Theorem frgpup1 19712

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 2730	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2730	. 2 ⊢ (+g‘𝐻) = (+g‘𝐻)
5		frgpup.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
6		frgpup.g	. . . 4 ⊢ 𝐺 = (freeGrp‘𝐼)
7	6	frgpgrp 19699	. . 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 19710	. 2 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵)
17		eqid 2730	. . . . . . . . . . 11 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o))
18	6, 17, 14	frgpval 19695	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
19	5, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
20		2on 8450	. . . . . . . . . . . . 13 ⊢ 2o ∈ On
21		xpexg 7729	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
22	5, 20, 21	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 × 2o) ∈ V)
23		wrdexg 14496	. . . . . . . . . . . 12 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
24		fvi 6940	. . . . . . . . . . . 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 2777	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o))
27		eqid 2730	. . . . . . . . . . . 12 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o)))
28	17, 27	frmdbas 18786	. . . . . . . . . . 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 2768	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o))))
31	14	fvexi 6875	. . . . . . . . . 10 ⊢ ∼ ∈ V
32	31	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ∼ ∈ V)
33		fvexd 6876	. . . . . . . . 9 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2o)) ∈ V)
34	19, 30, 32, 33	qusbas 17515	. . . . . . . 8 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺))
35	1, 34	eqtr4id 2784	. . . . . . 7 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ ))
36		eqimss 4008	. . . . . . 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 3949	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑐 ∈ 𝑋) → 𝑐 ∈ (𝑊 / ∼ ))
40		eqid 2730	. . . . 5 ⊢ (𝑊 / ∼ ) = (𝑊 / ∼ )
41		oveq2 7398	. . . . . . 7 ⊢ ([𝑢] ∼ = 𝑐 → (𝑎(+g‘𝐺)[𝑢] ∼ ) = (𝑎(+g‘𝐺)𝑐))
42	41	fveq2d 6865	. . . . . 6 ⊢ ([𝑢] ∼ = 𝑐 → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = (𝐸‘(𝑎(+g‘𝐺)𝑐)))
43		fveq2 6861	. . . . . . 7 ⊢ ([𝑢] ∼ = 𝑐 → (𝐸‘[𝑢] ∼ ) = (𝐸‘𝑐))
44	43	oveq2d 7406	. . . . . 6 ⊢ ([𝑢] ∼ = 𝑐 → ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘𝑐)))
45	42, 44	eqeq12d 2746	. . . . 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 3949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (𝑊 / ∼ ))
47	46	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (𝑊 / ∼ ))
48		fvoveq1 7413	. . . . . . . . 9 ⊢ ([𝑡] ∼ = 𝑎 → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )))
49		fveq2 6861	. . . . . . . . . 10 ⊢ ([𝑡] ∼ = 𝑎 → (𝐸‘[𝑡] ∼ ) = (𝐸‘𝑎))
50	49	oveq1d 7405	. . . . . . . . 9 ⊢ ([𝑡] ∼ = 𝑎 → ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
51	48, 50	eqeq12d 2746	. . . . . . . 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 6941	. . . . . . . . . . . . . . . 16 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
53	13, 52	eqsstri 3996	. . . . . . . . . . . . . . 15 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
54	53	sseli 3945	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑊 → 𝑡 ∈ Word (𝐼 × 2o))
55	53	sseli 3945	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ 𝑊 → 𝑢 ∈ Word (𝐼 × 2o))
56		ccatcl 14546	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ Word (𝐼 × 2o) ∧ 𝑢 ∈ Word (𝐼 × 2o)) → (𝑡 ++ 𝑢) ∈ Word (𝐼 × 2o))
57	54, 55, 56	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → (𝑡 ++ 𝑢) ∈ Word (𝐼 × 2o))
58	13	efgrcl 19652	. . . . . . . . . . . . . . 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 2832	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → (𝑡 ++ 𝑢) ∈ 𝑊)
62	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupval 19711	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ++ 𝑢) ∈ 𝑊) → (𝐸‘[(𝑡 ++ 𝑢)] ∼ ) = (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))))
63	61, 62	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘[(𝑡 ++ 𝑢)] ∼ ) = (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))))
64	54	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝑡 ∈ Word (𝐼 × 2o))
65	55	ad2antll 729	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝑢 ∈ Word (𝐼 × 2o))
66	2, 10, 11, 9, 5, 12	frgpuptf 19707	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵)
67	66	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝑇:(𝐼 × 2o)⟶𝐵)
68		ccatco 14808	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝑇 ∘ (𝑡 ++ 𝑢)) = ((𝑇 ∘ 𝑡) ++ (𝑇 ∘ 𝑢)))
70	69	oveq2d 7406	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))) = (𝐻 Σg ((𝑇 ∘ 𝑡) ++ (𝑇 ∘ 𝑢))))
71	9	grpmndd 18885	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ∈ Mnd)
72	71	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝐻 ∈ Mnd)
73		wrdco 14804	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ 𝑡) ∈ Word 𝐵)
74	54, 66, 73	syl2anr 597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑇 ∘ 𝑡) ∈ Word 𝐵)
75	74	adantrr 717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝑇 ∘ 𝑡) ∈ Word 𝐵)
76		wrdco 14804	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ 𝑢) ∈ Word 𝐵)
77	65, 67, 76	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝑇 ∘ 𝑢) ∈ Word 𝐵)
78	2, 4	gsumccat 18775	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 2769	. . . . . . . . . 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 19700	. . . . . . . . . . . 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 6865	. . . . . . . . . 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 19711	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐸‘[𝑡] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑡)))
85	84	adantrr 717	. . . . . . . . . . 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 19711	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑊) → (𝐸‘[𝑢] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑢)))
87	86	adantrl 716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘[𝑢] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑢)))
88	85, 87	oveq12d 7408	. . . . . . . . . 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 2775	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
90	89	anass1rs 655	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑡 ∈ 𝑊) → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
91	40, 51, 90	ectocld 8758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑎 ∈ (𝑊 / ∼ )) → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
92	47, 91	syldan 591	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑎 ∈ 𝑋) → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
93	92	an32s 652	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑢 ∈ 𝑊) → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
94	40, 45, 93	ectocld 8758	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑐 ∈ (𝑊 / ∼ )) → (𝐸‘(𝑎(+g‘𝐺)𝑐)) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘𝑐)))
95	39, 94	syldan 591	. . 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 19164	1 ⊢ (𝜑 → 𝐸 ∈ (𝐺 GrpHom 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  ifcif 4491  ⟨cop 4598   ↦ cmpt 5191   I cid 5535   × cxp 5639  ran crn 5642   ∘ ccom 5645  Oncon0 6335  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  2_oc2o 8431  [cec 8672   / cqs 8673  Word cword 14485   ++ cconcat 14542  Basecbs 17186  +_gcplusg 17227   Σ_g cgsu 17410   /_s cqus 17475  Mndcmnd 18668  freeMndcfrmd 18781  Grpcgrp 18872  inv_gcminusg 18873   GrpHom cghm 19151   ~_FG cefg 19643  freeGrpcfrgp 19644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-ec 8676  df-qs 8680  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-xnn0 12523  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-fzo 13623  df-seq 13974  df-hash 14303  df-word 14486  df-lsw 14535  df-concat 14543  df-s1 14568  df-substr 14613  df-pfx 14643  df-splice 14722  df-reverse 14731  df-s2 14821  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-0g 17411  df-gsum 17412  df-imas 17478  df-qus 17479  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-submnd 18718  df-frmd 18783  df-grp 18875  df-minusg 18876  df-ghm 19152  df-efg 19646  df-frgp 19647

This theorem is referenced by:  frgpup3lem  19714  frgpup3  19715