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Theorem gsumwspan 17430
 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 17412 . . . . 5 (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (ACS‘𝐵))
32acsmred 16364 . . . 4 (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
43adantr 480 . . 3 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
5 simpr 476 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → 𝑥𝐺)
65s1cld 13419 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → ⟨“𝑥”⟩ ∈ Word 𝐺)
7 ssel2 3631 . . . . . . . . . 10 ((𝐺𝐵𝑥𝐺) → 𝑥𝐵)
87adantll 750 . . . . . . . . 9 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → 𝑥𝐵)
91gsumws1 17423 . . . . . . . . 9 (𝑥𝐵 → (𝑀 Σg ⟨“𝑥”⟩) = 𝑥)
108, 9syl 17 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → (𝑀 Σg ⟨“𝑥”⟩) = 𝑥)
1110eqcomd 2657 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → 𝑥 = (𝑀 Σg ⟨“𝑥”⟩))
12 oveq2 6698 . . . . . . . . 9 (𝑤 = ⟨“𝑥”⟩ → (𝑀 Σg 𝑤) = (𝑀 Σg ⟨“𝑥”⟩))
1312eqeq2d 2661 . . . . . . . 8 (𝑤 = ⟨“𝑥”⟩ → (𝑥 = (𝑀 Σg 𝑤) ↔ 𝑥 = (𝑀 Σg ⟨“𝑥”⟩)))
1413rspcev 3340 . . . . . . 7 ((⟨“𝑥”⟩ ∈ Word 𝐺𝑥 = (𝑀 Σg ⟨“𝑥”⟩)) → ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤))
156, 11, 14syl2anc 694 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤))
16 vex 3234 . . . . . . 7 𝑥 ∈ V
17 eqid 2651 . . . . . . . 8 (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))
1817elrnmpt 5404 . . . . . . 7 (𝑥 ∈ V → (𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤)))
1916, 18ax-mp 5 . . . . . 6 (𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺𝑥 = (𝑀 Σg 𝑤))
2015, 19sylibr 224 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑥𝐺) → 𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
2120ex 449 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝑥𝐺𝑥 ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))))
2221ssrdv 3642 . . 3 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → 𝐺 ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
23 gsumwspan.k . . . . . . . . . . 11 𝐾 = (mrCls‘(SubMnd‘𝑀))
2423mrccl 16318 . . . . . . . . . 10 (((SubMnd‘𝑀) ∈ (Moore‘𝐵) ∧ 𝐺𝐵) → (𝐾𝐺) ∈ (SubMnd‘𝑀))
253, 24sylan 487 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝐾𝐺) ∈ (SubMnd‘𝑀))
2625adantr 480 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑤 ∈ Word 𝐺) → (𝐾𝐺) ∈ (SubMnd‘𝑀))
2723mrcssid 16324 . . . . . . . . . . 11 (((SubMnd‘𝑀) ∈ (Moore‘𝐵) ∧ 𝐺𝐵) → 𝐺 ⊆ (𝐾𝐺))
283, 27sylan 487 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → 𝐺 ⊆ (𝐾𝐺))
29 sswrd 13345 . . . . . . . . . 10 (𝐺 ⊆ (𝐾𝐺) → Word 𝐺 ⊆ Word (𝐾𝐺))
3028, 29syl 17 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → Word 𝐺 ⊆ Word (𝐾𝐺))
3130sselda 3636 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑤 ∈ Word 𝐺) → 𝑤 ∈ Word (𝐾𝐺))
32 gsumwsubmcl 17422 . . . . . . . 8 (((𝐾𝐺) ∈ (SubMnd‘𝑀) ∧ 𝑤 ∈ Word (𝐾𝐺)) → (𝑀 Σg 𝑤) ∈ (𝐾𝐺))
3326, 31, 32syl2anc 694 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ 𝑤 ∈ Word 𝐺) → (𝑀 Σg 𝑤) ∈ (𝐾𝐺))
3433, 17fmptd 6425 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)):Word 𝐺⟶(𝐾𝐺))
35 frn 6091 . . . . . 6 ((𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)):Word 𝐺⟶(𝐾𝐺) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ (𝐾𝐺))
3634, 35syl 17 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ (𝐾𝐺))
373, 23mrcssvd 16330 . . . . . 6 (𝑀 ∈ Mnd → (𝐾𝐺) ⊆ 𝐵)
3837adantr 480 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝐾𝐺) ⊆ 𝐵)
3936, 38sstrd 3646 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ⊆ 𝐵)
40 wrd0 13362 . . . . . 6 ∅ ∈ Word 𝐺
41 eqid 2651 . . . . . . . . 9 (0g𝑀) = (0g𝑀)
4241gsum0 17325 . . . . . . . 8 (𝑀 Σg ∅) = (0g𝑀)
4342eqcomi 2660 . . . . . . 7 (0g𝑀) = (𝑀 Σg ∅)
4443a1i 11 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (0g𝑀) = (𝑀 Σg ∅))
45 oveq2 6698 . . . . . . . 8 (𝑤 = ∅ → (𝑀 Σg 𝑤) = (𝑀 Σg ∅))
4645eqeq2d 2661 . . . . . . 7 (𝑤 = ∅ → ((0g𝑀) = (𝑀 Σg 𝑤) ↔ (0g𝑀) = (𝑀 Σg ∅)))
4746rspcev 3340 . . . . . 6 ((∅ ∈ Word 𝐺 ∧ (0g𝑀) = (𝑀 Σg ∅)) → ∃𝑤 ∈ Word 𝐺(0g𝑀) = (𝑀 Σg 𝑤))
4840, 44, 47sylancr 696 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ∃𝑤 ∈ Word 𝐺(0g𝑀) = (𝑀 Σg 𝑤))
49 fvex 6239 . . . . . 6 (0g𝑀) ∈ V
5017elrnmpt 5404 . . . . . 6 ((0g𝑀) ∈ V → ((0g𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺(0g𝑀) = (𝑀 Σg 𝑤)))
5149, 50ax-mp 5 . . . . 5 ((0g𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝐺(0g𝑀) = (𝑀 Σg 𝑤))
5248, 51sylibr 224 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (0g𝑀) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
53 ccatcl 13392 . . . . . . . . 9 ((𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺) → (𝑧 ++ 𝑣) ∈ Word 𝐺)
5453adantl 481 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → (𝑧 ++ 𝑣) ∈ Word 𝐺)
55 simpll 805 . . . . . . . . . 10 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑀 ∈ Mnd)
56 sswrd 13345 . . . . . . . . . . . 12 (𝐺𝐵 → Word 𝐺 ⊆ Word 𝐵)
5756ad2antlr 763 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → Word 𝐺 ⊆ Word 𝐵)
58 simprl 809 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑧 ∈ Word 𝐺)
5957, 58sseldd 3637 . . . . . . . . . 10 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑧 ∈ Word 𝐵)
60 simprr 811 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑣 ∈ Word 𝐺)
6157, 60sseldd 3637 . . . . . . . . . 10 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → 𝑣 ∈ Word 𝐵)
62 eqid 2651 . . . . . . . . . . 11 (+g𝑀) = (+g𝑀)
631, 62gsumccat 17425 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ 𝑧 ∈ Word 𝐵𝑣 ∈ Word 𝐵) → (𝑀 Σg (𝑧 ++ 𝑣)) = ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)))
6455, 59, 61, 63syl3anc 1366 . . . . . . . . 9 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → (𝑀 Σg (𝑧 ++ 𝑣)) = ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)))
6564eqcomd 2657 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg (𝑧 ++ 𝑣)))
66 oveq2 6698 . . . . . . . . . 10 (𝑤 = (𝑧 ++ 𝑣) → (𝑀 Σg 𝑤) = (𝑀 Σg (𝑧 ++ 𝑣)))
6766eqeq2d 2661 . . . . . . . . 9 (𝑤 = (𝑧 ++ 𝑣) → (((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg 𝑤) ↔ ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg (𝑧 ++ 𝑣))))
6867rspcev 3340 . . . . . . . 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))
6954, 65, 68syl2anc 694 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → ∃𝑤 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) = (𝑀 Σg 𝑤))
70 ovex 6718 . . . . . . . 8 ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ V
7117elrnmpt 5404 . . . . . . . 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)))
7270, 71ax-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))
7369, 72sylibr 224 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝐺𝐵) ∧ (𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺)) → ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
7473ralrimivva 3000 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ∀𝑧 ∈ Word 𝐺𝑣 ∈ Word 𝐺((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)) ∈ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
75 oveq2 6698 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑧))
7675cbvmptv 4783 . . . . . . . 8 (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
7776rneqi 5384 . . . . . . 7 ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
7877raleqi 3172 . . . . . 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)))
79 oveq2 6698 . . . . . . . . . . 11 (𝑤 = 𝑣 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑣))
8079cbvmptv 4783 . . . . . . . . . 10 (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
8180rneqi 5384 . . . . . . . . 9 ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) = ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
8281raleqi 3172 . . . . . . . 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)))
83 eqid 2651 . . . . . . . . . 10 (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑣))
84 oveq2 6698 . . . . . . . . . . 11 (𝑦 = (𝑀 Σg 𝑣) → (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑀)(𝑀 Σg 𝑣)))
8584eleq1d 2715 . . . . . . . . . 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))))
8683, 85ralrnmpt 6408 . . . . . . . . 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))))
87 ovexd 6720 . . . . . . . . 9 (𝑣 ∈ Word 𝐺 → (𝑀 Σg 𝑣) ∈ V)
8886, 87mprg 2955 . . . . . . . 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)))
8982, 88bitri 264 . . . . . . 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)))
9089ralbii 3009 . . . . . 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)))
91 eqid 2651 . . . . . . . 8 (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑧))
92 oveq1 6697 . . . . . . . . . 10 (𝑥 = (𝑀 Σg 𝑧) → (𝑥(+g𝑀)(𝑀 Σg 𝑣)) = ((𝑀 Σg 𝑧)(+g𝑀)(𝑀 Σg 𝑣)))
9392eleq1d 2715 . . . . . . . . 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))))
9493ralbidv 3015 . . . . . . . 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))))
9591, 94ralrnmpt 6408 . . . . . . 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))))
96 ovexd 6720 . . . . . . 7 (𝑧 ∈ Word 𝐺 → (𝑀 Σg 𝑧) ∈ V)
9795, 96mprg 2955 . . . . . 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)))
9878, 90, 973bitri 286 . . . . 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)))
9974, 98sylibr 224 . . . 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)))
1001, 41, 62issubm 17394 . . . . 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)))))
101100adantr 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)))))
10239, 52, 99, 101mpbir3and 1264 . . 3 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)) ∈ (SubMnd‘𝑀))
10323mrcsscl 16327 . . 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)))
1044, 22, 102, 103syl3anc 1366 . 2 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝐾𝐺) ⊆ ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
105104, 36eqssd 3653 1 ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝐾𝐺) = ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ⊆ wss 3607  ∅c0 3948   ↦ cmpt 4762  ran crn 5144  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  Word cword 13323   ++ cconcat 13325  ⟨“cs1 13326  Basecbs 15904  +gcplusg 15988  0gc0g 16147   Σg cgsu 16148  Moorecmre 16289  mrClscmrc 16290  Mndcmnd 17341  SubMndcsubmnd 17381 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-gsum 16150  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383 This theorem is referenced by:  psgneldm2  17970  psgnfitr  17983
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