Theorem gsumwspan 18780

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 18761	. . . . 5 ⊢ (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (ACS‘𝐵))
3	2	acsmred 17624	. . . 4 ⊢ (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
4	3	adantr 480	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
5		simpr 484	. . . . . . . 8 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐺)
6	5	s1cld 14575	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → ⟨“𝑥”⟩ ∈ Word 𝐺)
7		ssel2 3944	. . . . . . . . . 10 ⊢ ((𝐺 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐵)
8	7	adantll 714	. . . . . . . . 9 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐵)
9	1	gsumws1 18772	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → (𝑀 Σg ⟨“𝑥”⟩) = 𝑥)
10	8, 9	syl 17	. . . . . . . 8 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → (𝑀 Σg ⟨“𝑥”⟩) = 𝑥)
11	10	eqcomd 2736	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 = (𝑀 Σg ⟨“𝑥”⟩))
12		oveq2 7398	. . . . . . . 8 ⊢ (𝑤 = ⟨“𝑥”⟩ → (𝑀 Σg 𝑤) = (𝑀 Σg ⟨“𝑥”⟩))
13	12	rspceeqv 3614	. . . . . . 7 ⊢ ((⟨“𝑥”⟩ ∈ Word 𝐺 ∧ 𝑥 = (𝑀 Σg ⟨“𝑥”⟩)) → ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤))
14	6, 11, 13	syl2anc 584	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤))
15		eqid 2730	. . . . . . . 8 ⊢ (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))
16	15	elrnmpt 5925	. . . . . . 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 234	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
19	18	ex 412	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝑥 ∈ 𝐺 → 𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
20	19	ssrdv 3955	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → 𝐺 ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
21		gsumwspan.k	. . . . . . . . . 10 ⊢ 𝐾 = (mrCls‘(SubMnd‘𝑀))
22	21	mrccl 17579	. . . . . . . . 9 ⊢ (((SubMnd‘𝑀) ∈ (Moore‘𝐵) ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ∈ (SubMnd‘𝑀))
23	3, 22	sylan 580	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ∈ (SubMnd‘𝑀))
24	21	mrcssid 17585	. . . . . . . . . . 11 ⊢ (((SubMnd‘𝑀) ∈ (Moore‘𝐵) ∧ 𝐺 ⊆ 𝐵) → 𝐺 ⊆ (𝐾‘𝐺))
25	3, 24	sylan 580	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → 𝐺 ⊆ (𝐾‘𝐺))
26		sswrd 14494	. . . . . . . . . 10 ⊢ (𝐺 ⊆ (𝐾‘𝐺) → Word 𝐺 ⊆ Word (𝐾‘𝐺))
27	25, 26	syl 17	. . . . . . . . 9 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → Word 𝐺 ⊆ Word (𝐾‘𝐺))
28	27	sselda 3949	. . . . . . . 8 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑤 ∈ Word 𝐺) → 𝑤 ∈ Word (𝐾‘𝐺))
29		gsumwsubmcl 18771	. . . . . . . 8 ⊢ (((𝐾‘𝐺) ∈ (SubMnd‘𝑀) ∧ 𝑤 ∈ Word (𝐾‘𝐺)) → (𝑀 Σg 𝑤) ∈ (𝐾‘𝐺))
30	23, 28, 29	syl2an2r 685	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑤 ∈ Word 𝐺) → (𝑀 Σg 𝑤) ∈ (𝐾‘𝐺))
31	30	fmpttd 7090	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)):Word 𝐺⟶(𝐾‘𝐺))
32	31	frnd 6699	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ (𝐾‘𝐺))
33	3, 21	mrcssvd 17591	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (𝐾‘𝐺) ⊆ 𝐵)
34	33	adantr 480	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ⊆ 𝐵)
35	32, 34	sstrd 3960	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ 𝐵)
36		wrd0 14511	. . . . . 6 ⊢ ∅ ∈ Word 𝐺
37		eqid 2730	. . . . . . . . 9 ⊢ (0g‘𝑀) = (0g‘𝑀)
38	37	gsum0 18618	. . . . . . . 8 ⊢ (𝑀 Σg ∅) = (0g‘𝑀)
39	38	eqcomi 2739	. . . . . . 7 ⊢ (0g‘𝑀) = (𝑀 Σg ∅)
40	39	a1i 11	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (0g‘𝑀) = (𝑀 Σg ∅))
41		oveq2 7398	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑀 Σg 𝑤) = (𝑀 Σg ∅))
42	41	rspceeqv 3614	. . . . . 6 ⊢ ((∅ ∈ Word 𝐺 ∧ (0g‘𝑀) = (𝑀 Σg ∅)) → ∃𝑤 ∈ Word 𝐺(0g‘𝑀) = (𝑀 Σg 𝑤))
43	36, 40, 42	sylancr 587	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ∃𝑤 ∈ Word 𝐺(0g‘𝑀) = (𝑀 Σg 𝑤))
44		fvex 6874	. . . . . 6 ⊢ (0g‘𝑀) ∈ V
45	15	elrnmpt 5925	. . . . . 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 234	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (0g‘𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
48		ccatcl 14546	. . . . . . . 8 ⊢ ((𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺) → (𝑧 ++ 𝑣) ∈ Word 𝐺)
49		simpll 766	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑀 ∈ Mnd)
50		sswrd 14494	. . . . . . . . . . . 12 ⊢ (𝐺 ⊆ 𝐵 → Word 𝐺 ⊆ Word 𝐵)
51	50	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → Word 𝐺 ⊆ Word 𝐵)
52		simprl 770	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑧 ∈ Word 𝐺)
53	51, 52	sseldd 3950	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑧 ∈ Word 𝐵)
54		simprr 772	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑣 ∈ Word 𝐺)
55	51, 54	sseldd 3950	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑣 ∈ Word 𝐵)
56		eqid 2730	. . . . . . . . . . 11 ⊢ (+g‘𝑀) = (+g‘𝑀)
57	1, 56	gsumccat 18775	. . . . . . . . . 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 1373	. . . . . . . . 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 2736	. . . . . . . 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 7398	. . . . . . . . 9 ⊢ (𝑤 = (𝑧 ++ 𝑣) → (𝑀 Σg 𝑤) = (𝑀 Σg (𝑧 ++ 𝑣)))
61	60	rspceeqv 3614	. . . . . . . 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 686	. . . . . . 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 7423	. . . . . . . 8 ⊢ ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ V
64	15	elrnmpt 5925	. . . . . . . 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 234	. . . . . 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 3181	. . . . 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 7398	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑧))
69	68	cbvmptv 5214	. . . . . . . 8 ⊢ (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
70	69	rneqi 5904	. . . . . . 7 ⊢ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
71	70	raleqi 3299	. . . . . 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 7398	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑣))
73	72	cbvmptv 5214	. . . . . . . . . 10 ⊢ (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
74	73	rneqi 5904	. . . . . . . . 9 ⊢ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
75	74	raleqi 3299	. . . . . . . 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 2730	. . . . . . . . . 10 ⊢ (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
77		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑀 Σg 𝑣) → (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑀)(𝑀 Σg 𝑣)))
78	77	eleq1d 2814	. . . . . . . . . 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 7069	. . . . . . . . 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 7425	. . . . . . . . 9 ⊢ (𝑣 ∈ Word 𝐺 → (𝑀 Σg 𝑣) ∈ V)
81	79, 80	mprg 3051	. . . . . . . 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 275	. . . . . . 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 3076	. . . . . 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 2730	. . . . . . . 8 ⊢ (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
85		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑥 = (𝑀 Σg 𝑧) → (𝑥(+g‘𝑀)(𝑀 Σg 𝑣)) = ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)))
86	85	eleq1d 2814	. . . . . . . . 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 3157	. . . . . . . 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 7069	. . . . . . 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 7425	. . . . . . 7 ⊢ (𝑧 ∈ Word 𝐺 → (𝑀 Σg 𝑧) ∈ V)
90	88, 89	mprg 3051	. . . . . 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 297	. . . . 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 234	. . . 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 18737	. . . . 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 480	. . . 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 1343	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∈ (SubMnd‘𝑀))
96	21	mrcsscl 17588	. . 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 1373	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
98	97, 32	eqssd 3967	1 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) = ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ⊆ wss 3917  ∅c0 4299   ↦ cmpt 5191  ran crn 5642  ‘cfv 6514  (class class class)co 7390  Word cword 14485   ++ cconcat 14542  ⟨“cs1 14567  Basecbs 17186  +_gcplusg 17227  0_gc0g 17409   Σ_g cgsu 17410  Moorecmre 17550  mrClscmrc 17551  Mndcmnd 18668  SubMndcsubmnd 18716

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-op 4599  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-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  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-n0 12450  df-z 12537  df-uz 12801  df-fz 13476  df-fzo 13623  df-seq 13974  df-hash 14303  df-word 14486  df-concat 14543  df-s1 14568  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-0g 17411  df-gsum 17412  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-submnd 18718

This theorem is referenced by:  psgneldm2  19441  psgnfitr  19454