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Theorem cyc3genpm 32050
Description: The alternating group 𝐴 is generated by 3-cycles. Property (a) of [Lang] p. 32 . (Contributed by Thierry Arnoux, 27-Sep-2023.)
Hypotheses
Ref Expression
cyc3genpm.t 𝐢 = (𝑀 β€œ (β—‘β™― β€œ {3}))
cyc3genpm.a 𝐴 = (pmEvenβ€˜π·)
cyc3genpm.s 𝑆 = (SymGrpβ€˜π·)
cyc3genpm.n 𝑁 = (β™―β€˜π·)
cyc3genpm.m 𝑀 = (toCycβ€˜π·)
Assertion
Ref Expression
cyc3genpm (𝐷 ∈ Fin β†’ (𝑄 ∈ 𝐴 ↔ βˆƒπ‘€ ∈ Word 𝐢𝑄 = (𝑆 Ξ£g 𝑀)))
Distinct variable groups:   𝑀,𝐴   𝑀,𝐢   𝑀,𝐷   𝑀,𝑁   𝑀,𝑄   𝑀,𝑆
Allowed substitution hint:   𝑀(𝑀)

Proof of Theorem cyc3genpm
Dummy variables 𝑖 𝑒 𝑣 𝑐 𝑒 𝑓 𝑔 β„Ž 𝑗 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 768 . . . . 5 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ 𝑣 ∈ Word ran (pmTrspβ€˜π·))
2 lencl 14427 . . . . . . . 8 (𝑣 ∈ Word ran (pmTrspβ€˜π·) β†’ (β™―β€˜π‘£) ∈ β„•0)
32ad2antlr 726 . . . . . . 7 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ (β™―β€˜π‘£) ∈ β„•0)
43nn0zd 12530 . . . . . 6 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ (β™―β€˜π‘£) ∈ β„€)
5 simpr 486 . . . . . . . 8 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ 𝑄 = (𝑆 Ξ£g 𝑣))
65fveq2d 6847 . . . . . . 7 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ ((pmSgnβ€˜π·)β€˜π‘„) = ((pmSgnβ€˜π·)β€˜(𝑆 Ξ£g 𝑣)))
7 simplll 774 . . . . . . . 8 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ 𝐷 ∈ Fin)
8 simpllr 775 . . . . . . . . 9 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ 𝑄 ∈ 𝐴)
9 cyc3genpm.a . . . . . . . . 9 𝐴 = (pmEvenβ€˜π·)
108, 9eleqtrdi 2844 . . . . . . . 8 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ 𝑄 ∈ (pmEvenβ€˜π·))
11 cyc3genpm.s . . . . . . . . 9 𝑆 = (SymGrpβ€˜π·)
12 eqid 2733 . . . . . . . . 9 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
13 eqid 2733 . . . . . . . . 9 (pmSgnβ€˜π·) = (pmSgnβ€˜π·)
1411, 12, 13psgnevpm 21009 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝑄 ∈ (pmEvenβ€˜π·)) β†’ ((pmSgnβ€˜π·)β€˜π‘„) = 1)
157, 10, 14syl2anc 585 . . . . . . 7 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ ((pmSgnβ€˜π·)β€˜π‘„) = 1)
16 eqid 2733 . . . . . . . . 9 ran (pmTrspβ€˜π·) = ran (pmTrspβ€˜π·)
1711, 16, 13psgnvalii 19296 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) β†’ ((pmSgnβ€˜π·)β€˜(𝑆 Ξ£g 𝑣)) = (-1↑(β™―β€˜π‘£)))
187, 1, 17syl2anc 585 . . . . . . 7 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ ((pmSgnβ€˜π·)β€˜(𝑆 Ξ£g 𝑣)) = (-1↑(β™―β€˜π‘£)))
196, 15, 183eqtr3rd 2782 . . . . . 6 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ (-1↑(β™―β€˜π‘£)) = 1)
20 m1exp1 16263 . . . . . . 7 ((β™―β€˜π‘£) ∈ β„€ β†’ ((-1↑(β™―β€˜π‘£)) = 1 ↔ 2 βˆ₯ (β™―β€˜π‘£)))
2120biimpa 478 . . . . . 6 (((β™―β€˜π‘£) ∈ β„€ ∧ (-1↑(β™―β€˜π‘£)) = 1) β†’ 2 βˆ₯ (β™―β€˜π‘£))
224, 19, 21syl2anc 585 . . . . 5 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ 2 βˆ₯ (β™―β€˜π‘£))
23 oveq2 7366 . . . . . . . . . 10 (π‘₯ = βˆ… β†’ (𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g βˆ…))
2423eqeq1d 2735 . . . . . . . . 9 (π‘₯ = βˆ… β†’ ((𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑀) ↔ (𝑆 Ξ£g βˆ…) = (𝑆 Ξ£g 𝑀)))
2524rexbidv 3172 . . . . . . . 8 (π‘₯ = βˆ… β†’ (βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑀) ↔ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g βˆ…) = (𝑆 Ξ£g 𝑀)))
2625imbi2d 341 . . . . . . 7 (π‘₯ = βˆ… β†’ ((𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑀)) ↔ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g βˆ…) = (𝑆 Ξ£g 𝑀))))
27 oveq2 7366 . . . . . . . . . 10 (π‘₯ = 𝑒 β†’ (𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑒))
2827eqeq1d 2735 . . . . . . . . 9 (π‘₯ = 𝑒 β†’ ((𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑀) ↔ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀)))
2928rexbidv 3172 . . . . . . . 8 (π‘₯ = 𝑒 β†’ (βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑀) ↔ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀)))
3029imbi2d 341 . . . . . . 7 (π‘₯ = 𝑒 β†’ ((𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑀)) ↔ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀))))
31 oveq2 7366 . . . . . . . . . 10 (π‘₯ = (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©) β†’ (𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)))
3231eqeq1d 2735 . . . . . . . . 9 (π‘₯ = (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©) β†’ ((𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑀) ↔ (𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g 𝑀)))
3332rexbidv 3172 . . . . . . . 8 (π‘₯ = (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©) β†’ (βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑀) ↔ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g 𝑀)))
3433imbi2d 341 . . . . . . 7 (π‘₯ = (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©) β†’ ((𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑀)) ↔ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g 𝑀))))
35 oveq2 7366 . . . . . . . . . 10 (π‘₯ = 𝑣 β†’ (𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑣))
3635eqeq1d 2735 . . . . . . . . 9 (π‘₯ = 𝑣 β†’ ((𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑀) ↔ (𝑆 Ξ£g 𝑣) = (𝑆 Ξ£g 𝑀)))
3736rexbidv 3172 . . . . . . . 8 (π‘₯ = 𝑣 β†’ (βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑀) ↔ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑣) = (𝑆 Ξ£g 𝑀)))
3837imbi2d 341 . . . . . . 7 (π‘₯ = 𝑣 β†’ ((𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g π‘₯) = (𝑆 Ξ£g 𝑀)) ↔ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑣) = (𝑆 Ξ£g 𝑀))))
39 wrd0 14433 . . . . . . . . 9 βˆ… ∈ Word 𝐢
4039a1i 11 . . . . . . . 8 (𝐷 ∈ Fin β†’ βˆ… ∈ Word 𝐢)
41 simpr 486 . . . . . . . . . 10 ((𝐷 ∈ Fin ∧ 𝑀 = βˆ…) β†’ 𝑀 = βˆ…)
4241oveq2d 7374 . . . . . . . . 9 ((𝐷 ∈ Fin ∧ 𝑀 = βˆ…) β†’ (𝑆 Ξ£g 𝑀) = (𝑆 Ξ£g βˆ…))
4342eqeq2d 2744 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝑀 = βˆ…) β†’ ((𝑆 Ξ£g βˆ…) = (𝑆 Ξ£g 𝑀) ↔ (𝑆 Ξ£g βˆ…) = (𝑆 Ξ£g βˆ…)))
44 eqidd 2734 . . . . . . . 8 (𝐷 ∈ Fin β†’ (𝑆 Ξ£g βˆ…) = (𝑆 Ξ£g βˆ…))
4540, 43, 44rspcedvd 3582 . . . . . . 7 (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g βˆ…) = (𝑆 Ξ£g 𝑀))
46 ccatcl 14468 . . . . . . . . . . . . . 14 ((𝑣 ∈ Word 𝐢 ∧ 𝑐 ∈ Word 𝐢) β†’ (𝑣 ++ 𝑐) ∈ Word 𝐢)
4746ad5ant24 760 . . . . . . . . . . . . 13 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ (𝑣 ++ 𝑐) ∈ Word 𝐢)
48 oveq2 7366 . . . . . . . . . . . . . . 15 (𝑀 = (𝑣 ++ 𝑐) β†’ (𝑆 Ξ£g 𝑀) = (𝑆 Ξ£g (𝑣 ++ 𝑐)))
4948eqeq2d 2744 . . . . . . . . . . . . . 14 (𝑀 = (𝑣 ++ 𝑐) β†’ ((𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g 𝑀) ↔ (𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g (𝑣 ++ 𝑐))))
5049adantl 483 . . . . . . . . . . . . 13 (((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) ∧ 𝑀 = (𝑣 ++ 𝑐)) β†’ ((𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g 𝑀) ↔ (𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g (𝑣 ++ 𝑐))))
51 simpllr 775 . . . . . . . . . . . . . . 15 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣))
52 simpllr 775 . . . . . . . . . . . . . . . . . . 19 ((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) β†’ 𝐷 ∈ Fin)
5352ad2antrr 725 . . . . . . . . . . . . . . . . . 18 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ 𝐷 ∈ Fin)
5411symggrp 19187 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ Fin β†’ 𝑆 ∈ Grp)
55 grpmnd 18760 . . . . . . . . . . . . . . . . . 18 (𝑆 ∈ Grp β†’ 𝑆 ∈ Mnd)
5653, 54, 553syl 18 . . . . . . . . . . . . . . . . 17 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ 𝑆 ∈ Mnd)
5716, 11, 12symgtrf 19256 . . . . . . . . . . . . . . . . . . 19 ran (pmTrspβ€˜π·) βŠ† (Baseβ€˜π‘†)
5857a1i 11 . . . . . . . . . . . . . . . . . 18 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ ran (pmTrspβ€˜π·) βŠ† (Baseβ€˜π‘†))
59 simp-5r 785 . . . . . . . . . . . . . . . . . . 19 ((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) β†’ 𝑖 ∈ ran (pmTrspβ€˜π·))
6059ad2antrr 725 . . . . . . . . . . . . . . . . . 18 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ 𝑖 ∈ ran (pmTrspβ€˜π·))
6158, 60sseldd 3946 . . . . . . . . . . . . . . . . 17 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ 𝑖 ∈ (Baseβ€˜π‘†))
62 simp-6r 787 . . . . . . . . . . . . . . . . . 18 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ 𝑗 ∈ ran (pmTrspβ€˜π·))
6358, 62sseldd 3946 . . . . . . . . . . . . . . . . 17 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ 𝑗 ∈ (Baseβ€˜π‘†))
64 eqid 2733 . . . . . . . . . . . . . . . . . 18 (+gβ€˜π‘†) = (+gβ€˜π‘†)
6512, 64gsumws2 18657 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ Mnd ∧ 𝑖 ∈ (Baseβ€˜π‘†) ∧ 𝑗 ∈ (Baseβ€˜π‘†)) β†’ (𝑆 Ξ£g βŸ¨β€œπ‘–π‘—β€βŸ©) = (𝑖(+gβ€˜π‘†)𝑗))
6656, 61, 63, 65syl3anc 1372 . . . . . . . . . . . . . . . 16 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ (𝑆 Ξ£g βŸ¨β€œπ‘–π‘—β€βŸ©) = (𝑖(+gβ€˜π‘†)𝑗))
67 simpr 486 . . . . . . . . . . . . . . . 16 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐))
6866, 67eqtrd 2773 . . . . . . . . . . . . . . 15 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ (𝑆 Ξ£g βŸ¨β€œπ‘–π‘—β€βŸ©) = (𝑆 Ξ£g 𝑐))
6951, 68oveq12d 7376 . . . . . . . . . . . . . 14 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ ((𝑆 Ξ£g 𝑒)(+gβ€˜π‘†)(𝑆 Ξ£g βŸ¨β€œπ‘–π‘—β€βŸ©)) = ((𝑆 Ξ£g 𝑣)(+gβ€˜π‘†)(𝑆 Ξ£g 𝑐)))
70 sswrd 14416 . . . . . . . . . . . . . . . . 17 (ran (pmTrspβ€˜π·) βŠ† (Baseβ€˜π‘†) β†’ Word ran (pmTrspβ€˜π·) βŠ† Word (Baseβ€˜π‘†))
7158, 70syl 17 . . . . . . . . . . . . . . . 16 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ Word ran (pmTrspβ€˜π·) βŠ† Word (Baseβ€˜π‘†))
72 simp-7l 788 . . . . . . . . . . . . . . . 16 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ 𝑒 ∈ Word ran (pmTrspβ€˜π·))
7371, 72sseldd 3946 . . . . . . . . . . . . . . 15 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ 𝑒 ∈ Word (Baseβ€˜π‘†))
7461, 63s2cld 14766 . . . . . . . . . . . . . . 15 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ βŸ¨β€œπ‘–π‘—β€βŸ© ∈ Word (Baseβ€˜π‘†))
7512, 64gsumccat 18656 . . . . . . . . . . . . . . 15 ((𝑆 ∈ Mnd ∧ 𝑒 ∈ Word (Baseβ€˜π‘†) ∧ βŸ¨β€œπ‘–π‘—β€βŸ© ∈ Word (Baseβ€˜π‘†)) β†’ (𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = ((𝑆 Ξ£g 𝑒)(+gβ€˜π‘†)(𝑆 Ξ£g βŸ¨β€œπ‘–π‘—β€βŸ©)))
7656, 73, 74, 75syl3anc 1372 . . . . . . . . . . . . . 14 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ (𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = ((𝑆 Ξ£g 𝑒)(+gβ€˜π‘†)(𝑆 Ξ£g βŸ¨β€œπ‘–π‘—β€βŸ©)))
77 cyc3genpm.t . . . . . . . . . . . . . . . . . . . 20 𝐢 = (𝑀 β€œ (β—‘β™― β€œ {3}))
78 cyc3genpm.m . . . . . . . . . . . . . . . . . . . . 21 𝑀 = (toCycβ€˜π·)
7978imaeq1i 6011 . . . . . . . . . . . . . . . . . . . 20 (𝑀 β€œ (β—‘β™― β€œ {3})) = ((toCycβ€˜π·) β€œ (β—‘β™― β€œ {3}))
8077, 79eqtri 2761 . . . . . . . . . . . . . . . . . . 19 𝐢 = ((toCycβ€˜π·) β€œ (β—‘β™― β€œ {3}))
8180, 9cyc3evpm 32048 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ Fin β†’ 𝐢 βŠ† 𝐴)
8211, 12evpmss 21006 . . . . . . . . . . . . . . . . . . 19 (pmEvenβ€˜π·) βŠ† (Baseβ€˜π‘†)
839, 82eqsstri 3979 . . . . . . . . . . . . . . . . . 18 𝐴 βŠ† (Baseβ€˜π‘†)
8481, 83sstrdi 3957 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ Fin β†’ 𝐢 βŠ† (Baseβ€˜π‘†))
85 sswrd 14416 . . . . . . . . . . . . . . . . 17 (𝐢 βŠ† (Baseβ€˜π‘†) β†’ Word 𝐢 βŠ† Word (Baseβ€˜π‘†))
8653, 84, 853syl 18 . . . . . . . . . . . . . . . 16 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ Word 𝐢 βŠ† Word (Baseβ€˜π‘†))
87 simp-4r 783 . . . . . . . . . . . . . . . 16 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ 𝑣 ∈ Word 𝐢)
8886, 87sseldd 3946 . . . . . . . . . . . . . . 15 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ 𝑣 ∈ Word (Baseβ€˜π‘†))
89 simplr 768 . . . . . . . . . . . . . . . 16 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ 𝑐 ∈ Word 𝐢)
9086, 89sseldd 3946 . . . . . . . . . . . . . . 15 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ 𝑐 ∈ Word (Baseβ€˜π‘†))
9112, 64gsumccat 18656 . . . . . . . . . . . . . . 15 ((𝑆 ∈ Mnd ∧ 𝑣 ∈ Word (Baseβ€˜π‘†) ∧ 𝑐 ∈ Word (Baseβ€˜π‘†)) β†’ (𝑆 Ξ£g (𝑣 ++ 𝑐)) = ((𝑆 Ξ£g 𝑣)(+gβ€˜π‘†)(𝑆 Ξ£g 𝑐)))
9256, 88, 90, 91syl3anc 1372 . . . . . . . . . . . . . 14 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ (𝑆 Ξ£g (𝑣 ++ 𝑐)) = ((𝑆 Ξ£g 𝑣)(+gβ€˜π‘†)(𝑆 Ξ£g 𝑐)))
9369, 76, 923eqtr4d 2783 . . . . . . . . . . . . 13 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ (𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g (𝑣 ++ 𝑐)))
9447, 50, 93rspcedvd 3582 . . . . . . . . . . . 12 ((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑐 ∈ Word 𝐢) ∧ (𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐)) β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g 𝑀))
95 cyc3genpm.n . . . . . . . . . . . . . . 15 𝑁 = (β™―β€˜π·)
96 simp-6r 787 . . . . . . . . . . . . . . 15 ((((((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) ∧ 𝑔 ∈ 𝐷) ∧ β„Ž ∈ 𝐷) ∧ (𝑔 β‰  β„Ž ∧ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©))) β†’ 𝑒 ∈ 𝐷)
97 simp-5r 785 . . . . . . . . . . . . . . 15 ((((((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) ∧ 𝑔 ∈ 𝐷) ∧ β„Ž ∈ 𝐷) ∧ (𝑔 β‰  β„Ž ∧ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©))) β†’ 𝑓 ∈ 𝐷)
98 simpllr 775 . . . . . . . . . . . . . . 15 ((((((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) ∧ 𝑔 ∈ 𝐷) ∧ β„Ž ∈ 𝐷) ∧ (𝑔 β‰  β„Ž ∧ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©))) β†’ 𝑔 ∈ 𝐷)
99 simplr 768 . . . . . . . . . . . . . . 15 ((((((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) ∧ 𝑔 ∈ 𝐷) ∧ β„Ž ∈ 𝐷) ∧ (𝑔 β‰  β„Ž ∧ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©))) β†’ β„Ž ∈ 𝐷)
100 simp-4r 783 . . . . . . . . . . . . . . . 16 ((((((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) ∧ 𝑔 ∈ 𝐷) ∧ β„Ž ∈ 𝐷) ∧ (𝑔 β‰  β„Ž ∧ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©))) β†’ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©)))
101100simprd 497 . . . . . . . . . . . . . . 15 ((((((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) ∧ 𝑔 ∈ 𝐷) ∧ β„Ž ∈ 𝐷) ∧ (𝑔 β‰  β„Ž ∧ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©))) β†’ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))
102 simprr 772 . . . . . . . . . . . . . . 15 ((((((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) ∧ 𝑔 ∈ 𝐷) ∧ β„Ž ∈ 𝐷) ∧ (𝑔 β‰  β„Ž ∧ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©))) β†’ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©))
10352ad6antr 735 . . . . . . . . . . . . . . 15 ((((((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) ∧ 𝑔 ∈ 𝐷) ∧ β„Ž ∈ 𝐷) ∧ (𝑔 β‰  β„Ž ∧ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©))) β†’ 𝐷 ∈ Fin)
104100simpld 496 . . . . . . . . . . . . . . 15 ((((((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) ∧ 𝑔 ∈ 𝐷) ∧ β„Ž ∈ 𝐷) ∧ (𝑔 β‰  β„Ž ∧ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©))) β†’ 𝑒 β‰  𝑓)
105 simprl 770 . . . . . . . . . . . . . . 15 ((((((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) ∧ 𝑔 ∈ 𝐷) ∧ β„Ž ∈ 𝐷) ∧ (𝑔 β‰  β„Ž ∧ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©))) β†’ 𝑔 β‰  β„Ž)
10677, 9, 11, 95, 78, 64, 96, 97, 98, 99, 101, 102, 103, 104, 105cyc3genpmlem 32049 . . . . . . . . . . . . . 14 ((((((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) ∧ 𝑔 ∈ 𝐷) ∧ β„Ž ∈ 𝐷) ∧ (𝑔 β‰  β„Ž ∧ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©))) β†’ βˆƒπ‘ ∈ Word 𝐢(𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐))
107 simp-6r 787 . . . . . . . . . . . . . . 15 (((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) β†’ 𝐷 ∈ Fin)
108 simp-7r 789 . . . . . . . . . . . . . . 15 (((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) β†’ 𝑗 ∈ ran (pmTrspβ€˜π·))
10916, 78trsp2cyc 32021 . . . . . . . . . . . . . . 15 ((𝐷 ∈ Fin ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) β†’ βˆƒπ‘” ∈ 𝐷 βˆƒβ„Ž ∈ 𝐷 (𝑔 β‰  β„Ž ∧ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©)))
110107, 108, 109syl2anc 585 . . . . . . . . . . . . . 14 (((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) β†’ βˆƒπ‘” ∈ 𝐷 βˆƒβ„Ž ∈ 𝐷 (𝑔 β‰  β„Ž ∧ 𝑗 = (π‘€β€˜βŸ¨β€œπ‘”β„Žβ€βŸ©)))
111106, 110r19.29vva 3204 . . . . . . . . . . . . 13 (((((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©))) β†’ βˆƒπ‘ ∈ Word 𝐢(𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐))
11216, 78trsp2cyc 32021 . . . . . . . . . . . . . 14 ((𝐷 ∈ Fin ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) β†’ βˆƒπ‘’ ∈ 𝐷 βˆƒπ‘“ ∈ 𝐷 (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©)))
11352, 59, 112syl2anc 585 . . . . . . . . . . . . 13 ((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) β†’ βˆƒπ‘’ ∈ 𝐷 βˆƒπ‘“ ∈ 𝐷 (𝑒 β‰  𝑓 ∧ 𝑖 = (π‘€β€˜βŸ¨β€œπ‘’π‘“β€βŸ©)))
114111, 113r19.29vva 3204 . . . . . . . . . . . 12 ((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) β†’ βˆƒπ‘ ∈ Word 𝐢(𝑖(+gβ€˜π‘†)𝑗) = (𝑆 Ξ£g 𝑐))
11594, 114r19.29a 3156 . . . . . . . . . . 11 ((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g 𝑀))
116115adantl3r 749 . . . . . . . . . 10 (((((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀))) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐢) ∧ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣)) β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g 𝑀))
117 simpr 486 . . . . . . . . . . . 12 (((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀))) ∧ 𝐷 ∈ Fin) β†’ 𝐷 ∈ Fin)
118 simplr 768 . . . . . . . . . . . 12 (((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀))) ∧ 𝐷 ∈ Fin) β†’ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀)))
119117, 118mpd 15 . . . . . . . . . . 11 (((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀))) ∧ 𝐷 ∈ Fin) β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀))
120 oveq2 7366 . . . . . . . . . . . . 13 (𝑣 = 𝑀 β†’ (𝑆 Ξ£g 𝑣) = (𝑆 Ξ£g 𝑀))
121120eqeq2d 2744 . . . . . . . . . . . 12 (𝑣 = 𝑀 β†’ ((𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣) ↔ (𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀)))
122121cbvrexvw 3225 . . . . . . . . . . 11 (βˆƒπ‘£ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣) ↔ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀))
123119, 122sylibr 233 . . . . . . . . . 10 (((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀))) ∧ 𝐷 ∈ Fin) β†’ βˆƒπ‘£ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑣))
124116, 123r19.29a 3156 . . . . . . . . 9 (((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀))) ∧ 𝐷 ∈ Fin) β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g 𝑀))
125124ex 414 . . . . . . . 8 ((((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·)) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) ∧ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀))) β†’ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g 𝑀)))
126125ex3 1347 . . . . . . 7 ((𝑒 ∈ Word ran (pmTrspβ€˜π·) ∧ 𝑖 ∈ ran (pmTrspβ€˜π·) ∧ 𝑗 ∈ ran (pmTrspβ€˜π·)) β†’ ((𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑒) = (𝑆 Ξ£g 𝑀)) β†’ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g (𝑒 ++ βŸ¨β€œπ‘–π‘—β€βŸ©)) = (𝑆 Ξ£g 𝑀))))
12726, 30, 34, 38, 45, 126wrdt2ind 31856 . . . . . 6 ((𝑣 ∈ Word ran (pmTrspβ€˜π·) ∧ 2 βˆ₯ (β™―β€˜π‘£)) β†’ (𝐷 ∈ Fin β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑣) = (𝑆 Ξ£g 𝑀)))
128127imp 408 . . . . 5 (((𝑣 ∈ Word ran (pmTrspβ€˜π·) ∧ 2 βˆ₯ (β™―β€˜π‘£)) ∧ 𝐷 ∈ Fin) β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑣) = (𝑆 Ξ£g 𝑀))
1291, 22, 7, 128syl21anc 837 . . . 4 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑣) = (𝑆 Ξ£g 𝑀))
1305eqeq1d 2735 . . . . 5 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ (𝑄 = (𝑆 Ξ£g 𝑀) ↔ (𝑆 Ξ£g 𝑣) = (𝑆 Ξ£g 𝑀)))
131130rexbidv 3172 . . . 4 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ (βˆƒπ‘€ ∈ Word 𝐢𝑄 = (𝑆 Ξ£g 𝑀) ↔ βˆƒπ‘€ ∈ Word 𝐢(𝑆 Ξ£g 𝑣) = (𝑆 Ξ£g 𝑀)))
132129, 131mpbird 257 . . 3 ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrspβ€˜π·)) ∧ 𝑄 = (𝑆 Ξ£g 𝑣)) β†’ βˆƒπ‘€ ∈ Word 𝐢𝑄 = (𝑆 Ξ£g 𝑀))
13383sseli 3941 . . . 4 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜π‘†))
13411, 12, 16psgnfitr 19304 . . . . 5 (𝐷 ∈ Fin β†’ (𝑄 ∈ (Baseβ€˜π‘†) ↔ βˆƒπ‘£ ∈ Word ran (pmTrspβ€˜π·)𝑄 = (𝑆 Ξ£g 𝑣)))
135134biimpa 478 . . . 4 ((𝐷 ∈ Fin ∧ 𝑄 ∈ (Baseβ€˜π‘†)) β†’ βˆƒπ‘£ ∈ Word ran (pmTrspβ€˜π·)𝑄 = (𝑆 Ξ£g 𝑣))
136133, 135sylan2 594 . . 3 ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) β†’ βˆƒπ‘£ ∈ Word ran (pmTrspβ€˜π·)𝑄 = (𝑆 Ξ£g 𝑣))
137132, 136r19.29a 3156 . 2 ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) β†’ βˆƒπ‘€ ∈ Word 𝐢𝑄 = (𝑆 Ξ£g 𝑀))
138 simpr 486 . . . 4 (((𝐷 ∈ Fin ∧ 𝑀 ∈ Word 𝐢) ∧ 𝑄 = (𝑆 Ξ£g 𝑀)) β†’ 𝑄 = (𝑆 Ξ£g 𝑀))
13911altgnsg 32047 . . . . . . . . 9 (𝐷 ∈ Fin β†’ (pmEvenβ€˜π·) ∈ (NrmSGrpβ€˜π‘†))
1409, 139eqeltrid 2838 . . . . . . . 8 (𝐷 ∈ Fin β†’ 𝐴 ∈ (NrmSGrpβ€˜π‘†))
141 nsgsubg 18965 . . . . . . . 8 (𝐴 ∈ (NrmSGrpβ€˜π‘†) β†’ 𝐴 ∈ (SubGrpβ€˜π‘†))
142 subgsubm 18955 . . . . . . . 8 (𝐴 ∈ (SubGrpβ€˜π‘†) β†’ 𝐴 ∈ (SubMndβ€˜π‘†))
143140, 141, 1423syl 18 . . . . . . 7 (𝐷 ∈ Fin β†’ 𝐴 ∈ (SubMndβ€˜π‘†))
144143adantr 482 . . . . . 6 ((𝐷 ∈ Fin ∧ 𝑀 ∈ Word 𝐢) β†’ 𝐴 ∈ (SubMndβ€˜π‘†))
145 sswrd 14416 . . . . . . . 8 (𝐢 βŠ† 𝐴 β†’ Word 𝐢 βŠ† Word 𝐴)
14681, 145syl 17 . . . . . . 7 (𝐷 ∈ Fin β†’ Word 𝐢 βŠ† Word 𝐴)
147146sselda 3945 . . . . . 6 ((𝐷 ∈ Fin ∧ 𝑀 ∈ Word 𝐢) β†’ 𝑀 ∈ Word 𝐴)
148 gsumwsubmcl 18652 . . . . . 6 ((𝐴 ∈ (SubMndβ€˜π‘†) ∧ 𝑀 ∈ Word 𝐴) β†’ (𝑆 Ξ£g 𝑀) ∈ 𝐴)
149144, 147, 148syl2anc 585 . . . . 5 ((𝐷 ∈ Fin ∧ 𝑀 ∈ Word 𝐢) β†’ (𝑆 Ξ£g 𝑀) ∈ 𝐴)
150149adantr 482 . . . 4 (((𝐷 ∈ Fin ∧ 𝑀 ∈ Word 𝐢) ∧ 𝑄 = (𝑆 Ξ£g 𝑀)) β†’ (𝑆 Ξ£g 𝑀) ∈ 𝐴)
151138, 150eqeltrd 2834 . . 3 (((𝐷 ∈ Fin ∧ 𝑀 ∈ Word 𝐢) ∧ 𝑄 = (𝑆 Ξ£g 𝑀)) β†’ 𝑄 ∈ 𝐴)
152151r19.29an 3152 . 2 ((𝐷 ∈ Fin ∧ βˆƒπ‘€ ∈ Word 𝐢𝑄 = (𝑆 Ξ£g 𝑀)) β†’ 𝑄 ∈ 𝐴)
153137, 152impbida 800 1 (𝐷 ∈ Fin β†’ (𝑄 ∈ 𝐴 ↔ βˆƒπ‘€ ∈ Word 𝐢𝑄 = (𝑆 Ξ£g 𝑀)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆƒwrex 3070   βŠ† wss 3911  βˆ…c0 4283  {csn 4587   class class class wbr 5106  β—‘ccnv 5633  ran crn 5635   β€œ cima 5637  β€˜cfv 6497  (class class class)co 7358  Fincfn 8886  1c1 11057  -cneg 11391  2c2 12213  3c3 12214  β„•0cn0 12418  β„€cz 12504  β†‘cexp 13973  β™―chash 14236  Word cword 14408   ++ cconcat 14464  βŸ¨β€œcs2 14736   βˆ₯ cdvds 16141  Basecbs 17088  +gcplusg 17138   Ξ£g cgsu 17327  Mndcmnd 18561  SubMndcsubmnd 18605  Grpcgrp 18753  SubGrpcsubg 18927  NrmSGrpcnsg 18928  SymGrpcsymg 19153  pmTrspcpmtr 19228  pmSgncpsgn 19276  pmEvencevpm 19277  toCycctocyc 32004
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-reg 9533  ax-ac2 10404  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134  ax-addf 11135  ax-mulf 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-xor 1511  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-tpos 8158  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-inf 9384  df-card 9880  df-ac 10057  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-xnn0 12491  df-z 12505  df-dec 12624  df-uz 12769  df-rp 12921  df-fz 13431  df-fzo 13574  df-fl 13703  df-mod 13781  df-seq 13913  df-exp 13974  df-hash 14237  df-word 14409  df-lsw 14457  df-concat 14465  df-s1 14490  df-substr 14535  df-pfx 14565  df-splice 14644  df-reverse 14653  df-csh 14683  df-s2 14743  df-s3 14744  df-dvds 16142  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-starv 17153  df-tset 17157  df-ple 17158  df-ds 17160  df-unif 17161  df-0g 17328  df-gsum 17329  df-mre 17471  df-mrc 17472  df-acs 17474  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-submnd 18607  df-efmnd 18684  df-grp 18756  df-minusg 18757  df-sbg 18758  df-subg 18930  df-nsg 18931  df-ghm 19011  df-gim 19054  df-oppg 19129  df-symg 19154  df-pmtr 19229  df-psgn 19278  df-evpm 19279  df-cmn 19569  df-abl 19570  df-mgp 19902  df-ur 19919  df-ring 19971  df-cring 19972  df-oppr 20054  df-dvdsr 20075  df-unit 20076  df-invr 20106  df-dvr 20117  df-drng 20199  df-cnfld 20813  df-tocyc 32005
This theorem is referenced by: (None)
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