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Theorem gsumwspan 17996
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 ∧ 𝐺𝐵) → (𝐾𝐺) = 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.
StepHypRef Expression
1 gsumwspan.b . . . . . 6 𝐵 = (Base‘𝑀)
21submacs 17976 . . . . 5 (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (ACS‘𝐵))
32acsmred 16919 . . . 4 (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
43adantr 481 . . 3 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
5 simpr 485 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → 𝑥𝐺)
65s1cld 13950 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → ⟨“𝑥”⟩ ∈ Word 𝐺)
7 ssel2 3965 . . . . . . . . . 10 ((𝐺𝐵𝑥𝐺) → 𝑥𝐵)
87adantll 710 . . . . . . . . 9 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → 𝑥𝐵)
91gsumws1 17987 . . . . . . . . 9 (𝑥𝐵 → (𝑀 Σg ⟨“𝑥”⟩) = 𝑥)
108, 9syl 17 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → (𝑀 Σg ⟨“𝑥”⟩) = 𝑥)
1110eqcomd 2831 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → 𝑥 = (𝑀 Σg ⟨“𝑥”⟩))
12 oveq2 7159 . . . . . . . 8 (𝑤 = ⟨“𝑥”⟩ → (𝑀 Σg 𝑤) = (𝑀 Σg ⟨“𝑥”⟩))
1312rspceeqv 3641 . . . . . . 7 ((⟨“𝑥”⟩ ∈ Word 𝐺𝑥 = (𝑀 Σg ⟨“𝑥”⟩)) → ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤))
146, 11, 13syl2anc 584 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤))
15 eqid 2825 . . . . . . . 8 (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))
1615elrnmpt 5826 . . . . . . 7 (𝑥 ∈ V → (𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤)))
1716elv 3504 . . . . . 6 (𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤))
1814, 17sylibr 235 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → 𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
1918ex 413 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝑥𝐺𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
2019ssrdv 3976 . . 3 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → 𝐺 ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
21 gsumwspan.k . . . . . . . . . 10 𝐾 = (mrCls‘(SubMnd‘𝑀))
2221mrccl 16874 . . . . . . . . 9 (((SubMnd‘𝑀) ∈ (Moore‘𝐵) ∧ 𝐺𝐵) → (𝐾𝐺) ∈ (SubMnd‘𝑀))
233, 22sylan 580 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝐾𝐺) ∈ (SubMnd‘𝑀))
2421mrcssid 16880 . . . . . . . . . . 11 (((SubMnd‘𝑀) ∈ (Moore‘𝐵) ∧ 𝐺𝐵) → 𝐺 ⊆ (𝐾𝐺))
253, 24sylan 580 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → 𝐺 ⊆ (𝐾𝐺))
26 sswrd 13862 . . . . . . . . . 10 (𝐺 ⊆ (𝐾𝐺) → Word 𝐺 ⊆ Word (𝐾𝐺))
2725, 26syl 17 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → Word 𝐺 ⊆ Word (𝐾𝐺))
2827sselda 3970 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑤 ∈ Word 𝐺) → 𝑤 ∈ Word (𝐾𝐺))
29 gsumwsubmcl 17986 . . . . . . . 8 (((𝐾𝐺) ∈ (SubMnd‘𝑀) ∧ 𝑤 ∈ Word (𝐾𝐺)) → (𝑀 Σg 𝑤) ∈ (𝐾𝐺))
3023, 28, 29syl2an2r 681 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑤 ∈ Word 𝐺) → (𝑀 Σg 𝑤) ∈ (𝐾𝐺))
3130fmpttd 6874 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)):Word 𝐺⟶(𝐾𝐺))
3231frnd 6517 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ (𝐾𝐺))
333, 21mrcssvd 16886 . . . . . 6 (𝑀 ∈ Mnd → (𝐾𝐺) ⊆ 𝐵)
3433adantr 481 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝐾𝐺) ⊆ 𝐵)
3532, 34sstrd 3980 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ 𝐵)
36 wrd0 13882 . . . . . 6 ∅ ∈ Word 𝐺
37 eqid 2825 . . . . . . . . 9 (0g𝑀) = (0g𝑀)
3837gsum0 17885 . . . . . . . 8 (𝑀 Σg ∅) = (0g𝑀)
3938eqcomi 2834 . . . . . . 7 (0g𝑀) = (𝑀 Σg ∅)
4039a1i 11 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (0g𝑀) = (𝑀 Σg ∅))
41 oveq2 7159 . . . . . . 7 (𝑤 = ∅ → (𝑀 Σg 𝑤) = (𝑀 Σg ∅))
4241rspceeqv 3641 . . . . . 6 ((∅ ∈ Word 𝐺 ∧ (0g𝑀) = (𝑀 Σg ∅)) → ∃𝑤 ∈ Word 𝐺(0g𝑀) = (𝑀 Σg 𝑤))
4336, 40, 42sylancr 587 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ∃𝑤 ∈ Word 𝐺(0g𝑀) = (𝑀 Σg 𝑤))
44 fvex 6679 . . . . . 6 (0g𝑀) ∈ V
4515elrnmpt 5826 . . . . . 6 ((0g𝑀) ∈ V → ((0g𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺(0g𝑀) = (𝑀 Σg 𝑤)))
4644, 45ax-mp 5 . . . . 5 ((0g𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺(0g𝑀) = (𝑀 Σg 𝑤))
4743, 46sylibr 235 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (0g𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
48 ccatcl 13919 . . . . . . . 8 ((𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺) → (𝑧 ++ 𝑣) ∈ Word 𝐺)
49 simpll 763 . . . . . . . . . 10 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑀 ∈ Mnd)
50 sswrd 13862 . . . . . . . . . . . 12 (𝐺𝐵 → Word 𝐺 ⊆ Word 𝐵)
5150ad2antlr 723 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → Word 𝐺 ⊆ Word 𝐵)
52 simprl 767 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑧 ∈ Word 𝐺)
5351, 52sseldd 3971 . . . . . . . . . 10 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑧 ∈ Word 𝐵)
54 simprr 769 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑣 ∈ Word 𝐺)
5551, 54sseldd 3971 . . . . . . . . . 10 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑣 ∈ Word 𝐵)
56 eqid 2825 . . . . . . . . . . 11 (+g𝑀) = (+g𝑀)
571, 56gsumccat 17991 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ 𝑧 ∈ Word 𝐵𝑣 ∈ Word 𝐵) → (𝑀 Σg (𝑧 ++ 𝑣)) = ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)))
5849, 53, 55, 57syl3anc 1365 . . . . . . . . 9 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → (𝑀 Σg (𝑧 ++ 𝑣)) = ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)))
5958eqcomd 2831 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg (𝑧 ++ 𝑣)))
60 oveq2 7159 . . . . . . . . 9 (𝑤 = (𝑧 ++ 𝑣) → (𝑀 Σg 𝑤) = (𝑀 Σg (𝑧 ++ 𝑣)))
6160rspceeqv 3641 . . . . . . . 8 (((𝑧 ++ 𝑣) ∈ Word 𝐺 ∧ ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg (𝑧 ++ 𝑣))) → ∃𝑤 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg 𝑤))
6248, 59, 61syl2an2 682 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → ∃𝑤 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg 𝑤))
63 ovex 7184 . . . . . . . 8 ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ V
6415elrnmpt 5826 . . . . . . . 8 (((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ V → (((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg 𝑤)))
6563, 64ax-mp 5 . . . . . . 7 (((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg 𝑤))
6662, 65sylibr 235 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
6766ralrimivva 3195 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ∀𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
68 oveq2 7159 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑧))
6968cbvmptv 5165 . . . . . . . 8 (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
7069rneqi 5805 . . . . . . 7 ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
7170raleqi 3418 . . . . . 6 (∀𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g𝑀)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(+g𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
72 oveq2 7159 . . . . . . . . . . 11 (𝑤 = 𝑣 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑣))
7372cbvmptv 5165 . . . . . . . . . 10 (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
7473rneqi 5805 . . . . . . . . 9 ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
7574raleqi 3418 . . . . . . . 8 (∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑦 ∈ ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))(𝑥(+g𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
76 eqid 2825 . . . . . . . . . 10 (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
77 oveq2 7159 . . . . . . . . . . 11 (𝑦 = (𝑀 Σg 𝑣) → (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑀)(𝑀 Σg 𝑣)))
7877eleq1d 2901 . . . . . . . . . 10 (𝑦 = (𝑀 Σg 𝑣) → ((𝑥(+g𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ (𝑥(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
7976, 78ralrnmptw 6855 . . . . . . . . 9 (∀𝑣 ∈ Word 𝐺(𝑀 Σg 𝑣) ∈ V → (∀𝑦 ∈ ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))(𝑥(+g𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑣 ∈ Word 𝐺(𝑥(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
80 ovexd 7186 . . . . . . . . 9 (𝑣 ∈ Word 𝐺 → (𝑀 Σg 𝑣) ∈ V)
8179, 80mprg 3156 . . . . . . . 8 (∀𝑦 ∈ ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))(𝑥(+g𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑣 ∈ Word 𝐺(𝑥(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
8275, 81bitri 276 . . . . . . 7 (∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑣 ∈ Word 𝐺(𝑥(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
8382ralbii 3169 . . . . . 6 (∀𝑥 ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑥 ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))∀𝑣 ∈ Word 𝐺(𝑥(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
84 eqid 2825 . . . . . . . 8 (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
85 oveq1 7158 . . . . . . . . . 10 (𝑥 = (𝑀 Σg 𝑧) → (𝑥(+g𝑀)(𝑀 Σg 𝑣)) = ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)))
8685eleq1d 2901 . . . . . . . . 9 (𝑥 = (𝑀 Σg 𝑧) → ((𝑥(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
8786ralbidv 3201 . . . . . . . 8 (𝑥 = (𝑀 Σg 𝑧) → (∀𝑣 ∈ Word 𝐺(𝑥(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑣 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
8884, 87ralrnmptw 6855 . . . . . . 7 (∀𝑧 ∈ Word 𝐺(𝑀 Σg 𝑧) ∈ V → (∀𝑥 ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))∀𝑣 ∈ Word 𝐺(𝑥(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
89 ovexd 7186 . . . . . . 7 (𝑧 ∈ Word 𝐺 → (𝑀 Σg 𝑧) ∈ V)
9088, 89mprg 3156 . . . . . 6 (∀𝑥 ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))∀𝑣 ∈ Word 𝐺(𝑥(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
9171, 83, 903bitri 298 . . . . 5 (∀𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∀𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
9267, 91sylibr 235 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ∀𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
931, 37, 56issubm 17958 . . . . 5 (𝑀 ∈ Mnd → (ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∈ (SubMnd‘𝑀) ↔ (ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ 𝐵 ∧ (0g𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∧ ∀𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))))
9493adantr 481 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∈ (SubMnd‘𝑀) ↔ (ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ 𝐵 ∧ (0g𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∧ ∀𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))∀𝑦 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))(𝑥(+g𝑀)𝑦) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))))
9535, 47, 92, 94mpbir3and 1336 . . 3 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∈ (SubMnd‘𝑀))
9621mrcsscl 16883 . . 3 (((SubMnd‘𝑀) ∈ (Moore‘𝐵) ∧ 𝐺 ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∧ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∈ (SubMnd‘𝑀)) → (𝐾𝐺) ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
974, 20, 95, 96syl3anc 1365 . 2 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝐾𝐺) ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
9897, 32eqssd 3987 1 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝐾𝐺) = ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3142  wrex 3143  Vcvv 3499  wss 3939  c0 4294  cmpt 5142  ran crn 5554  cfv 6351  (class class class)co 7151  Word cword 13854   ++ cconcat 13915  ⟨“cs1 13942  Basecbs 16475  +gcplusg 16557  0gc0g 16705   Σg cgsu 16706  Moorecmre 16845  mrClscmrc 16846  Mndcmnd 17902  SubMndcsubmnd 17945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12886  df-fzo 13027  df-seq 13363  df-hash 13684  df-word 13855  df-concat 13916  df-s1 13943  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-gsum 16708  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-submnd 17947
This theorem is referenced by:  psgneldm2  18554  psgnfitr  18567
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