Theorem gsumwspan 19048

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 19029	. . . . 5 ⊢ (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (ACS‘𝐵))
3	2	acsmred 17836	. . . 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 14754	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → ⟨“𝑥”⟩ ∈ Word 𝐺)
7		ssel2 3903	. . . . . . . . . 10 ⊢ ((𝐺 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐵)
8	7	adantll 727	. . . . . . . . 9 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐵)
9	1	gsumws1 19040	. . . . . . . . 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 2766	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 = (𝑀 Σg ⟨“𝑥”⟩))
12		oveq2 7429	. . . . . . . 8 ⊢ (𝑤 = ⟨“𝑥”⟩ → (𝑀 Σg 𝑤) = (𝑀 Σg ⟨“𝑥”⟩))
13	12	rspceeqv 3598	. . . . . . 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 2760	. . . . . . . 8 ⊢ (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))
16	15	elrnmpt 5939	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤)))
17	16	elv 3455	. . . . . 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 3914	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → 𝐺 ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
21		gsumwspan.k	. . . . . . . . . 10 ⊢ 𝐾 = (mrCls‘(SubMnd‘𝑀))
22	21	mrccl 17791	. . . . . . . . 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 17797	. . . . . . . . . . 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 14671	. . . . . . . . . 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 3908	. . . . . . . 8 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑤 ∈ Word 𝐺) → 𝑤 ∈ Word (𝐾‘𝐺))
29		gsumwsubmcl 19039	. . . . . . . 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 7116	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)):Word 𝐺⟶(𝐾‘𝐺))
32	31	frnd 6719	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ (𝐾‘𝐺))
33	3, 21	mrcssvd 17803	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (𝐾‘𝐺) ⊆ 𝐵)
34	33	adantr 486	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ⊆ 𝐵)
35	32, 34	sstrd 3918	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ 𝐵)
36		wrd0 14688	. . . . . 6 ⊢ ∅ ∈ Word 𝐺
37		eqid 2760	. . . . . . . . 9 ⊢ (0g‘𝑀) = (0g‘𝑀)
38	37	gsum0 18879	. . . . . . . 8 ⊢ (𝑀 Σg ∅) = (0g‘𝑀)
39	38	eqcomi 2769	. . . . . . 7 ⊢ (0g‘𝑀) = (𝑀 Σg ∅)
40	39	a1i 11	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (0g‘𝑀) = (𝑀 Σg ∅))
41		oveq2 7429	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑀 Σg 𝑤) = (𝑀 Σg ∅))
42	41	rspceeqv 3598	. . . . . 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 6899	. . . . . 6 ⊢ (0g‘𝑀) ∈ V
45	15	elrnmpt 5939	. . . . . 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 14723	. . . . . . . 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 14671	. . . . . . . . . . . 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 3909	. . . . . . . . . 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 3909	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑣 ∈ Word 𝐵)
56		eqid 2760	. . . . . . . . . . 11 ⊢ (+g‘𝑀) = (+g‘𝑀)
57	1, 56	gsumccat 19043	. . . . . . . . . 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 2766	. . . . . . . 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 7429	. . . . . . . . 9 ⊢ (𝑤 = (𝑧 ++ 𝑣) → (𝑀 Σg 𝑤) = (𝑀 Σg (𝑧 ++ 𝑣)))
61	60	rspceeqv 3598	. . . . . . . 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 7454	. . . . . . . 8 ⊢ ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ V
64	15	elrnmpt 5939	. . . . . . . 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 3205	. . . . 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 7429	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑧))
69	68	cbvmptv 5208	. . . . . . . 8 ⊢ (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
70	69	rneqi 5918	. . . . . . 7 ⊢ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
71	70	raleqi 3317	. . . . . 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 7429	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑣))
73	72	cbvmptv 5208	. . . . . . . . . 10 ⊢ (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
74	73	rneqi 5918	. . . . . . . . 9 ⊢ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
75	74	raleqi 3317	. . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
77		oveq2 7429	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑀 Σg 𝑣) → (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑀)(𝑀 Σg 𝑣)))
78	77	eleq1d 2845	. . . . . . . . . 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 7095	. . . . . . . . 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 7456	. . . . . . . . 9 ⊢ (𝑣 ∈ Word 𝐺 → (𝑀 Σg 𝑣) ∈ V)
81	79, 80	mprg 3082	. . . . . . . 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 3108	. . . . . 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 2760	. . . . . . . 8 ⊢ (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
85		oveq1 7428	. . . . . . . . . 10 ⊢ (𝑥 = (𝑀 Σg 𝑧) → (𝑥(+g‘𝑀)(𝑀 Σg 𝑣)) = ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)))
86	85	eleq1d 2845	. . . . . . . . 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 3185	. . . . . . . 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 7095	. . . . . . 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 7456	. . . . . . 7 ⊢ (𝑧 ∈ Word 𝐺 → (𝑀 Σg 𝑧) ∈ V)
90	88, 89	mprg 3082	. . . . . 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 19004	. . . . 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 17800	. . 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 3925	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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  ∅c0 4278   ↦ cmpt 5185  ran crn 5651  ‘cfv 6539  (class class class)co 7421  Word cword 14662   ++ cconcat 14719  ⟨“cs1 14746  Basecbs 17393  +_gcplusg 17434  0_gc0g 17616   Σ_g cgsu 17617  Moorecmre 17758  mrClscmrc 17759  Mndcmnd 18929  SubMndcsubmnd 18983

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-2o 8481  df-er 8721  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-card 10024  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-nn 12340  df-2 12409  df-n0 12611  df-z 12698  df-uz 12970  df-fz 13644  df-fzo 13793  df-seq 14149  df-hash 14479  df-word 14663  df-concat 14720  df-s1 14747  df-sets 17348  df-slot 17366  df-ndx 17378  df-base 17394  df-ress 17415  df-plusg 17447  df-0g 17618  df-gsum 17619  df-mre 17762  df-mrc 17763  df-acs 17765  df-mgm 18822  df-sgrp 18914  df-mnd 18930  df-submnd 18985

This theorem is used by:  psgneldm2  19724  psgnfitr  19737