Theorem gsumwspan 18928

Description: The submonoid generated by a set of elements is precisely the set of elements which can be expressed as finite products of the generator. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Hypotheses

Ref	Expression
gsumwspan.b	⊢ 𝐵 = (Base‘𝑀)
gsumwspan.k	⊢ 𝐾 = (mrCls‘(SubMnd‘𝑀))

Assertion

Ref	Expression
gsumwspan	⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) = ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))

Distinct variable groups:   𝑤,𝐺   𝑤,𝐵   𝑤,𝑀   𝑤,𝐾

Proof of Theorem gsumwspan

Dummy variables 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumwspan.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
2	1	submacs 18909	. . . . 5 ⊢ (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (ACS‘𝐵))
3	2	acsmred 17729	. . . 4 ⊢ (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
4	3	adantr 486	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
5		simpr 490	. . . . . . . 8 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐺)
6	5	s1cld 14655	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → ⟨“𝑥”⟩ ∈ Word 𝐺)
7		ssel2 3933	. . . . . . . . . 10 ⊢ ((𝐺 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐵)
8	7	adantll 727	. . . . . . . . 9 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐵)
9	1	gsumws1 18920	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → (𝑀 Σg ⟨“𝑥”⟩) = 𝑥)
10	8, 9	syl 18	. . . . . . . 8 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → (𝑀 Σg ⟨“𝑥”⟩) = 𝑥)
11	10	eqcomd 2771	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 = (𝑀 Σg ⟨“𝑥”⟩))
12		oveq2 7424	. . . . . . . 8 ⊢ (𝑤 = ⟨“𝑥”⟩ → (𝑀 Σg 𝑤) = (𝑀 Σg ⟨“𝑥”⟩))
13	12	rspceeqv 3606	. . . . . . 7 ⊢ ((⟨“𝑥”⟩ ∈ Word 𝐺 ∧ 𝑥 = (𝑀 Σg ⟨“𝑥”⟩)) → ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤))
14	6, 11, 13	syl2anc 596	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤))
15		eqid 2765	. . . . . . . 8 ⊢ (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))
16	15	elrnmpt 5950	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤)))
17	16	elv 3462	. . . . . 6 ⊢ (𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤))
18	14, 17	sylibr 237	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
19	18	ex 418	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝑥 ∈ 𝐺 → 𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
20	19	ssrdv 3944	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → 𝐺 ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
21		gsumwspan.k	. . . . . . . . . 10 ⊢ 𝐾 = (mrCls‘(SubMnd‘𝑀))
22	21	mrccl 17684	. . . . . . . . 9 ⊢ (((SubMnd‘𝑀) ∈ (Moore‘𝐵) ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ∈ (SubMnd‘𝑀))
23	3, 22	sylan 592	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ∈ (SubMnd‘𝑀))
24	21	mrcssid 17690	. . . . . . . . . . 11 ⊢ (((SubMnd‘𝑀) ∈ (Moore‘𝐵) ∧ 𝐺 ⊆ 𝐵) → 𝐺 ⊆ (𝐾‘𝐺))
25	3, 24	sylan 592	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → 𝐺 ⊆ (𝐾‘𝐺))
26		sswrd 14572	. . . . . . . . . 10 ⊢ (𝐺 ⊆ (𝐾‘𝐺) → Word 𝐺 ⊆ Word (𝐾‘𝐺))
27	25, 26	syl 18	. . . . . . . . 9 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → Word 𝐺 ⊆ Word (𝐾‘𝐺))
28	27	sselda 3938	. . . . . . . 8 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑤 ∈ Word 𝐺) → 𝑤 ∈ Word (𝐾‘𝐺))
29		gsumwsubmcl 18919	. . . . . . . 8 ⊢ (((𝐾‘𝐺) ∈ (SubMnd‘𝑀) ∧ 𝑤 ∈ Word (𝐾‘𝐺)) → (𝑀 Σg 𝑤) ∈ (𝐾‘𝐺))
30	23, 28, 29	syl2an2r 698	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑤 ∈ Word 𝐺) → (𝑀 Σg 𝑤) ∈ (𝐾‘𝐺))
31	30	fmpttd 7114	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)):Word 𝐺⟶(𝐾‘𝐺))
32	31	frnd 6718	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ (𝐾‘𝐺))
33	3, 21	mrcssvd 17696	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (𝐾‘𝐺) ⊆ 𝐵)
34	33	adantr 486	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ⊆ 𝐵)
35	32, 34	sstrd 3948	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ 𝐵)
36		wrd0 14589	. . . . . 6 ⊢ ∅ ∈ Word 𝐺
37		eqid 2765	. . . . . . . . 9 ⊢ (0g‘𝑀) = (0g‘𝑀)
38	37	gsum0 18763	. . . . . . . 8 ⊢ (𝑀 Σg ∅) = (0g‘𝑀)
39	38	eqcomi 2774	. . . . . . 7 ⊢ (0g‘𝑀) = (𝑀 Σg ∅)
40	39	a1i 11	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (0g‘𝑀) = (𝑀 Σg ∅))
41		oveq2 7424	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑀 Σg 𝑤) = (𝑀 Σg ∅))
42	41	rspceeqv 3606	. . . . . 6 ⊢ ((∅ ∈ Word 𝐺 ∧ (0g‘𝑀) = (𝑀 Σg ∅)) → ∃𝑤 ∈ Word 𝐺(0g‘𝑀) = (𝑀 Σg 𝑤))
43	36, 40, 42	sylancr 599	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ∃𝑤 ∈ Word 𝐺(0g‘𝑀) = (𝑀 Σg 𝑤))
44		fvex 6898	. . . . . 6 ⊢ (0g‘𝑀) ∈ V
45	15	elrnmpt 5950	. . . . . 6 ⊢ ((0g‘𝑀) ∈ V → ((0g‘𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺(0g‘𝑀) = (𝑀 Σg 𝑤)))
46	44, 45	ax-mp 5	. . . . 5 ⊢ ((0g‘𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺(0g‘𝑀) = (𝑀 Σg 𝑤))
47	43, 46	sylibr 237	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (0g‘𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
48		ccatcl 14624	. . . . . . . 8 ⊢ ((𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺) → (𝑧 ++ 𝑣) ∈ Word 𝐺)
49		simpll 779	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑀 ∈ Mnd)
50		sswrd 14572	. . . . . . . . . . . 12 ⊢ (𝐺 ⊆ 𝐵 → Word 𝐺 ⊆ Word 𝐵)
51	50	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → Word 𝐺 ⊆ Word 𝐵)
52		simprl 783	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑧 ∈ Word 𝐺)
53	51, 52	sseldd 3939	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑧 ∈ Word 𝐵)
54		simprr 785	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑣 ∈ Word 𝐺)
55	51, 54	sseldd 3939	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑣 ∈ Word 𝐵)
56		eqid 2765	. . . . . . . . . . 11 ⊢ (+g‘𝑀) = (+g‘𝑀)
57	1, 56	gsumccat 18923	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mnd ∧ 𝑧 ∈ Word 𝐵 ∧ 𝑣 ∈ Word 𝐵) → (𝑀 Σg (𝑧 ++ 𝑣)) = ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)))
58	49, 53, 55, 57	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → (𝑀 Σg (𝑧 ++ 𝑣)) = ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)))
59	58	eqcomd 2771	. . . . . . . 8 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg (𝑧 ++ 𝑣)))
60		oveq2 7424	. . . . . . . . 9 ⊢ (𝑤 = (𝑧 ++ 𝑣) → (𝑀 Σg 𝑤) = (𝑀 Σg (𝑧 ++ 𝑣)))
61	60	rspceeqv 3606	. . . . . . . 8 ⊢ (((𝑧 ++ 𝑣) ∈ Word 𝐺 ∧ ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg (𝑧 ++ 𝑣))) → ∃𝑤 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg 𝑤))
62	48, 59, 61	syl2an2 699	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → ∃𝑤 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg 𝑤))
63		ovex 7449	. . . . . . . 8 ⊢ ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ V
64	15	elrnmpt 5950	. . . . . . . 8 ⊢ (((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ V → (((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg 𝑤)))
65	63, 64	ax-mp 5	. . . . . . 7 ⊢ (((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg 𝑤))
66	62, 65	sylibr 237	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
67	66	ralrimivva 3210	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ∀𝑧 ∈ Word 𝐺∀𝑣 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
68		oveq2 7424	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑧))
69	68	cbvmptv 5217	. . . . . . . 8 ⊢ (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
70	69	rneqi 5929	. . . . . . 7 ⊢ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
71	70	raleqi 3323	. . . . . 6 ⊢ (∀𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g‘𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑥 ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g‘𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
72		oveq2 7424	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑣))
73	72	cbvmptv 5217	. . . . . . . . . 10 ⊢ (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
74	73	rneqi 5929	. . . . . . . . 9 ⊢ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
75	74	raleqi 3323	. . . . . . . 8 ⊢ (∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g‘𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑦 ∈ ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))(𝑥(+g‘𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
76		eqid 2765	. . . . . . . . . 10 ⊢ (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
77		oveq2 7424	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑀 Σg 𝑣) → (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑀)(𝑀 Σg 𝑣)))
78	77	eleq1d 2850	. . . . . . . . . 10 ⊢ (𝑦 = (𝑀 Σg 𝑣) → ((𝑥(+g‘𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ (𝑥(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
79	76, 78	ralrnmptw 7093	. . . . . . . . 9 ⊢ (∀𝑣 ∈ Word 𝐺(𝑀 Σg 𝑣) ∈ V → (∀𝑦 ∈ ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))(𝑥(+g‘𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑣 ∈ Word 𝐺(𝑥(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
80		ovexd 7451	. . . . . . . . 9 ⊢ (𝑣 ∈ Word 𝐺 → (𝑀 Σg 𝑣) ∈ V)
81	79, 80	mprg 3087	. . . . . . . 8 ⊢ (∀𝑦 ∈ ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))(𝑥(+g‘𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑣 ∈ Word 𝐺(𝑥(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
82	75, 81	bitri 278	. . . . . . 7 ⊢ (∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g‘𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑣 ∈ Word 𝐺(𝑥(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
83	82	ralbii 3113	. . . . . 6 ⊢ (∀𝑥 ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g‘𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑥 ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))∀𝑣 ∈ Word 𝐺(𝑥(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
84		eqid 2765	. . . . . . . 8 ⊢ (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
85		oveq1 7423	. . . . . . . . . 10 ⊢ (𝑥 = (𝑀 Σg 𝑧) → (𝑥(+g‘𝑀)(𝑀 Σg 𝑣)) = ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)))
86	85	eleq1d 2850	. . . . . . . . 9 ⊢ (𝑥 = (𝑀 Σg 𝑧) → ((𝑥(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
87	86	ralbidv 3190	. . . . . . . 8 ⊢ (𝑥 = (𝑀 Σg 𝑧) → (∀𝑣 ∈ Word 𝐺(𝑥(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑣 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
88	84, 87	ralrnmptw 7093	. . . . . . 7 ⊢ (∀𝑧 ∈ Word 𝐺(𝑀 Σg 𝑧) ∈ V → (∀𝑥 ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))∀𝑣 ∈ Word 𝐺(𝑥(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑧 ∈ Word 𝐺∀𝑣 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
89		ovexd 7451	. . . . . . 7 ⊢ (𝑧 ∈ Word 𝐺 → (𝑀 Σg 𝑧) ∈ V)
90	88, 89	mprg 3087	. . . . . 6 ⊢ (∀𝑥 ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))∀𝑣 ∈ Word 𝐺(𝑥(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑧 ∈ Word 𝐺∀𝑣 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
91	71, 83, 90	3bitri 300	. . . . 5 ⊢ (∀𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g‘𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑧 ∈ Word 𝐺∀𝑣 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
92	67, 91	sylibr 237	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ∀𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g‘𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
93	1, 37, 56	issubm 18884	. . . . 5 ⊢ (𝑀 ∈ Mnd → (ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∈ (SubMnd‘𝑀) ↔ (ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ 𝐵 ∧ (0g‘𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∧ ∀𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g‘𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))))
94	93	adantr 486	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∈ (SubMnd‘𝑀) ↔ (ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ 𝐵 ∧ (0g‘𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∧ ∀𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g‘𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))))
95	35, 47, 92, 94	mpbir3and 1361	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∈ (SubMnd‘𝑀))
96	21	mrcsscl 17693	. . 3 ⊢ (((SubMnd‘𝑀) ∈ (Moore‘𝐵) ∧ 𝐺 ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∧ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∈ (SubMnd‘𝑀)) → (𝐾‘𝐺) ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
97	4, 20, 95, 96	syl3anc 1398	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
98	97, 32	eqssd 3955	1 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) = ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  ∅c0 4286   ↦ cmpt 5194  ran crn 5664  ‘cfv 6540  (class class class)co 7416  Word cword 14563   ++ cconcat 14620  ⟨“cs1 14647  Basecbs 17286  +_gcplusg 17327  0_gc0g 17509   Σ_g cgsu 17510  Moorecmre 17651  mrClscmrc 17652  Mndcmnd 18813  SubMndcsubmnd 18863

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-card 9937  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-n0 12516  df-z 12603  df-uz 12874  df-fz 13547  df-fzo 13695  df-seq 14051  df-hash 14380  df-word 14564  df-concat 14621  df-s1 14648  df-sets 17241  df-slot 17259  df-ndx 17271  df-base 17287  df-ress 17308  df-plusg 17340  df-0g 17511  df-gsum 17512  df-mre 17655  df-mrc 17656  df-acs 17658  df-mgm 18715  df-sgrp 18798  df-mnd 18814  df-submnd 18865

This theorem is used by:  psgneldm2  19597  psgnfitr  19610