Theorem frgpup1 19761

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 2736	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2736	. 2 ⊢ (+g‘𝐻) = (+g‘𝐻)
5		frgpup.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
6		frgpup.g	. . . 4 ⊢ 𝐺 = (freeGrp‘𝐼)
7	6	frgpgrp 19748	. . 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 19759	. 2 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵)
17		eqid 2736	. . . . . . . . . . 11 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o))
18	6, 17, 14	frgpval 19744	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
19	5, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
20		2on 8499	. . . . . . . . . . . . 13 ⊢ 2o ∈ On
21		xpexg 7749	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
22	5, 20, 21	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 × 2o) ∈ V)
23		wrdexg 14547	. . . . . . . . . . . 12 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
24		fvi 6960	. . . . . . . . . . . 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 2783	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o))
27		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o)))
28	17, 27	frmdbas 18835	. . . . . . . . . . 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 2774	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o))))
31	14	fvexi 6895	. . . . . . . . . 10 ⊢ ∼ ∈ V
32	31	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ∼ ∈ V)
33		fvexd 6896	. . . . . . . . 9 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2o)) ∈ V)
34	19, 30, 32, 33	qusbas 17564	. . . . . . . 8 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺))
35	1, 34	eqtr4id 2790	. . . . . . 7 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ ))
36		eqimss 4022	. . . . . . 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 3963	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑋) ∧ 𝑐 ∈ 𝑋) → 𝑐 ∈ (𝑊 / ∼ ))
40		eqid 2736	. . . . 5 ⊢ (𝑊 / ∼ ) = (𝑊 / ∼ )
41		oveq2 7418	. . . . . . 7 ⊢ ([𝑢] ∼ = 𝑐 → (𝑎(+g‘𝐺)[𝑢] ∼ ) = (𝑎(+g‘𝐺)𝑐))
42	41	fveq2d 6885	. . . . . 6 ⊢ ([𝑢] ∼ = 𝑐 → (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )) = (𝐸‘(𝑎(+g‘𝐺)𝑐)))
43		fveq2 6881	. . . . . . 7 ⊢ ([𝑢] ∼ = 𝑐 → (𝐸‘[𝑢] ∼ ) = (𝐸‘𝑐))
44	43	oveq2d 7426	. . . . . 6 ⊢ ([𝑢] ∼ = 𝑐 → ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘𝑐)))
45	42, 44	eqeq12d 2752	. . . . 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 3963	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (𝑊 / ∼ ))
47	46	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑊) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (𝑊 / ∼ ))
48		fvoveq1 7433	. . . . . . . . 9 ⊢ ([𝑡] ∼ = 𝑎 → (𝐸‘([𝑡] ∼ (+g‘𝐺)[𝑢] ∼ )) = (𝐸‘(𝑎(+g‘𝐺)[𝑢] ∼ )))
49		fveq2 6881	. . . . . . . . . 10 ⊢ ([𝑡] ∼ = 𝑎 → (𝐸‘[𝑡] ∼ ) = (𝐸‘𝑎))
50	49	oveq1d 7425	. . . . . . . . 9 ⊢ ([𝑡] ∼ = 𝑎 → ((𝐸‘[𝑡] ∼ )(+g‘𝐻)(𝐸‘[𝑢] ∼ )) = ((𝐸‘𝑎)(+g‘𝐻)(𝐸‘[𝑢] ∼ )))
51	48, 50	eqeq12d 2752	. . . . . . . 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 6961	. . . . . . . . . . . . . . . 16 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
53	13, 52	eqsstri 4010	. . . . . . . . . . . . . . 15 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
54	53	sseli 3959	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑊 → 𝑡 ∈ Word (𝐼 × 2o))
55	53	sseli 3959	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ 𝑊 → 𝑢 ∈ Word (𝐼 × 2o))
56		ccatcl 14597	. . . . . . . . . . . . . 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 19701	. . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊) → (𝑡 ++ 𝑢) ∈ 𝑊)
62	2, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15	frgpupval 19760	. . . . . . . . . . . 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 19756	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵)
67	66	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝑇:(𝐼 × 2o)⟶𝐵)
68		ccatco 14859	. . . . . . . . . . . . 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 7426	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))) = (𝐻 Σg ((𝑇 ∘ 𝑡) ++ (𝑇 ∘ 𝑢))))
71	9	grpmndd 18934	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ∈ Mnd)
72	71	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊)) → 𝐻 ∈ Mnd)
73		wrdco 14855	. . . . . . . . . . . . . 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 14855	. . . . . . . . . . . . 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 18824	. . . . . . . . . . . 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 2775	. . . . . . . . . 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 19749	. . . . . . . . . . . 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 6885	. . . . . . . . . 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 19760	. . . . . . . . . . . 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 19760	. . . . . . . . . . . 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 7428	. . . . . . . . . 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 2781	. . . . . . . . 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 8803	. . . . . . 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 8803	. . . 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 19213	1 ⊢ (𝜑 → 𝐸 ∈ (𝐺 GrpHom 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ⊆ wss 3931  ∅c0 4313  ifcif 4505  ⟨cop 4612   ↦ cmpt 5206   I cid 5552   × cxp 5657  ran crn 5660   ∘ ccom 5663  Oncon0 6357  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  2_oc2o 8479  [cec 8722   / cqs 8723  Word cword 14536   ++ cconcat 14593  Basecbs 17233  +_gcplusg 17276   Σ_g cgsu 17459   /_s cqus 17524  Mndcmnd 18717  freeMndcfrmd 18830  Grpcgrp 18921  inv_gcminusg 18922   GrpHom cghm 19200   ~_FG cefg 19692  freeGrpcfrgp 19693

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8724  df-ec 8726  df-qs 8730  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9459  df-inf 9460  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-xnn0 12580  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-fzo 13677  df-seq 14025  df-hash 14354  df-word 14537  df-lsw 14586  df-concat 14594  df-s1 14619  df-substr 14664  df-pfx 14694  df-splice 14773  df-reverse 14782  df-s2 14872  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-mulr 17290  df-sca 17292  df-vsca 17293  df-ip 17294  df-tset 17295  df-ple 17296  df-ds 17298  df-0g 17460  df-gsum 17461  df-imas 17527  df-qus 17528  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-submnd 18767  df-frmd 18832  df-grp 18924  df-minusg 18925  df-ghm 19201  df-efg 19695  df-frgp 19696

This theorem is referenced by:  frgpup3lem  19763  frgpup3  19764