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Theorem gsumwspan 18872
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 18853 . . . . 5 (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (ACS‘𝐵))
32acsmred 17701 . . . 4 (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
43adantr 480 . . 3 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
5 simpr 484 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → 𝑥𝐺)
65s1cld 14638 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → ⟨“𝑥”⟩ ∈ Word 𝐺)
7 ssel2 3990 . . . . . . . . . 10 ((𝐺𝐵𝑥𝐺) → 𝑥𝐵)
87adantll 714 . . . . . . . . 9 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → 𝑥𝐵)
91gsumws1 18864 . . . . . . . . 9 (𝑥𝐵 → (𝑀 Σg ⟨“𝑥”⟩) = 𝑥)
108, 9syl 17 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → (𝑀 Σg ⟨“𝑥”⟩) = 𝑥)
1110eqcomd 2741 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → 𝑥 = (𝑀 Σg ⟨“𝑥”⟩))
12 oveq2 7439 . . . . . . . 8 (𝑤 = ⟨“𝑥”⟩ → (𝑀 Σg 𝑤) = (𝑀 Σg ⟨“𝑥”⟩))
1312rspceeqv 3645 . . . . . . 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 2735 . . . . . . . 8 (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))
1615elrnmpt 5972 . . . . . . 7 (𝑥 ∈ V → (𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤)))
1716elv 3483 . . . . . 6 (𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤))
1814, 17sylibr 234 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → 𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
1918ex 412 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝑥𝐺𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
2019ssrdv 4001 . . 3 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → 𝐺 ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
21 gsumwspan.k . . . . . . . . . 10 𝐾 = (mrCls‘(SubMnd‘𝑀))
2221mrccl 17656 . . . . . . . . 9 (((SubMnd‘𝑀) ∈ (Moore‘𝐵) ∧ 𝐺𝐵) → (𝐾𝐺) ∈ (SubMnd‘𝑀))
233, 22sylan 580 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝐾𝐺) ∈ (SubMnd‘𝑀))
2421mrcssid 17662 . . . . . . . . . . 11 (((SubMnd‘𝑀) ∈ (Moore‘𝐵) ∧ 𝐺𝐵) → 𝐺 ⊆ (𝐾𝐺))
253, 24sylan 580 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → 𝐺 ⊆ (𝐾𝐺))
26 sswrd 14557 . . . . . . . . . 10 (𝐺 ⊆ (𝐾𝐺) → Word 𝐺 ⊆ Word (𝐾𝐺))
2725, 26syl 17 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → Word 𝐺 ⊆ Word (𝐾𝐺))
2827sselda 3995 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑤 ∈ Word 𝐺) → 𝑤 ∈ Word (𝐾𝐺))
29 gsumwsubmcl 18863 . . . . . . . 8 (((𝐾𝐺) ∈ (SubMnd‘𝑀) ∧ 𝑤 ∈ Word (𝐾𝐺)) → (𝑀 Σg 𝑤) ∈ (𝐾𝐺))
3023, 28, 29syl2an2r 685 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑤 ∈ Word 𝐺) → (𝑀 Σg 𝑤) ∈ (𝐾𝐺))
3130fmpttd 7135 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)):Word 𝐺⟶(𝐾𝐺))
3231frnd 6745 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ (𝐾𝐺))
333, 21mrcssvd 17668 . . . . . 6 (𝑀 ∈ Mnd → (𝐾𝐺) ⊆ 𝐵)
3433adantr 480 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝐾𝐺) ⊆ 𝐵)
3532, 34sstrd 4006 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ 𝐵)
36 wrd0 14574 . . . . . 6 ∅ ∈ Word 𝐺
37 eqid 2735 . . . . . . . . 9 (0g𝑀) = (0g𝑀)
3837gsum0 18710 . . . . . . . 8 (𝑀 Σg ∅) = (0g𝑀)
3938eqcomi 2744 . . . . . . 7 (0g𝑀) = (𝑀 Σg ∅)
4039a1i 11 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (0g𝑀) = (𝑀 Σg ∅))
41 oveq2 7439 . . . . . . 7 (𝑤 = ∅ → (𝑀 Σg 𝑤) = (𝑀 Σg ∅))
4241rspceeqv 3645 . . . . . 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 6920 . . . . . 6 (0g𝑀) ∈ V
4515elrnmpt 5972 . . . . . 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 234 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (0g𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
48 ccatcl 14609 . . . . . . . 8 ((𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺) → (𝑧 ++ 𝑣) ∈ Word 𝐺)
49 simpll 767 . . . . . . . . . 10 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑀 ∈ Mnd)
50 sswrd 14557 . . . . . . . . . . . 12 (𝐺𝐵 → Word 𝐺 ⊆ Word 𝐵)
5150ad2antlr 727 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → Word 𝐺 ⊆ Word 𝐵)
52 simprl 771 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑧 ∈ Word 𝐺)
5351, 52sseldd 3996 . . . . . . . . . 10 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑧 ∈ Word 𝐵)
54 simprr 773 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑣 ∈ Word 𝐺)
5551, 54sseldd 3996 . . . . . . . . . 10 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑣 ∈ Word 𝐵)
56 eqid 2735 . . . . . . . . . . 11 (+g𝑀) = (+g𝑀)
571, 56gsumccat 18867 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ 𝑧 ∈ Word 𝐵𝑣 ∈ Word 𝐵) → (𝑀 Σg (𝑧 ++ 𝑣)) = ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)))
5849, 53, 55, 57syl3anc 1370 . . . . . . . . 9 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → (𝑀 Σg (𝑧 ++ 𝑣)) = ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)))
5958eqcomd 2741 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg (𝑧 ++ 𝑣)))
60 oveq2 7439 . . . . . . . . 9 (𝑤 = (𝑧 ++ 𝑣) → (𝑀 Σg 𝑤) = (𝑀 Σg (𝑧 ++ 𝑣)))
6160rspceeqv 3645 . . . . . . . 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 686 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → ∃𝑤 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg 𝑤))
63 ovex 7464 . . . . . . . 8 ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ V
6415elrnmpt 5972 . . . . . . . 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 234 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
6766ralrimivva 3200 . . . . 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 7439 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑧))
6968cbvmptv 5261 . . . . . . . 8 (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
7069rneqi 5951 . . . . . . 7 ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
7170raleqi 3322 . . . . . 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 7439 . . . . . . . . . . 11 (𝑤 = 𝑣 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑣))
7372cbvmptv 5261 . . . . . . . . . 10 (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
7473rneqi 5951 . . . . . . . . 9 ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
7574raleqi 3322 . . . . . . . 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 2735 . . . . . . . . . 10 (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
77 oveq2 7439 . . . . . . . . . . 11 (𝑦 = (𝑀 Σg 𝑣) → (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑀)(𝑀 Σg 𝑣)))
7877eleq1d 2824 . . . . . . . . . 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 7114 . . . . . . . . 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 7466 . . . . . . . . 9 (𝑣 ∈ Word 𝐺 → (𝑀 Σg 𝑣) ∈ V)
8179, 80mprg 3065 . . . . . . . 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 275 . . . . . . 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 3091 . . . . . 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 2735 . . . . . . . 8 (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
85 oveq1 7438 . . . . . . . . . 10 (𝑥 = (𝑀 Σg 𝑧) → (𝑥(+g𝑀)(𝑀 Σg 𝑣)) = ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)))
8685eleq1d 2824 . . . . . . . . 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 3176 . . . . . . . 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 7114 . . . . . . 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 7466 . . . . . . 7 (𝑧 ∈ Word 𝐺 → (𝑀 Σg 𝑧) ∈ V)
9088, 89mprg 3065 . . . . . 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 297 . . . . 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 234 . . . 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 18829 . . . . 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 480 . . . 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 1341 . . 3 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∈ (SubMnd‘𝑀))
9621mrcsscl 17665 . . 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 1370 . 2 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝐾𝐺) ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
9897, 32eqssd 4013 1 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝐾𝐺) = ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963  c0 4339  cmpt 5231  ran crn 5690  cfv 6563  (class class class)co 7431  Word cword 14549   ++ cconcat 14605  ⟨“cs1 14630  Basecbs 17245  +gcplusg 17298  0gc0g 17486   Σg cgsu 17487  Moorecmre 17627  mrClscmrc 17628  Mndcmnd 18760  SubMndcsubmnd 18808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-gsum 17489  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810
This theorem is referenced by:  psgneldm2  19537  psgnfitr  19550
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