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Theorem gsumwspan 18760
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 𝐡 = (Baseβ€˜π‘€)
gsumwspan.k 𝐾 = (mrClsβ€˜(SubMndβ€˜π‘€))
Assertion
Ref Expression
gsumwspan ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ (πΎβ€˜πΊ) = ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
Distinct variable groups:   𝑀,𝐺   𝑀,𝐡   𝑀,𝑀   𝑀,𝐾
Proof of Theorem gsumwspan
Dummy variables 𝑣 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumwspan.b . . . . . 6 𝐡 = (Baseβ€˜π‘€)
21submacs 18741 . . . . 5 (𝑀 ∈ Mnd β†’ (SubMndβ€˜π‘€) ∈ (ACSβ€˜π΅))
32acsmred 17598 . . . 4 (𝑀 ∈ Mnd β†’ (SubMndβ€˜π‘€) ∈ (Mooreβ€˜π΅))
43adantr 480 . . 3 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ (SubMndβ€˜π‘€) ∈ (Mooreβ€˜π΅))
5 simpr 484 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ π‘₯ ∈ 𝐺) β†’ π‘₯ ∈ 𝐺)
65s1cld 14549 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ π‘₯ ∈ 𝐺) β†’ βŸ¨β€œπ‘₯β€βŸ© ∈ Word 𝐺)
7 ssel2 3969 . . . . . . . . . 10 ((𝐺 βŠ† 𝐡 ∧ π‘₯ ∈ 𝐺) β†’ π‘₯ ∈ 𝐡)
87adantll 711 . . . . . . . . 9 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ π‘₯ ∈ 𝐺) β†’ π‘₯ ∈ 𝐡)
91gsumws1 18752 . . . . . . . . 9 (π‘₯ ∈ 𝐡 β†’ (𝑀 Ξ£g βŸ¨β€œπ‘₯β€βŸ©) = π‘₯)
108, 9syl 17 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ π‘₯ ∈ 𝐺) β†’ (𝑀 Ξ£g βŸ¨β€œπ‘₯β€βŸ©) = π‘₯)
1110eqcomd 2730 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ π‘₯ ∈ 𝐺) β†’ π‘₯ = (𝑀 Ξ£g βŸ¨β€œπ‘₯β€βŸ©))
12 oveq2 7409 . . . . . . . 8 (𝑀 = βŸ¨β€œπ‘₯β€βŸ© β†’ (𝑀 Ξ£g 𝑀) = (𝑀 Ξ£g βŸ¨β€œπ‘₯β€βŸ©))
1312rspceeqv 3625 . . . . . . 7 ((βŸ¨β€œπ‘₯β€βŸ© ∈ Word 𝐺 ∧ π‘₯ = (𝑀 Ξ£g βŸ¨β€œπ‘₯β€βŸ©)) β†’ βˆƒπ‘€ ∈ Word 𝐺π‘₯ = (𝑀 Ξ£g 𝑀))
146, 11, 13syl2anc 583 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ π‘₯ ∈ 𝐺) β†’ βˆƒπ‘€ ∈ Word 𝐺π‘₯ = (𝑀 Ξ£g 𝑀))
15 eqid 2724 . . . . . . . 8 (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) = (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))
1615elrnmpt 5945 . . . . . . 7 (π‘₯ ∈ V β†’ (π‘₯ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆƒπ‘€ ∈ Word 𝐺π‘₯ = (𝑀 Ξ£g 𝑀)))
1716elv 3472 . . . . . 6 (π‘₯ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆƒπ‘€ ∈ Word 𝐺π‘₯ = (𝑀 Ξ£g 𝑀))
1814, 17sylibr 233 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ π‘₯ ∈ 𝐺) β†’ π‘₯ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
1918ex 412 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ (π‘₯ ∈ 𝐺 β†’ π‘₯ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))))
2019ssrdv 3980 . . 3 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ 𝐺 βŠ† ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
21 gsumwspan.k . . . . . . . . . 10 𝐾 = (mrClsβ€˜(SubMndβ€˜π‘€))
2221mrccl 17553 . . . . . . . . 9 (((SubMndβ€˜π‘€) ∈ (Mooreβ€˜π΅) ∧ 𝐺 βŠ† 𝐡) β†’ (πΎβ€˜πΊ) ∈ (SubMndβ€˜π‘€))
233, 22sylan 579 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ (πΎβ€˜πΊ) ∈ (SubMndβ€˜π‘€))
2421mrcssid 17559 . . . . . . . . . . 11 (((SubMndβ€˜π‘€) ∈ (Mooreβ€˜π΅) ∧ 𝐺 βŠ† 𝐡) β†’ 𝐺 βŠ† (πΎβ€˜πΊ))
253, 24sylan 579 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ 𝐺 βŠ† (πΎβ€˜πΊ))
26 sswrd 14468 . . . . . . . . . 10 (𝐺 βŠ† (πΎβ€˜πΊ) β†’ Word 𝐺 βŠ† Word (πΎβ€˜πΊ))
2725, 26syl 17 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ Word 𝐺 βŠ† Word (πΎβ€˜πΊ))
2827sselda 3974 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ 𝑀 ∈ Word 𝐺) β†’ 𝑀 ∈ Word (πΎβ€˜πΊ))
29 gsumwsubmcl 18751 . . . . . . . 8 (((πΎβ€˜πΊ) ∈ (SubMndβ€˜π‘€) ∧ 𝑀 ∈ Word (πΎβ€˜πΊ)) β†’ (𝑀 Ξ£g 𝑀) ∈ (πΎβ€˜πΊ))
3023, 28, 29syl2an2r 682 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ 𝑀 ∈ Word 𝐺) β†’ (𝑀 Ξ£g 𝑀) ∈ (πΎβ€˜πΊ))
3130fmpttd 7106 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)):Word 𝐺⟢(πΎβ€˜πΊ))
3231frnd 6715 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) βŠ† (πΎβ€˜πΊ))
333, 21mrcssvd 17565 . . . . . 6 (𝑀 ∈ Mnd β†’ (πΎβ€˜πΊ) βŠ† 𝐡)
3433adantr 480 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ (πΎβ€˜πΊ) βŠ† 𝐡)
3532, 34sstrd 3984 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) βŠ† 𝐡)
36 wrd0 14485 . . . . . 6 βˆ… ∈ Word 𝐺
37 eqid 2724 . . . . . . . . 9 (0gβ€˜π‘€) = (0gβ€˜π‘€)
3837gsum0 18606 . . . . . . . 8 (𝑀 Ξ£g βˆ…) = (0gβ€˜π‘€)
3938eqcomi 2733 . . . . . . 7 (0gβ€˜π‘€) = (𝑀 Ξ£g βˆ…)
4039a1i 11 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ (0gβ€˜π‘€) = (𝑀 Ξ£g βˆ…))
41 oveq2 7409 . . . . . . 7 (𝑀 = βˆ… β†’ (𝑀 Ξ£g 𝑀) = (𝑀 Ξ£g βˆ…))
4241rspceeqv 3625 . . . . . 6 ((βˆ… ∈ Word 𝐺 ∧ (0gβ€˜π‘€) = (𝑀 Ξ£g βˆ…)) β†’ βˆƒπ‘€ ∈ Word 𝐺(0gβ€˜π‘€) = (𝑀 Ξ£g 𝑀))
4336, 40, 42sylancr 586 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ βˆƒπ‘€ ∈ Word 𝐺(0gβ€˜π‘€) = (𝑀 Ξ£g 𝑀))
44 fvex 6894 . . . . . 6 (0gβ€˜π‘€) ∈ V
4515elrnmpt 5945 . . . . . 6 ((0gβ€˜π‘€) ∈ V β†’ ((0gβ€˜π‘€) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆƒπ‘€ ∈ Word 𝐺(0gβ€˜π‘€) = (𝑀 Ξ£g 𝑀)))
4644, 45ax-mp 5 . . . . 5 ((0gβ€˜π‘€) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆƒπ‘€ ∈ Word 𝐺(0gβ€˜π‘€) = (𝑀 Ξ£g 𝑀))
4743, 46sylibr 233 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ (0gβ€˜π‘€) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
48 ccatcl 14520 . . . . . . . 8 ((𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺) β†’ (𝑧 ++ 𝑣) ∈ Word 𝐺)
49 simpll 764 . . . . . . . . . 10 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) β†’ 𝑀 ∈ Mnd)
50 sswrd 14468 . . . . . . . . . . . 12 (𝐺 βŠ† 𝐡 β†’ Word 𝐺 βŠ† Word 𝐡)
5150ad2antlr 724 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) β†’ Word 𝐺 βŠ† Word 𝐡)
52 simprl 768 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) β†’ 𝑧 ∈ Word 𝐺)
5351, 52sseldd 3975 . . . . . . . . . 10 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) β†’ 𝑧 ∈ Word 𝐡)
54 simprr 770 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) β†’ 𝑣 ∈ Word 𝐺)
5551, 54sseldd 3975 . . . . . . . . . 10 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) β†’ 𝑣 ∈ Word 𝐡)
56 eqid 2724 . . . . . . . . . . 11 (+gβ€˜π‘€) = (+gβ€˜π‘€)
571, 56gsumccat 18755 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ 𝑧 ∈ Word 𝐡 ∧ 𝑣 ∈ Word 𝐡) β†’ (𝑀 Ξ£g (𝑧 ++ 𝑣)) = ((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)))
5849, 53, 55, 57syl3anc 1368 . . . . . . . . 9 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) β†’ (𝑀 Ξ£g (𝑧 ++ 𝑣)) = ((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)))
5958eqcomd 2730 . . . . . . . 8 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) β†’ ((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) = (𝑀 Ξ£g (𝑧 ++ 𝑣)))
60 oveq2 7409 . . . . . . . . 9 (𝑀 = (𝑧 ++ 𝑣) β†’ (𝑀 Ξ£g 𝑀) = (𝑀 Ξ£g (𝑧 ++ 𝑣)))
6160rspceeqv 3625 . . . . . . . 8 (((𝑧 ++ 𝑣) ∈ Word 𝐺 ∧ ((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) = (𝑀 Ξ£g (𝑧 ++ 𝑣))) β†’ βˆƒπ‘€ ∈ Word 𝐺((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) = (𝑀 Ξ£g 𝑀))
6248, 59, 61syl2an2 683 . . . . . . 7 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) β†’ βˆƒπ‘€ ∈ Word 𝐺((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) = (𝑀 Ξ£g 𝑀))
63 ovex 7434 . . . . . . . 8 ((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ V
6415elrnmpt 5945 . . . . . . . 8 (((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ V β†’ (((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆƒπ‘€ ∈ Word 𝐺((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) = (𝑀 Ξ£g 𝑀)))
6563, 64ax-mp 5 . . . . . . 7 (((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆƒπ‘€ ∈ Word 𝐺((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) = (𝑀 Ξ£g 𝑀))
6662, 65sylibr 233 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) ∧ (𝑧 ∈ Word 𝐺 ∧ 𝑣 ∈ Word 𝐺)) β†’ ((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
6766ralrimivva 3192 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ βˆ€π‘§ ∈ Word πΊβˆ€π‘£ ∈ Word 𝐺((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
68 oveq2 7409 . . . . . . . . 9 (𝑀 = 𝑧 β†’ (𝑀 Ξ£g 𝑀) = (𝑀 Ξ£g 𝑧))
6968cbvmptv 5251 . . . . . . . 8 (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑧))
7069rneqi 5926 . . . . . . 7 ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) = ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑧))
7170raleqi 3315 . . . . . 6 (βˆ€π‘₯ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))βˆ€π‘¦ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))(π‘₯(+gβ€˜π‘€)𝑦) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆ€π‘₯ ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑧))βˆ€π‘¦ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))(π‘₯(+gβ€˜π‘€)𝑦) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
72 oveq2 7409 . . . . . . . . . . 11 (𝑀 = 𝑣 β†’ (𝑀 Ξ£g 𝑀) = (𝑀 Ξ£g 𝑣))
7372cbvmptv 5251 . . . . . . . . . 10 (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑣))
7473rneqi 5926 . . . . . . . . 9 ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) = ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑣))
7574raleqi 3315 . . . . . . . 8 (βˆ€π‘¦ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))(π‘₯(+gβ€˜π‘€)𝑦) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆ€π‘¦ ∈ ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑣))(π‘₯(+gβ€˜π‘€)𝑦) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
76 eqid 2724 . . . . . . . . . 10 (𝑣 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑣)) = (𝑣 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑣))
77 oveq2 7409 . . . . . . . . . . 11 (𝑦 = (𝑀 Ξ£g 𝑣) β†’ (π‘₯(+gβ€˜π‘€)𝑦) = (π‘₯(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)))
7877eleq1d 2810 . . . . . . . . . 10 (𝑦 = (𝑀 Ξ£g 𝑣) β†’ ((π‘₯(+gβ€˜π‘€)𝑦) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ (π‘₯(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))))
7976, 78ralrnmptw 7085 . . . . . . . . 9 (βˆ€π‘£ ∈ Word 𝐺(𝑀 Ξ£g 𝑣) ∈ V β†’ (βˆ€π‘¦ ∈ ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑣))(π‘₯(+gβ€˜π‘€)𝑦) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆ€π‘£ ∈ Word 𝐺(π‘₯(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))))
80 ovexd 7436 . . . . . . . . 9 (𝑣 ∈ Word 𝐺 β†’ (𝑀 Ξ£g 𝑣) ∈ V)
8179, 80mprg 3059 . . . . . . . 8 (βˆ€π‘¦ ∈ ran (𝑣 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑣))(π‘₯(+gβ€˜π‘€)𝑦) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆ€π‘£ ∈ Word 𝐺(π‘₯(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
8275, 81bitri 275 . . . . . . 7 (βˆ€π‘¦ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))(π‘₯(+gβ€˜π‘€)𝑦) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆ€π‘£ ∈ Word 𝐺(π‘₯(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
8382ralbii 3085 . . . . . 6 (βˆ€π‘₯ ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑧))βˆ€π‘¦ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))(π‘₯(+gβ€˜π‘€)𝑦) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆ€π‘₯ ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑧))βˆ€π‘£ ∈ Word 𝐺(π‘₯(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
84 eqid 2724 . . . . . . . 8 (𝑧 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑧)) = (𝑧 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑧))
85 oveq1 7408 . . . . . . . . . 10 (π‘₯ = (𝑀 Ξ£g 𝑧) β†’ (π‘₯(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) = ((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)))
8685eleq1d 2810 . . . . . . . . 9 (π‘₯ = (𝑀 Ξ£g 𝑧) β†’ ((π‘₯(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ ((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))))
8786ralbidv 3169 . . . . . . . 8 (π‘₯ = (𝑀 Ξ£g 𝑧) β†’ (βˆ€π‘£ ∈ Word 𝐺(π‘₯(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆ€π‘£ ∈ Word 𝐺((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))))
8884, 87ralrnmptw 7085 . . . . . . 7 (βˆ€π‘§ ∈ Word 𝐺(𝑀 Ξ£g 𝑧) ∈ V β†’ (βˆ€π‘₯ ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑧))βˆ€π‘£ ∈ Word 𝐺(π‘₯(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆ€π‘§ ∈ Word πΊβˆ€π‘£ ∈ Word 𝐺((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))))
89 ovexd 7436 . . . . . . 7 (𝑧 ∈ Word 𝐺 β†’ (𝑀 Ξ£g 𝑧) ∈ V)
9088, 89mprg 3059 . . . . . 6 (βˆ€π‘₯ ∈ ran (𝑧 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑧))βˆ€π‘£ ∈ Word 𝐺(π‘₯(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆ€π‘§ ∈ Word πΊβˆ€π‘£ ∈ Word 𝐺((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
9171, 83, 903bitri 297 . . . . 5 (βˆ€π‘₯ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))βˆ€π‘¦ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))(π‘₯(+gβ€˜π‘€)𝑦) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ↔ βˆ€π‘§ ∈ Word πΊβˆ€π‘£ ∈ Word 𝐺((𝑀 Ξ£g 𝑧)(+gβ€˜π‘€)(𝑀 Ξ£g 𝑣)) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
9267, 91sylibr 233 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ βˆ€π‘₯ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))βˆ€π‘¦ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))(π‘₯(+gβ€˜π‘€)𝑦) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
931, 37, 56issubm 18717 . . . . 5 (𝑀 ∈ Mnd β†’ (ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ∈ (SubMndβ€˜π‘€) ↔ (ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) βŠ† 𝐡 ∧ (0gβ€˜π‘€) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ∧ βˆ€π‘₯ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))βˆ€π‘¦ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))(π‘₯(+gβ€˜π‘€)𝑦) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))))
9493adantr 480 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ (ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ∈ (SubMndβ€˜π‘€) ↔ (ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) βŠ† 𝐡 ∧ (0gβ€˜π‘€) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ∧ βˆ€π‘₯ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))βˆ€π‘¦ ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀))(π‘₯(+gβ€˜π‘€)𝑦) ∈ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))))
9535, 47, 92, 94mpbir3and 1339 . . 3 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ∈ (SubMndβ€˜π‘€))
9621mrcsscl 17562 . . 3 (((SubMndβ€˜π‘€) ∈ (Mooreβ€˜π΅) ∧ 𝐺 βŠ† ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ∧ ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)) ∈ (SubMndβ€˜π‘€)) β†’ (πΎβ€˜πΊ) βŠ† ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
974, 20, 95, 96syl3anc 1368 . 2 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ (πΎβ€˜πΊ) βŠ† ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
9897, 32eqssd 3991 1 ((𝑀 ∈ Mnd ∧ 𝐺 βŠ† 𝐡) β†’ (πΎβ€˜πΊ) = ran (𝑀 ∈ Word 𝐺 ↦ (𝑀 Ξ£g 𝑀)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βˆƒwrex 3062  Vcvv 3466   βŠ† wss 3940  βˆ…c0 4314   ↦ cmpt 5221  ran crn 5667  β€˜cfv 6533  (class class class)co 7401  Word cword 14460   ++ cconcat 14516  βŸ¨β€œcs1 14541  Basecbs 17142  +gcplusg 17195  0gc0g 17383   Ξ£g cgsu 17384  Moorecmre 17524  mrClscmrc 17525  Mndcmnd 18656  SubMndcsubmnd 18701
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-card 9929  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-word 14461  df-concat 14517  df-s1 14542  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17143  df-ress 17172  df-plusg 17208  df-0g 17385  df-gsum 17386  df-mre 17528  df-mrc 17529  df-acs 17531  df-mgm 18562  df-sgrp 18641  df-mnd 18657  df-submnd 18703
This theorem is referenced by:  psgneldm2  19413  psgnfitr  19426
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