑊)
62	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupval 18540	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ++ 𝑢) ∈ 𝑊) → (𝐸‘[(𝑡 ++ 𝑢)] ∼ ) = (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))))
63	61, 62	sylan2 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘[(𝑡 ++ 𝑢)] ∼ ) = (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))))
64	54	ad2antrl 719	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝑡 ∈ Word (𝐼 × 2o))
65	55	ad2antll 720	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝑢 ∈ Word (𝐼 × 2o))
66	2, 10, 11, 9, 5, 12	frgpuptf 18536	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵)
67	66	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝑇:(𝐼 × 2o)⟶𝐵)
68		ccatco 13956	. . . . . . . . . . . . 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 1494	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝑇 ∘ (𝑡 ++ 𝑢)) = ((𝑇 ∘ 𝑡) ++ (𝑇 ∘ 𝑢)))
70	69	oveq2d 6921	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))) = (𝐻 Σg ((𝑇 ∘ 𝑡) ++ (𝑇 ∘ 𝑢))))
71		grpmnd 17783	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
72	9, 71	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ∈ Mnd)
73	72	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝐻 ∈ Mnd)
74		wrdco 13952	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ 𝑡) ∈ Word 𝐵)
75	54, 66, 74	syl2anr 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑇 ∘ 𝑡) ∈ Word 𝐵)
76	75	adantrr 708	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝑇 ∘ 𝑡) ∈ Word 𝐵)
77		wrdco 13952	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ 𝑢) ∈ Word 𝐵)
78	65, 67, 77	syl2anc 579	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝑇 ∘ 𝑢) ∈ Word 𝐵)
79	2, 4	gsumccat 17731	. . . . . . . . . . . 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))))
80	73, 76, 78, 79	syl3anc 1494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐻 Σg ((𝑇 ∘ 𝑡) ++ (𝑇 ∘ 𝑢))) = ((𝐻 Σg (𝑇 ∘ 𝑡))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ 𝑢))))
81	63, 70, 80	3eqtrd 2865	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘[(𝑡 ++ 𝑢)] ∼ ) = ((𝐻 Σg (𝑇 ∘ 𝑡))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ 𝑢))))
82	13, 6, 14, 3	frgpadd 18529	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → ([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ ) = [(𝑡 ++ 𝑢)] ∼ )
83	82	adantl 475	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → ([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ ) = [(𝑡 ++ 𝑢)] ∼ )
84	83	fveq2d 6437	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = (𝐸‘[(𝑡 ++ 𝑢)] ∼ ))
85	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupval 18540	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐸‘[𝑡] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑡)))
86	85	adantrr 708	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘[𝑡] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑡)))
87	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupval 18540	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑊) → (𝐸‘[𝑢] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑢)))
88	87	adantrl 707	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘[𝑢] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑢)))
89	86, 88	oveq12d 6923	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )) = ((𝐻 Σg (𝑇 ∘ 𝑡))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ 𝑢))))
90	81, 84, 89	3eqtr4d 2871	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
91	90	anass1rs 645	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑡 ∈ 𝑊) → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
92	40, 51, 91	ectocld 8079	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑎 ∈ (𝑊 / ∼ )) → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
93	47, 92	syldan 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑎 ∈ 𝑋) → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
94	93	an32s 642	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑢 ∈ 𝑊) → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
95	40, 45, 94	ectocld 8079	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑐 ∈ (𝑊 / ∼ )) → (𝐸‘(𝑎(+g‘𝐺)𝑐)) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘𝑐)))
96	39, 95	syldan 585	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑐 ∈ 𝑋) → (𝐸‘(𝑎(+g‘𝐺)𝑐)) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘𝑐)))
97	96	anasss 460	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐸‘(𝑎(+g‘𝐺)𝑐)) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘𝑐)))
98	1, 2, 3, 4, 8, 9, 16, 97	isghmd 18020	1 ⊢ (𝜑 → 𝐸 ∈ (𝐺 GrpHom 𝐻))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164  Vcvv 3414   ⊆ wss 3798  ∅c0 4144  ifcif 4306  ⟨cop 4403   ↦ cmpt 4952   I cid 5249   × cxp 5340  ran crn 5343   ∘ ccom 5346  Oncon0 5963  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905   ↦ cmpt2 6907  2_oc2o 7820  [cec 8007   / cqs 8008  Word cword 13574   ++ cconcat 13630  Basecbs 16222  +_gcplusg 16305   Σ_g cgsu 16454   /_s cqus 16518  Mndcmnd 17647  freeMndcfrmd 17738  Grpcgrp 17776  inv_gcminusg 17777   GrpHom cghm 18008   ~_FG cefg 18470  freeGrpcfrgp 18471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-ot 4406  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-ec 8011  df-qs 8015  df-map 8124  df-pm 8125  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-sup 8617  df-inf 8618  df-card 9078  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-xnn0 11691  df-z 11705  df-dec 11822  df-uz 11969  df-fz 12620  df-fzo 12761  df-seq 13096  df-hash 13411  df-word 13575  df-lsw 13623  df-concat 13631  df-s1 13656  df-substr 13701  df-pfx 13750  df-splice 13857  df-reverse 13875  df-s2 13969  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-sets 16229  df-ress 16230  df-plusg 16318  df-mulr 16319  df-sca 16321  df-vsca 16322  df-ip 16323  df-tset 16324  df-ple 16325  df-ds 16327  df-0g 16455  df-gsum 16456  df-imas 16521  df-qus 16522  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-submnd 17689  df-frmd 17740  df-grp 17779  df-minusg 17780  df-ghm 18009  df-efg 18473  df-frgp 18474

This theorem is referenced by:  frgpup3lem  18543  frgpup3  18544