Theorem gsumwspan 18485

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 18465	. . . . 5 ⊢ (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (ACS‘𝐵))
3	2	acsmred 17365	. . . 4 ⊢ (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
4	3	adantr 481	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
5		simpr 485	. . . . . . . 8 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐺)
6	5	s1cld 14308	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → ⟨“𝑥”⟩ ∈ Word 𝐺)
7		ssel2 3916	. . . . . . . . . 10 ⊢ ((𝐺 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐵)
8	7	adantll 711	. . . . . . . . 9 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐵)
9	1	gsumws1 18476	. . . . . . . . 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 2744	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 = (𝑀 Σg ⟨“𝑥”⟩))
12		oveq2 7283	. . . . . . . 8 ⊢ (𝑤 = ⟨“𝑥”⟩ → (𝑀 Σg 𝑤) = (𝑀 Σg ⟨“𝑥”⟩))
13	12	rspceeqv 3575	. . . . . . 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 2738	. . . . . . . 8 ⊢ (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))
16	15	elrnmpt 5865	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤)))
17	16	elv 3438	. . . . . 6 ⊢ (𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤))
18	14, 17	sylibr 233	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
19	18	ex 413	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝑥 ∈ 𝐺 → 𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
20	19	ssrdv 3927	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → 𝐺 ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
21		gsumwspan.k	. . . . . . . . . 10 ⊢ 𝐾 = (mrCls‘(SubMnd‘𝑀))
22	21	mrccl 17320	. . . . . . . . 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 17326	. . . . . . . . . . 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 14225	. . . . . . . . . 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 3921	. . . . . . . 8 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑤 ∈ Word 𝐺) → 𝑤 ∈ Word (𝐾‘𝐺))
29		gsumwsubmcl 18475	. . . . . . . 8 ⊢ (((𝐾‘𝐺) ∈ (SubMnd‘𝑀) ∧ 𝑤 ∈ Word (𝐾‘𝐺)) → (𝑀 Σg 𝑤) ∈ (𝐾‘𝐺))
30	23, 28, 29	syl2an2r 682	. . . . . . 7 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ 𝑤 ∈ Word 𝐺) → (𝑀 Σg 𝑤) ∈ (𝐾‘𝐺))
31	30	fmpttd 6989	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)):Word 𝐺⟶(𝐾‘𝐺))
32	31	frnd 6608	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ (𝐾‘𝐺))
33	3, 21	mrcssvd 17332	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (𝐾‘𝐺) ⊆ 𝐵)
34	33	adantr 481	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ⊆ 𝐵)
35	32, 34	sstrd 3931	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ 𝐵)
36		wrd0 14242	. . . . . 6 ⊢ ∅ ∈ Word 𝐺
37		eqid 2738	. . . . . . . . 9 ⊢ (0g‘𝑀) = (0g‘𝑀)
38	37	gsum0 18368	. . . . . . . 8 ⊢ (𝑀 Σg ∅) = (0g‘𝑀)
39	38	eqcomi 2747	. . . . . . 7 ⊢ (0g‘𝑀) = (𝑀 Σg ∅)
40	39	a1i 11	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (0g‘𝑀) = (𝑀 Σg ∅))
41		oveq2 7283	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑀 Σg 𝑤) = (𝑀 Σg ∅))
42	41	rspceeqv 3575	. . . . . 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 6787	. . . . . 6 ⊢ (0g‘𝑀) ∈ V
45	15	elrnmpt 5865	. . . . . 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 233	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (0g‘𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
48		ccatcl 14277	. . . . . . . 8 ⊢ ((𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺) → (𝑧 ++ 𝑣) ∈ Word 𝐺)
49		simpll 764	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑀 ∈ Mnd)
50		sswrd 14225	. . . . . . . . . . . 12 ⊢ (𝐺 ⊆ 𝐵 → Word 𝐺 ⊆ Word 𝐵)
51	50	ad2antlr 724	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → Word 𝐺 ⊆ Word 𝐵)
52		simprl 768	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑧 ∈ Word 𝐺)
53	51, 52	sseldd 3922	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑧 ∈ Word 𝐵)
54		simprr 770	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑣 ∈ Word 𝐺)
55	51, 54	sseldd 3922	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) → 𝑣 ∈ Word 𝐵)
56		eqid 2738	. . . . . . . . . . 11 ⊢ (+g‘𝑀) = (+g‘𝑀)
57	1, 56	gsumccat 18480	. . . . . . . . . 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 1370	. . . . . . . . 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 2744	. . . . . . . 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 7283	. . . . . . . . 9 ⊢ (𝑤 = (𝑧 ++ 𝑣) → (𝑀 Σg 𝑤) = (𝑀 Σg (𝑧 ++ 𝑣)))
61	60	rspceeqv 3575	. . . . . . . 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 683	. . . . . . 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 7308	. . . . . . . 8 ⊢ ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)) ∈ V
64	15	elrnmpt 5865	. . . . . . . 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 233	. . . . . 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 3123	. . . . 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 7283	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑧))
69	68	cbvmptv 5187	. . . . . . . 8 ⊢ (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
70	69	rneqi 5846	. . . . . . 7 ⊢ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
71	70	raleqi 3346	. . . . . 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 7283	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑣))
73	72	cbvmptv 5187	. . . . . . . . . 10 ⊢ (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
74	73	rneqi 5846	. . . . . . . . 9 ⊢ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
75	74	raleqi 3346	. . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
77		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑀 Σg 𝑣) → (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑀)(𝑀 Σg 𝑣)))
78	77	eleq1d 2823	. . . . . . . . . 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 6970	. . . . . . . . 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 7310	. . . . . . . . 9 ⊢ (𝑣 ∈ Word 𝐺 → (𝑀 Σg 𝑣) ∈ V)
81	79, 80	mprg 3078	. . . . . . . 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 274	. . . . . . 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 3092	. . . . . 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 2738	. . . . . . . 8 ⊢ (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
85		oveq1 7282	. . . . . . . . . 10 ⊢ (𝑥 = (𝑀 Σg 𝑧) → (𝑥(+g‘𝑀)(𝑀 Σg 𝑣)) = ((𝑀 Σg 𝑧)(+g‘𝑀)(𝑀 Σg 𝑣)))
86	85	eleq1d 2823	. . . . . . . . 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 3112	. . . . . . . 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 6970	. . . . . . 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 7310	. . . . . . 7 ⊢ (𝑧 ∈ Word 𝐺 → (𝑀 Σg 𝑧) ∈ V)
90	88, 89	mprg 3078	. . . . . 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 233	. . . 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 18442	. . . . 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 481	. . . 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 1341	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∈ (SubMnd‘𝑀))
96	21	mrcsscl 17329	. . 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 1370	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
98	97, 32	eqssd 3938	1 ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) = ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  ∅c0 4256   ↦ cmpt 5157  ran crn 5590  ‘cfv 6433  (class class class)co 7275  Word cword 14217   ++ cconcat 14273  ⟨“cs1 14300  Basecbs 16912  +_gcplusg 16962  0_gc0g 17150   Σ_g cgsu 17151  Moorecmre 17291  mrClscmrc 17292  Mndcmnd 18385  SubMndcsubmnd 18429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722  df-hash 14045  df-word 14218  df-concat 14274  df-s1 14301  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-0g 17152  df-gsum 17153  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431

This theorem is referenced by:  psgneldm2  19112  psgnfitr  19125